the story of numbers

Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 1 CHAPTER4:THE STORY OF NUMBERS §1:NUMBERS AS YOU KNOW THEM; Variables are like common nouns in Mathematics. If ‘x’ denotes a variable in the interval [2,3[. (all the numbers between 2 and three including 2 and not including 3), ‘x’ can be thought of assuming any value in this interval, but one at a time .It is a symbol like a place holder to contain a number .The symbols are formidable and have wielded tremendous power on which the grand edifice of Mathematics . A constant a number such as 2, -(4) etc or a symbol such as a, b, c, etc. which denote one particular value as long as we are inside a problem or dealing a particular topic .It is like a proper noun in Mathematics. An Interval of numbers [2, 3] represents every real number between 2 and 3, both of them inclusive and is called a closed interval. If one of them is not included , say, 2, we call it a semi-open interval which is written as ]2, 3] or )2, 3] . Similarly [2, 3[ or [2, 3( is a semi-open interval where 3 is not included. An open interval such as ]-2, -1[ or ( -2 , -1 ( is one where none of the ending numbers -2 , -1 are included. §2:Peano’s theory of natural numbers. The true spirit of Mathematics or logic lies in taking minimum number of assumptions or undefined terms, framing a set of operations , or rules of the game, and developing the subject as the interplay of these operations on the objects taken or assumed. An elegant example for all times to come is Peano’s axioms of natural numbers. The Mathematician had a knack for reducing all Mathematics into logical statements. The set of natural numbers and basics operations like addition, subtraction, multiplication and division etc. on them were reintroduced to the world by Peano as follows : Axiom 1: – 1 is a natural number. Axiom 2: – For every natural number ‘n’, there is a (unique) natural number n+( ‘b’, say, b = n +1; if you are not comfortable with the symbol n+ ) which is called its (immediate ) successor. Axiom 3: – 1 is not successor of any natural number. Axiom 4: – two natural numbers whose successors are equal, must be equal to each other. In symbols, if m+ = n+ , then m = n. Axiom 5: – Any set which contains 1 and contains successor of any of its members, must contain the entire set of natural numbers. There is no big thing to ‘explain ‘ or ‘understand’ the axioms or postulates; they appear so trivial. But the beauty is, that the system of entire natural numbers, with the operations on it, addition, Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 2 subtraction , multiplication, division etc., and their properties, i.e., closure, associativity, commutativity, etc. in their proper context, all are finished with these only five postulates. ‘Proper context’ means, that commutativity , closure etc. do not hold for processes of subtraction, division etc. For verification of this statement, we give examples below as to how the properties and results are demonstrated to be derived from these few assumptions only. In its totality, it may be observed as if The Principle Of Mathematical Induction has been applied to the set of natural numbers or as if the set of natural numbers with the set of Peano’s postulates exemplify and justify the philosophy of Mathematical Induction. §3:The Open Mind : We Learn Until We Are Dead; The process of induction; scientific methods. Simplicity is a necessary mindset for deciphering apparently complicated mysteries of Nature. Failed midway in some endeavours, one must do-it-all-over-again readily; and in the process, more often than not, new directions are envisioned. A story about Einstein goes like this : a small girl discussed with him about the Earth going round the Sun . it was certainly difficult for her to understand, for she perceived the Sun going round the Earth. When Einstein tried to explain her that the astronomical measurements would be same in both ways; no matter whether the Earth went round the Sun or otherwise, she interrupted asking whether it might just be possible that nobody goes round the other. Simple enough is the question; but it made the big head meditate and the General Theory of Relativity was ushered in. Simplicity is the way up to the truth. What truth is, nobody knows. But it has no double standards. What tastes sweat to you must do so to me or anybody else. All religions and ethics endorse this. That is the proper way to live and let live; to protect the environment, to prosper and to save the planet in brief. A grain of rice may be small and negligible for a rice merchant but never so for spectroscopic study and never so for an ant either. But Mathematics describes smallness for everybody, applicable everywhere and at all times. What Calculus studies is only this smallness or largeness. Development of number system in the following articles displays one example open mind or simplicity of thought or ability to learn until death – call it whatever you please. Can there be some number whose square is negative ; when we know that squares of both positive and negative numbers both are positive only? Why should one accept that numbers could only be either positive or negative? Doesn’t that sound simple enough ? Yes, it is. Great geniuses who foresee formidable results are sometimes unable to prove them. Many examples may be cited, like the four colour theorem due to Euler ; it states that four colours are required at most for clouring any number of regions in a map. But the theorem was proved only after half a century later. More fundamental the result is, more difficult is becomes for proving; and is accepted as an axiom for a long period of time until somebody formally proves it. But such situation does not arise too often; though it is not rare. The reason is; we have a powerful logic or method of Mathematical induction to verify if something is true for as many number of cases as one may count. In Mathematics, if a million examples are given to support validity of some statement; the statement is not necessarily proved; but a single example to the contrary is Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 3 sufficient to disprove the statement. Very true. The method of Mathematical induction ensures that a single exception shall not be found. And thus it is a proof, full-fledged , rigorously mathematical in spirit We try to formally state what it means : §4:a) Principle Of Mathematical Induction : If a statement is true for n = 1, or equivalently, if P(1) is true, and it is true for n = m + 1 P(m + 1) is true whenever it is true for n = m; whenever P(m) is true, then it is true for all natural numbers then it is true for all n  N. In short, to accept the truth of some statement, we must exclude possibility of even a single exception !The reader must clearly see the mathematical rigor in the principle of mathematical induction ; in contrast to the case of the following example : “Ram shall die one day. Hari shall die one day. Gopal shall die one day . Thus it is deduced that any man shall die one day.” The statement is very true. But we find it difficult to see whether the statement as such, excludes the possibility of even a single exception. Thus mathematical induction is rigorously different from layman’s generalization. By proceeding step by step from a starting point an assurance for validity of the next step leaves no room for a single exception. §4:b)Principle of Mathematical Induction in another different way in which the principle of mathematical induction is elucidated is given below. If a statement is true for n = 1, and it can be proved to be true for n = m when it is true for all n < m, the statement is true for any n  N. The reader can understand the equivalence of the two different formulations. Evidently the principle of mathematical induction can only be applied to statements which admit of successive cases corresponding to the order of natural numbers 1, 2, 3, … etc. It is in fact, the basis acceptance of some rule after experimental verification . And this makes mathematics an experimental science. Example 1 : To prove 2 n ) 1 n ( 2   for all n ≥ 6 , n  N It is not correct to say that the statement has exceptions of 5 cases 1 ≤ n ≤ 5 . Rather the statement 2 n ) 1 n ( 2   for all n  N has 5 exceptions . Hence the former statement comes under jurisdiction of principle of mathematical induction. The steps of induction process are :Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 4 Step 1 : To see whether 2 6 ) 1 6 ( 2   which is obviously true as 64 > 49. Step 2 : Assume 2 m ) 1 m ( 2   for some m ≥ 6 and to prove that 2 1 m ) 1 1 m ( 2     Now multiplying both sides of 2 m ) 1 m ( 2   by 2, we have, 2 m 4 m 2 ) 1 m ( 2 2 2 2 1 m       ; Or,  2 2 2 2 2 2 1 m 2 m 4 m 4 m ) 2 m ( 4 m 4 m ) 2 m ( m 4 m 2                As m2 – 2 > 0 as m > 6. Thus the proof is complete as the statement is proved for n = M + 1. Note : The statement 2 n ) 1 n ( 2   for all n ≥ 6 , n  N, enjoys the status of only a conjecture until it is proved; by way of induction or otherwise. Example 2 : Prove by induction (and otherwise) that the number of all the subsets of a set having n number of members is 2n. By induction : Let us assume it for n = m, where m is a particular natural number . Now take a set A with n = m + 1 members. To count its subsets let us do in two steps; one, let us pick up a particular member ‘b’ and count all the subsets containing b . In the second step let us count all the subsets which do not contain b and finish counting in this way. Let B = A – {b}, and it has only m number of members. In the first step, we can take any subset of B and then include b to it. Since B has only m number of members, we count 2m subsets in this way, by assumption. In the second step, we count all the subsets of A which do not include b. this can be done by taking all the subsets counted in the first step and excluding b from every one of them. Thus the total count in second step is 2m also. There is not a subset in common among the collections in two steps. So our total count in both steps should be 2 x 2m = 2m+1 . So we have proved the statement for n = m + 1, while we assumed it for n = m. Now we have to see if the statement is true for n = 1, 2 etc. Clearly for n = 0, A =  , it has only one subset  itself .the total number of subsets is 1 = 20, which satisfies the statement. For n = 1, A = {a}, say; and the subsets are  and A, a total 2 = 21 subsets, thus satisfying the statement. For n = 2, let A = {a, b}. the subsets are : , {a}, {b} and {a, b} , a total 4 or 22 of them, satisfying the statement. We have already seen that if the statement is true for n = 2, then it must be true for n = 2 + 1 = 3; and thus we proceed to n = 3, 4 5…. Etc as long as we please. Otherwise: If the set has n members, we make subsets by picking up 0 at a time, 1 at a time, two at a time, and so on up to all at a time. If we don’t pick up a member we get  and we can do it only once ( Aren’t all ’s the same). We can pick up one member out of n members in n different ways, 2 different members in nC2 ways, , 3 Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 5 different members in nC3 ways and so on. So the total count would be nC0 + nC1 + nC2 + nC3 +………………. nCn different ways. By a result of Binomial theorem the expression comes upto 2n. §4:c)Why after all we require some sort of proof : As intelligent and logical persons, we need verify that a particular statement applies in every case without exception before we accept it. The difficulty or impossibility of verifying every case of application of the statement forces us to look for a proof. For example, to prove that man is mortal, how many cases can we count or how long could we do so ? Hence we require a logical proof. The principle of mathematical induction gives us just as a shortcut to the process of counting and verifying the applicability of a certain statement case by case; assuming it for some case and deducing the validity for the next case. And ultimately, verifying validity of the statement for a particular case say for n = m, extends the validity of the statement to all the subsequent cases. The other statement of the principle of mathematical induction also essentially extends verification of the statement to the case n = m when the validity of it is assumed for all cases n < m. §4:d)Exclusion of exceptions; or contradiction of contradiction is also a method of proof : Naturally. When ‘proof’ is ‘verification of a certain statement case by case’ and ‘without exception’, a method of proof can be given just by guarantying that not a single case of exception hold good. In other words, when contradiction of the statement is disproved, the statement stands automatically proved. Like the method of induction, which is more often applied to theorems of fundamental importance , the proof of which otherwise is difficult; in the same manner, the method of contradiction of contradiction is applied for proving theorems of fundamental importance ,the proof of which is otherwise difficult. Examples of proofs by excluding possibility of contradiction of the statement under consideration abound in Mathematics and shall be encountered as we proceed. Vide examples under section irrational numbers below for examples. §4:e)Deduction – a method of proof : As we have stated above, logical deduction is a device or tool for verification of a statement and thus it is accepted as a method of proof. One example would suffice to illustrate the conjecture. A polynomial or a rational integral algebraic expression is like a0xn + a1xn -1 + a2xn -2 +…… an-1x +an where n is a positive integer and the coefficients are real or complex numbers. The fundamental theorem of Algebra states that it has n roots, real or complex. The statement is equivalent to a polynomial of degree n has at least one root, real or complex. Either of these two statements can be logically deduced from the other. If we assume that a polynomial of degree has n roots, then it Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 6 follows that it has at least a root and thus the second statement follows from the first. Conversely, if the polynomial has a root at all, say x =  then a0n + a1n -1 + a2n -2 +…… an-1 +an = 0 so that a0xn + a1xn -1 + a2xn -2 +…… an-1x +an = a0xn + a1xn -1 + a2xn -2 +…… an-1x +an -a0n -a1n -1 -a2n -2 -…… an-1 -an = a0 (xn -n ) + a1 (xn – 1 -n – 1)+ a2 (xn -2 -n -2 ) +…… an-1 (x -) and it is easily seen that (x -) is a factor of the polynomial as it is a factor of the expression on the right. When we divide the expression by x – α , we are left with a quotient, called reduced equation of degree n – 1 , which has also a root, β, say and thus a factor x – β ; so x – β is also a factor of the original expression. In this way the expression may be seen to be having r roots. Thus the two statements are deduced from each other and are equivalent. §5:Recapitulation of Permutations, Combinations as a special counting These concepts are nothing but special techniques of counting . perhaps the most important basic purpose of study of Mathematics is ‘counting’ and this poses the toughest of problems in advanced study. Let us satisfy ourselves with only a few preliminaries. How many patterns we get by setting 4 things in 4 adjacent places so that no two patterns are same? This is called Permutations of 4 things taking 4 at a time. We note that the first place can be filled up in 4 different ways. After the first place is filled up in any one way, the second place can be filled up in 3 different ways, as only three things now remain to choose one from. So the first and the second places together can be filled up in 4x3 different ways. After the first and the second places have been filled up in any one way, the third place can be filled up in 2 ways as only two things are now left to choose one from. So the first, second and the third places can be filled up in 4x3x2 different ways. After the three places have been filled up in one of the 4x3x2 ways, the fourth place can be automatically filled up with the only one thing left, i.e., in only one way. So this is the total number of permutations. More formally, 4x3x2x1. We denote this by 4P4 or 4! . By n! we mean, product of natural numbers from 1 upto n and it is spelt as ‘factorial n’, also it is denoted as |n in a different notation. Note that n! = n(n-1)! = n(n-1)(n-2)! etc. Especially, putting n = 1 in n! = n(n-1)! We get 0! = 1………(A) ( Putting -1 in it is impossible for reason of division by 0.) The above argument can be repeated and we get nPr = n(n-1)n-2)…..up to (n-r+1) (by nPr we denote, the number of permutations or different patterns produced by choosing r things out of n things at a time.)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 7 Please note nPr = n(n-1)n-2)...upto(n-r+1)(n-r)(n-r-1) x3x2x1/(n-r)(n-r-1).3x2x1 = n! /(n-r)!. nPr = n! /(n-r)!.................. ..........................................(B) Let us find out nPr in another way. Let r places are filled up taking r things out of n things at a time in nCr ways, when we do not mind the order of things in any arrangement. By nCr we mean the number of ways in which we may pick out r things out of n things in whatever sequence or order. Now, for each arrangement or combination, if we do mind for the order of things, we get rPr or r! permutations as, in each one combination, we can change only the order of the things choosing r things out of r things at a time and get a different permutation each time.. So total permutations are, nPr = nCr x r! , Or, nCr = nPr /r!...............................(C) When we are taking r things out of n things for any particular arrangement, we are automatically leaving n-r things out of n things. So the number of times we are taking r things = the number of things we are leaving n-r things. So nCr = nCn -r ………………………………….(D) Another useful result from this section required in later sections may be easily proved and noted n+1Cr = nCr + nCr-1 ……………….…………………...(E) We begin,     )! 1 r ( ! ) 1 r ( n ! n ! r ! r n ! n C C 1 r n r n                               1 r n 1 r1 )! 1 r ( ! r n ! n )! 1 r ( ! ) 1 r n ! n ! r ! r n ! n   } r )! 1 r ){( 1 r n ( )! r n ( ) 1 n ( ! n ) 1 r n ( r r 1 r n )! 1 r ( ! r n ! n                     r 1 n C ! r )! r 1 n ( )! 1 n ( ! n ! r )! 1 r n ( )! 1 n ( ! n           When we have already arrived at nCr number of combinations , and then given another thing B say, to be added to the n things we already have. We have nCr number of combinations without taking the new thing B.(For we have to choose r things only, even now). If we count the combinations in each of which, B is taken we get the total answer. Every time B is taken, we have to choose only r – 1 things, out of n things , of Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 8 course, giving us nCr-1 combinations. Total number of combinations is, those taking B and those not taking B i.e., nCr-1 + nCr .But the total number of combinations is n+1Cr. Hence n+1Cr = nCr + nCr-1 . §6:Binomial Theorem revisited Before more about numbers we must be able to work with numbers , binomials or two member expressions. The identities got so would help us understand expressions better. If n is a +ve integer, we have, (a + b)n = (a + b) (a +b) (a + b)……………n factors. For taking the product, we choose a’ s from r factors and b’ s from remaining n-r factors. So we get a term arbn-r in the product. How many times we get this for a particular r ?, Certainly as many times we could choose a’s from r factors , i.e., nCr number of ways and add them. Once only we could choose “no a’s” or “no b’s” also. So we get (a + b)n = n C n a n + n C n-1 a n-1b + n C n-2 an-2 b2 +……………n C 0bn = n C 0 a n + n C 1 a n-1b + n C 2 an-2 b2 +………………n C n b n (as nCr = nCn-r ) or, (a + b)n = a n +n a n-1b +[n(n-1)/2!] an-2 b2+ [n(n-1)(n-2)/3!]an-3b3…b n…. (A) Also for n being negative or rational number (fraction) and |x| < 1, (1+x)n = 1 + nx + [n(n-1)/2!]x……….upto  ………….………..(B) In particular, for n = -1, and |x| < 1, 1 /(1-x) = 1+x+x2+x3+ ,……………….upto  and 1 /(1+x) = 1-x+x2 -x3+ ,……………….upto  (Just divide 1 by 1 -x or 1 + x as we do in Arithmetic) Putting a = 1 = b in (a + b)n we get, 2n = (1+1)n= n C 0+ n C 1 + n C 2 +……n C n …………… …….. ……(C) Putting a = 1 and b = -1 in (a + b)n we get, 0 = (1-1)n= n C 0-n C 1 + n C 2 -n C 3 ………………(-1)n. n C n Or, nC0 + nC2+ nC4 ……=2n/2 =2n-1 and nC1+nC3+ nC5+…= 2n/2 =2n-1…..…(D)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 9 §7:INFINITESIMALS: Variables which ultimately tend to 0 are infinitesimals. At first it appears we have nothing to do with infinitesimals; but suppose we take the ratio of two infinitesimals say yx  where 2 y x    . Both tend to 0 all right but we cannot neglect yx  for it is always equal to 2, a finite number. In any problem involving infinitesimals, we can neglect them only at the end , not at any middle step and we have to be very careful while dealing with them. Further often we need know, out of two infinitesimals which tends to 0 faster, so that we can get rid of it earlier than the one which tend to 0 later. For example if we have to calculate 6 1 x x         , we need binomial expansion of the expression in the bracket ; 6 2 3 2 3 1 1 6 .......... x x x x x x x x                        . We need not calculate ,   etc. and put the final result at 1 6 x x   and neglect higher powers of x  outright. This makes calculation easier. Similar is the concept of infinity. Sum of two infinite variables is definitely infinity and product. But nothing could be said in general about difference or division of two infinite variables. They may happen to be constants also. For example 2 2 1 x x  must tend to 1 as x tends to infinity. If 2, y x   then 2 y x   even if both x and y tend to infinity. What are infinitesimals and how to deal with them, we shall discuss in the chapter for limits . Here this much would suffice to say that entire Calculus is algebra of infinitesimals and infinites.. In the figure, note that the number of steps are doubled when width and height of each step is halved, whereas the total height and total width of the steps is unaffected. We could go on reducing the width and height of the steps to any extent we please , increasing the total number of steps accordingly. We could Infinitesimals :width/height of steps go on decreasing but their totals are constant. Thus infinite number of infinitesimals are added to give finite number. Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 10 carry on the process until when neglecting two or four steps does not matter .But these infinitesimally small heights and widths always add up to a definite sum ,no matter how small they individually are. Infinitesimals do add up to finite sums even if they are individually negligible under certain conditions, of course.( you cannot make the series 1+1/2+1/3+1/4…….. converging, although after a large number of terms the remaining terms become only infinitesimals). Infinitesimals are small variable numbers. As another example, let us cover the distance from the school to home in several installments. First let us cover one half; the other half remains. Next let us cover half of the remaining; i.e., one fourth of the total distance. It remains only a quarter to be covered then. But now let us cover one half of the remaining distance; now we are still left with half of a quarter to be covered. We can continue the scheme indefinitely leaving negligible portion left to be covered; and we can make that as small as we please and finally reach home! So all those infinitesimal distances after a large number of installments make up to cover the whole distance. We assign some symbols like  and  etc. so that we can easily make calculations involving them and throw them away at the end. To deal with such infinitesimal quantities Calculus was developed. Without Calculus (perhaps) the Hindus did study Astronomy and could predict eclipses etc. fairly accurately; even before discovery of Uranus, Neptune ,Pluto and other planets and even assuming the Earth to be at the centre of the universe. May be, they could do so by assigning some corrective weights to the points Rahu and Ketu ,the points of intersection of the Moon’s orbit with the plane of the orbit of the Earth. Their approach was in digital from the very beginning and they have gifted the world with 10 digits; every calculation was with the help of digits or ratios of digits. Corrections were accounted for by what we call continued fractions today, giving birth to surds and irratio0nal numbers. But study of Astronomy was limited to a privileged few. When Calculus was developed by Newton (And Leibniz separately) for study of Astronomy, the subjects like Mechanics, Physics etc. virtually all the disciplines which studied changes of quantities and rates of changes ,exposed themselves to the less scholarly, and errors of measurement were made under control, and better approximations were achieved. In fact, Mechanics and Physics etc. were qualified to be called sciences only after the groundwork was provided by Calculus. (For a long time credit of discovering Calculus went to Newton although limiting processes were already developed by Fermat, Centuries earlier. Copernicus had also changed the world view of Astronomy. But it was Newton who gave a systematic background to it. Thanks to his wondering the apple fall and then the knowledge about nature began to be called sciences.) Infinitesimals are no problem; but such exceptional behavior of ‘zero’ alone 1necessitated development of this branch of Mathematics which we call Calculus. The edifices of Physics or any discipline which deals with rates of changes of variables must stand on the foundation of Calculus. Until the advent of Atomic Physics, or quantum Physics, Calculus-based Physics reigned supreme. Calculus continues its supremacy in the realm of Quantum Physics inasmuch as probabilities took the place of exactitude and these probability equations stood on the groundwork of Calculus. So utility of Calculus has stood the test of time. 1 No number, even 0 is divisible by 0.( see next article)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 11 Exercise1 To have a taste of infinitesimals at work, try to find out 3 7 r  We know 13 = 1 and 23 = 8. So r must lie between 1 and 2. Now take to divisions 1.1, 1.2, 1.3… etc between 1 and 2 and take their cubes to see r lies between which two. In this way we can narrow down r to any degree of approximation we desire, but never reach it; although it could be exactly cut off a straight line on paper. The fact is we cannot exactly represent r with the 10 digits in closed form in the manner we are able to represent any rational number say 1/7. That is not necessary either, as we can freely use the number r wherever and in whatever manner we desire. If you think r is quite expressed completely with the cube root symbol, it is not; it is just a symbol just like Roman numerals and the number of symbols would go on increasing if are satisfied only with symbols . §8:Development Of Number Systems. Everybody is familiar with the numbers used in counting. Counting starts at one and goes on increasing by 1 as far as one can count. In advanced Mathematics, there is seldom any other challenge bigger than the problem of counting. The set of Natural Numbers is denoted by the set N ={ 1,2,3,4,5, …… up to  } Any set having (Infinite) members having one-to-one correspondence with this set is called countable ( or denumerable). By ‘countable’ we certainly do not mean finite sets. ‘one to one correspondence with natural numbers’ – this is because we count only with the help of natural numbers. If the set of natural numbers were uncountable, how on earth , that would count other sets? For example the set of even natural numbers { 2,4,6,8,10,12 ….  ) is seen to be countable as we could associate 1 with 2, 2 with 4, 3 with 6 and so on. By one-to-one correspondence we mean that for every member of N we have one and only one member of the set of even numbers and vice versa. So also the set of odd numbers {1,3,5,7,9,…..up to ) is countable. The set of Integers I = { - …… -3,-2,-1,0,1,2,3,4 ……. } is also countable, as we could associate 1 with 0, 2 with 1, 3 with -1, 4 with 2, 5 with -2 and so on. The set of integers has expanded our number concept . Integers were discovered when we simply chose to do away with the restriction on subtracting a number from itself or from a smaller number ,i.e., when we just simply agreed to subtract a larger number from a smaller one. We could start from a point, proceed 10 steps towards the East and return 12 steps backwards to the West. This is just possible if we do not meet a wall at the starting point! Discovery of zero, by agreeing to subtract any number from itself is biggest contribution of India to the world. 0 contains everything along with its negative, a whole world of mystery, like a black hole containing a whole universe inside it. The place value concept enabled us to write an integer completely, however large it may be, just with 10 digits. Again this is contribution from India only.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 12 The set of natural numbers is said to be closed with respect to the processes of addition or multiplication. The meaning is, addition or multiplication of any two natural numbers gives a natural number again; i.e., we can not get out of the natural number set just by adding numbers or multiplying them. So also the sets of integers, rational numbers and the number systems we shall discuss afterwards are all closed with respect to the process of addition and multiplication. We may verify that the set of integers is closed with respect to subtraction process also. That was precisely the purpose of accepting the set of integers. Thus we see that the direct processes like addition and multiplication pose no problem; but the reverse process presents us with a restriction. When we agree to discard this restriction, our number system is expanded or extended. But one thing is for sure; the set of integers contains exactly the same number of numbers or elements or members as the set of natural number does ! For, we have just shown that the two sets are equivalent to each other , having a one to one and onto relation between them. The restriction encountered in the reverse process may be viewed in another manner. We can say that the equation x + a = 0 where a is a natural number, may not have a solution in the set of natural numbers for every a  N , but it must have a solution in the set of integers I. By calling the process of repeated addition as a separate process multiplication we faced with no difficulty. But its reverse process posed a problem .We could not divide a number ‘a’ by a number ‘b’ unless ‘a’ was an exact multiple of ‘b’. We again preferred to do away with this restriction and our number system was extended to what we call Rational Numbers ,which included the integers as well as fractions in the form a/b (a and b being integers, so all the terminating and non-terminating recurring decimals are included). Fractions can easily be converted to a terminating or non-terminating recurring decimal and vice versa. Of course the restriction that nothing could be divided by 0 remained and you can guarantee that the principle shall remain in force till the end of the earth. Viewing in the other way round, we can always have a root of the eqn. bx + a = 0 where and b are any integers and b  0 In simple words, the set of rational numbers are all fractions built out of integers in the numerator and denominator ( except 0) which is equivalent to the set of all integers and terminating decimals and nontermiinatin decimals. The set of rational numbers is precisely all the decimals which can be converted to fractions.( See cosmic song of numbers below to see how decimals are converted to fractions. To convert a fraction to decimal form, just divide the numerator by the denominator and get the decimal form.) §9:Scales of notation. With the example of residue classes congruent modulo 7, we could imagine a number system with seven digits, 0, 1, 2, 3, 4, 5, and 6. Seven can be written as 10, just as we write 10 for ten in decimal system when the digits ( symbols they are, after all) 0 , 1…to 9 are exhausted, forty nine can be written as 100 etc.. Similarly the fraction 1/7 can be written as 0.1 as 1/10 is written in decimal system, 0.01 may be written for the fraction 1/49 and so on in what is called the scale of 7. A good system of arithmetic i.e., addition, subtraction, multiplication , division, squaring, cubing , taking roots, etc. can be developed quite similar to the Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 13 decimal system arithmetic and it would be as easy as the decimal system, with place value 7 instead of ten. A terminating fraction in decimal system may turn out to be non-terminating in that system and a nontermiinatin fraction such as 1/7 may be terminating, i.e. 0.01.In fact a number is written as abcd in decimal system is actually 4 3 2 .10 .10 .10 a b c d    . The very same thing abcd written in scale of seven is 4 3 2 .7 .7 .7 a b c d    . The names for scales of 2, 3…..upto scale of 11 used are binary, ternary, quaternary, quinary, senary, septenary , octenary, nonary, denary = decimal , undenary, duodenary respectively and the scale of 16 is called hexadecimal. The base number 2, 3 etc. of the scale is called radix. Of these scales of notation, much useful is the binary system, having only digits, 0 and 1 only, which is the very theoretical basis of any computer programme and hardware. The ICs or integrated circuits which consist of large number of electrical circuits have only two states of individual circuits , 0 or 1, i.e. closed or open. A decimal number entered into the keyboard for calculation is first converted by interpreter to octenary or hexadecimal and then converted to binary. Actual calculations take place in binary mode and the result is again displayed through the interpreter again in decimal. Ordinary arithmetic calculations can be performed easily in any scale of notation without converting it into the decimal system, bearing in mind that the successive places from the right are not multiples of ten each time by multiples of the radix of the scale used. The decimal point in the right is replaced by what is called the radix point. To convert a number N, say in decimal system, into a scale of r, we may write 1 2 1 2 1 0 .......... n n n n n n N a r a r a r a r a          , where the coefficients 1 2 1 0 , , ,.... , n n n a a a a a   have to be determined. Evidently, 0 a is the remainder when N is divided by r . Also 1 2 3 1 2 3 1 .......... n n n n n n a r a r a r a          is the quotient. We can divide it by r to determine the remainder 1 a and also note the quotient would be 2 3 4 2 3 4 2 .......... n n n n n n a r a r a r a          . Continuing the process, we get the representation. In the similar manner a fraction F may be expressed in a scale of r by successive multiplication by r. See how it is. Let the fraction be 3 1 22 3 ............. a a a F r r r     Multiplying by r, 3 2 1 2 ............. a a Fr a r r     , so 1 a is integral part of Fr . Next, let the fractional part of Fr be 3 2 1 2 ............. a a F r r    .Multiplying this with r, we get 3 3 1 2 ............. a a F r a r r    Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 14 So 2 a . In this way by successive multiplication, the fraction may be expressed in terms of another scale of notation. If at any stage the product n F r is a whole number , the process terminates there and the fraction is expressed as a terminating one in that scale of notation. Example1 To convert a decimal number 55647 into scale of seven. Ans. Divide 55647 by 7, the quotient is 7949 and remainder is 4 = 0 a . Now divide 7949 by 7, the quotient is 1135 and the remainder is 4 = 1 a . Then divide1135 by 7, the quotient is 162 and the remainder is 1 = 2 a . Then divide 162 by 7, the quotient is 23 and the remainder is 1 = 3 a .The divide 23 by 7, the quotient is 3 and the remainder is 2 = 4 a . The last quotient is 3 = 5 a . The required septenary representation of 55647 as (341144)7.the division processes may also be condensed by synthetic division, discussed in the section of theory of equation. A word of caution – when a number is mixed one, i.e., a whole number and a fraction, the whole number part and the fraction part must be separately considered for conversion. Exercise2 Express 122 in binary system . Hint . If division by 2 leaves 0 as remainder, put 0 a =0 and proceed. Exercise3 Express 37 32 in terms of scale 6 Ans. Separating the whole number and the fractional part, 37 5 1 32 32  ; Now 1 in decimal system is 1 in scale of 6 also. Then 5 30 15 .6 32 32 16   , integer part is 0 15 45 5 .6 5 16 8 8   , the integer part is 5Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 15 5 15 3 .6 3 8 4 4   , the integer part is 3, 3 1 .6 4 4 2  , the integer part is 4. 1 .6 3 2  , and the process terminates. Thus the fractional part after the radix point (analogous to decimal point) is .05343 and integer is already known to be 1. Hence the number 37 32 in decimal system is 1.05343 in senary system. Exercise4 Show that if any number whose sum of digits is divisible by 1 r  in a scale of notation whose radix is r, must be divisible by 1 r  . Hint. Let 1 2 1 2 1 0 .......... n n n n n n N a r a r a r a r a          , 1 1 0 ................ n n S a a a a      Show that N S  is divisible by 1 r  Exercise5 If difference of sums of digits in odd and even places is d , of any number N in a scale of notation r, then show that N +d and N – d both are divisible by r +1. If d = 0, the number must be divisible by r +1. Exercise6 If the radix is odd in a scale of notation, show that the sum of digits in an odd number is odd and that of an even number is even. §10:The Cosmic Songs of Numbers. Any non-terminating repetitive decimal like 437 20 . 2 may be converted to p/q form as follows. Write S= 2.20437437437437437……………..……(1) Or, 100S= 220.437437437437437………………(2) (multiplying both sides by 100) Or, 100000S= 220437.437437437437…….……(3) (multiplying both sides by 1000) Or, 99900S = 220437 – 220,subtracting (2) from (3), which gives Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 16 437 20 . 2 = S = 99900220 220437  (look at the power in a symbol) which may be taken as a formula for converting a repetitive non-terminating decimal in p/q form. To convert any p/q in decimal form, just divide p by q and the result would either be an integer, or a terminating decimal or a non-terminating repetitive decimal. In literature they call songs as ‘numbers’ and dances for that matter. This is truly so; as a song can be represented by a single rational number p/q along with its music. Write out the musical notes replaced by digits 0, 1, 2, 3, 4, 5 and 6; each representing a note in the octave scale. Let the entire song be written as abcdef……..k. Repeat the number infinite times and put a decimal point before it so that it becomes k ......... abcdef . 0 which can be converted to p/q form. Thus the entire song is stored in a single rational number p/q and can be played out from computer if only seven octave tones are recorded in the computer; just by typing p/q without actually recording the song. One could play the song with harmonium or flute or with a whole orchestra even if the octave tones only are input. One can use hexadecimal system or other scale of notation in place of decimal system also. Thus one can design a song with numbers and listen to it without anyone singing it. Or a whole lot of rational numbers can be tested for what music they produce. Even known irrational numbers could be listened as a non-stop song comparable to an unending prose. Radio waves sent by distant stars may be converted to music and one can listen to the cosmic music, interpret it as signals from other civilizations. We can lie on a beach and listen to the nonstop song of the sea waves breaking ashore. After all you observe the rhythm of any song or dance – it is repeat ion of a simple combination of numbers only , 1,2,3,4.. etc. The terms ‘real number’ , ‘rational number’, ‘irrational number’, ‘imaginary number’ are unfortunately misnomers . Doesn’t an irrational number stand to reason ? Definitely no. They carry a finer reason with them. If natural numbers stood for counting, irrational numbers helped measurement and have enhanced our freedom in working with numbers. Do the imaginary numbers exist only in imagination ? Definitely no. They have expanded our minds from a line to an entire plain, presented us with a reasonable theory of equations and erected a discipline of Trigonometry. We are calling them wrong names quite for some centuries and it does not seem whether we would rectify ourselves in near future. We satisfy ourselves by calling ‘irrational ‘, ‘imaginary’, ‘complex’ etc as “reserved words” in the discipline of Mathematics, as in any other discipline. Ex 7:  Scale of notation – In decimal system we use digits from 0 to 9, ten digits and the next number after 9 is written as 10. If we use a scale of seven instead we use digits from 0 to 6 only.10 denotes seven, 100 denotes forty-nine, 0.1 stands for 1/7 in decimal system, 0.01 for 1/49 in decimal system and so on. Arithmetic operations are done exactly as in decimal system. Computer uses binary system consisting of only two digits, 0 and 1. Note that a terminating fraction 0.1 in scale of seven is 1/7 in decimal system which is non-terminating and recurring.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 17 Find integer values satisfying 1 2 1 2 2 n n n n   . Ans. :         3,3 , 2,4 , 0,0 , 1,1   Ex8: If   S n be the number of ordered pairs of positive integers   , x y satisfying 1 1 1 x y n   show that   3 S n  for any prime number n. Also find   6 S . Ans. 9 Ex9: Show that the number of ordered pairs of positive integers   , x y satisfying 2 2 2 1 x y   is only one. §11:Division by zero is an impossible operation. The attempt to divide anything by 0, even 0 by 0 results in a fallacy as illustrated below: 5/0 = p(say);  5 = 0xp= 0, is evidently false. Again, observe that 0 = 5 x 0 = 7 x 0;This should imply 5 = 7, which is false, otherwise 0/0 could be anything we choose, 5/7 or 7/5, which is again not admissible. So canceling 0 from both sides leads us to fallacy. Even 0 cannot be divided by 0. In plain words this may be said that anything multiplied by 0 gives 0 so division by 0 of 0 gives nothing in definite. The result is said to be indeterminate instead of impossible. We say division by 0 an impossible operation for two reasons. First, the division does not give unique quotient as we have just seen. Secondly, dividing a nonzero number say a by 0 does not exhaust it; does not reduce it at least; for a = ax0 + a, whereas we expect the remainder to be 0. Plainly speaking, if a piece of cake is distributed among some people giving each of them nothing at all, the cake shall never be exhausted even if distributed among infinite number of people are given their share amounting to nothing. Entire Calculus is concerned with division by numbers approaching 0. We would take on the topic starting from the chapter on limits. §12:The Power of the division process Often we use a symbol to express the ratio of numbers approximating to 0 each. Though we do not have a meaning of P/Q when P and Q are 0, there is no harm in considering the ratio when both P and Q approach 0. When such a denominator and numerator approach to a certain ratio after division, 0/0 is given a new symbol QP lim 0 Q , 0 P   . This is permissible as neither of the numerator not the denominator is not 0 even if the values taken may be as near to 0 as one pleases. The essence of Calculus is to analyse such situations with the process of taking limits of values which we would see later on as a magic wand. We do derive a world of utilities out of 0 in this way, as any number can be expressed as a limit of two numbers each of which approximating to 0. Even without the study of calculus or limiting process, we are already aware of the vast Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 18 utility of the process of division in trigonometry. After all entire trigonometry is about ratios of sides of a right-angled triangle. Division process would further be discussed while studying complex numbers , vectors, quaternions , and division rings . Quaternions were originally discovered as ‘quotient’ of two vectors and later were seen to be a ‘complex’ of a scalar ‘a’ with a vector bi + cj + dk. Further see the topic of ordered pairs of numbers for understanding of division processes. §13:The set of rational numbers is countable. We can write the set of rational numbers in rows like, n /1 in the 1st row, -n /1 in the 2nd row, n/2 in the 3rd row, -n /2 in 4th row, n /3 in 5th row, -n /3 in the 6th row etc. and so on; where we exclude any number from the list if it has appeared before. We have not left any rational number in the list as such. We can give a scheme of counting the rational numbers as arranged in the above table in this manner: 1  0, 2  1, 3  -1, 4  ½, 5  -2, 6  2, 7  3, 8  -3 , 9  3 /2, 10  -½, 11  1/3, 12  -3 /2, 13  5 /2, 14  -4, 15  4 and so on. It is easily seen that the set of natural numbers is in one-to-one correspondence with the set of rational numbers in this way. Alternatively, the scheme of counting may be like : (1,1),(1,2),(2,1),(3,1),(2,2),(1,3),(1,4),(1,5),(2,4).(3,3).(4,2),(5,1).(6,1)…. and so on. where the ordered pairs of numbers represent row and column numbers in order of counting. Repeated multiplication or raising to powers of numbers posed no problems .But the reverse process , extracting of roots limited us in finding square roots and cube roots of numbers that were not exact squares or cubes. Note : Another counting scheme could be considered . First take the Positive rational numbers (fractions reduced to having no common factor in the numerator and denominator) the sum of whose numerator and 0 1 2 3 4 5 6 7 8 9 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/2 -1/2 -3/2 -5/2 -7/2 -9/2 -11/2 -13/2 -15/2 -17/2 -19/2 1/3 2/3 4/3 5/3 7/3 8/3 10/3 11/3 13/3 14/3 -1/3 -2/3 -4/3 -5/3 -7/3 -8/3 -10/3 -11/3 -13/3 -14/3 1/4 3/4 5/4 7/4 9/4 11/4 13/4 15/4 17/4 19/4Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 19 denominator is 2, such as 11. Next take those , the sum of whose numerator and denominator is 3, such as 1 2 , 2 1 . Next, for all those for which this sum is 4, such as 1 2 3 , , 3 2 1 and so on, in increasing order each time. Putting it all together , and excluding repetitions, we get 1 1 1 2 3 1 1, , 2, ,3, , , ,4, , 5.............. 2 3 4 3 2 5 The difference between two schemes is that in the previous scheme, we proceeded progressively listing the elements , 1st whose denominators are 1, then whose denominators are 2, 3 and so on. In the 2nd scheme, we noted the elements 1st, whose numerator are 1, then those whose numerators are 2, and so on. To be practical we had to do away with this restrictions on extracting roots. We could very well construct numbers like √2, √3, etc according to Pythagoras theorem. For example take a square of one unit of length and breadth, its diagonal would be of √2 units. A diagonal of a rectangle of length of one unit and breadth two units would be of √3 units. Such numbers we called surds or irrational numberss in general (all irrational numbers are not surds) .They are non-terminating and non-repetitive decimals. They are not countable (as shown below). Every non-terminating repetitive decimal can be converted to a fraction of type p/q ,and every p/q , (p integer and q nonzero +ve integer) can be expanded to a decimal number, terminating or non-terminating repetitive. The number 2 is no rational number. If it is a rational number at all, let 2 = p/q , where we may suppose that we have already reduced the fraction i.e., there is no common factor between p and q to cancel out. It gives, p2 = 2q2 implying that p2 is even, and let it be 2k . This gives 4k2 = 2q2 , or, 2k2 = q2 , which gives q2 is even ,so that q must be even (otherwise q2 won’t be even) . Thus our supposition that there are no common factors between p and q is contradicted. Similarly one could prove √3 is irrational using the fact that if k is divisible by 3, so also k2 is and vice versa. Irrational numbers involving integral or fractional roots are called surds like a + b , a + 3b etc. The surds involving square roots only are called quadratic surds . ). Irrationals or surds of type a + b admit of a conjugate a -b so that product of the two is a2 – b, a rational number. Other surds such as a1/3 + b1/3 have rationalizing factors like a2/3 -a1/3b1/3 + b2/3 so that their product a + b is a rational number. In the equation analogy we may say that an equation of the type xa – b = 0 where b is any positive rational number and a is a nonzero integer , must always have a solution; the solution must be in the set of surds if it is not in the set of rational numbers. Another class of numbers are transcendental numbers are of the type log m n , where either m or n has a prime factor which the other one does not totally have, meaning thereby log28 is not included in our consideration which is 3, a rational number; as they are not having different prime factors. Now suppose log23 = m/n , where m and n are positive integers. This gives 2m/n =3, or 2m =3n . This is a contradiction as there are no common factors in left and right side. Hence log23 cannot be a rational number.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 20 In fact, if A is an integer which is not a perfect n-th power, then there is no rational number p/q in lowest terms ( meaning thereby there is no common factor between them except 1) such that (p/q)n = A. For pn/qn shall be in lowest terms if p/q is in lowest terms and as such, cannot be equal to an integer and A is taken to be an integer. Are there other irrational numbers , other than surds ? Yes; take for example, a real number x, such that x = 75 32 4 3 3 4 3 5 ( a           , which could hardly be called a surd. If this eqn. is simplified, we would arrive at a polynomial eqn.s like xn + axn – 1+….+ px + d = 0 where the coefficients a, p, d etc are taken to be rational numbers and n is taken to be a natural number and whose one of the roots is the complicated number we are considering. Certainly the number we are considering it does not belong to the set of rational numbers. This discussion is about what types of numbers may be irrational, bearing in mind always that they are nontermiinatin , non-repeating decimals any way. Can there be still other types of irrational numbers other than those which are roots of polynomials of finite degree? The surds may be viewed as roots of some polynomial equations also as can be easily verified. For example, let x = a + b1/3. this implies x – a = b1/3  (x – a )3 = b and simplifying the same we observe that x is nothing but a root of a polynomial equation of finite degree with rational coefficients. Such numbers which are roots of a polynomial or are algebraically derived from rational numbers are said to be algebraic numbers. Algebraic numbers are not only algebraic irrationals as in the above example, but surds, rational numbers and integers too are algebraic numbers. It is easy to see that these numbers can be easily algebraically derived from rational numbers or equivalently, may be thought of as roots of polynomials. By polynomials we generally mean finite power series with rational coefficients. A number is called algebraic of order n if it is a root of a polynomial of degree n and it is not a root of any polynomial of any degree less than n. Rational numbers are algebraic of order 1 as it may be seen that they are solutions of the eqn.s 1st degree, of the type ax + b = 0 and so also are integers. The coefficient a, should be 1 or should be a divisor of b if x has to be an integer. Quadratic surds are algebraic numbers of order two, as may be seen from the fact that they are roots of quadratic equations like ax2 +bx + c =0. As per rational root theorem (see footnote against the word ‘polynomial’ for proof) if a rational number p/q is a root of a polynomial, a0xn +a1xn-1……….an = 0, then q must divide a0 and p must divide an. in addition if a0 = 1, q is bound to be 1, for 1 has no other divisor except `1. this reduces p/q into an integer. Simply stated, if a polynomial has an integer root, the leading coefficient must be 1. During this discussion we are considering real roots only. What if the roots are not real and we call the roots of the polynomial integers when the leading coefficient is one ?. certainly the heavens won’t fall and Mathematicians do call such numbers as algebraic integers. For example,  and 2 , the cube roots of unity are solutions of x2 +x + 1 =0 and as such, are algebraic integers. ‘Algebraic integers ‘ sounds a little hard to believe; as irrationals are also included in it, equally difficult to believe would be to learn that algebraic numbers are countable!Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 21 Are there other irrationals which are not roots of a polynomial ? A power series with infinite number of terms and converging to a limit may have a real number as a root which is also a root of a polynomial ( the root is an algebraic number in that case) . Or, the root of the power series may not be a root of any polynomial at all. Then this number is called a transcendental number. Let us take a number like 0.101001000100001…..which is a convergent power series of 1/10 such that , in the n-th place, the coefficient is 1 if n = 1, 1 +2, 1 +2 +3, 1 +2 +3 +4 etc. and 0 otherwise. Surely this is a converging power series and is not a rational number, as a block of digits after the decimal point is never repeated. Can any algebraic expression in 1/10 represent this number? May not be. This is a transcendental number in that way. But if we truncate this number or take a number which is same as this only upto a certain place after the decimal point, the number so arrived at is easily seen to be a rational number as soon as we convert that into a fraction. Similarly take a number like 0.101000000100000…..which is a convergent power series of 1/10 such that there is 1 after 1, 2!, 3!, 4! places and 0 otherwise, no algebraic expression of 1/10 can represent this and it is not an algebraic number. Transcendental numbers are roots of transcendental functions. In fact a transcendental function might have an algebraic number as one of its root. But a transcendental number is one, which is a root of a transcendental function and it is not a root of any algebraic function. Is the eqn. 5 x 1 1   having a rational root ? Yes, it is 4/5; (in fact, the only one root) . But this eqn. may be written as 5 = 1 + x + x2 + x3…….upto  number of terms . such functions are transcendental functions but having an algebraic number as a root. . If an algebraic numbers.. is irrational then we call it algebraic irrational number to distinguish it from rational numbers. The ratio of circumference of any circle to its diameter  is such a transcendental number. The Napierian number e which is the amount of principal of one rupee with 100% interest per annum at the end of one year when interest is payable every moment, is another example of a transcendental number. Can x - =0 be regarded as an algebraic equation. Definitely not; as coefficients must be rational in order that the eqn. may be algebraic. Exercise10 Can a rational number be the sum of two irrational numbers? Difference, product or quotient ?  Some authors call rational numbers as commensurable and irrational numbers as incommensurable. Both the terms irrational and incommensurable are misnomers. An irrational or incommensurable number is as exact as any other number could be, definite points on the number line. The fact that we express it in successive approximations after the decimal point has nothing to do with the definiteness of the point, rather an inadequacy in our system of writing. The infinite converging series representation should be understood to be as natural as so called closed forms like  ,2 etc. which are mere names given to the numbers they refer to. Therefore the real numbers are defined as a ‘cut’ in the number line. ( See Dedikind’s property of real numbers). The same comments are justified for imaginary numbers or complex numbers too. They are as real as the real numbers are, only at right angles to the real numbers!. Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 22 Exercise11 If a real number x satisfies an equation 1 2 1 2 ........... 0 n n n n x a x a x a        where 1 2 , ........... n a a a are all integers, then x cannot be a rational number other than an integer. ( it may be an irrational number; any problem?) Assume the contrary – let x be a rational number /h k , (where gcd of h and k i.e. ,   , 1 h k  ), not an irrational and not an integer. Putting /x h k  in the given equation,   1 2 3 2 1 1 2 3 ............. n n n n n n h k a h a h k a h k a k          . If 1 k  , then any prime divisor p of k would divide n h , therefore h ( fundamental theorem of arithmetic).This clearly contradicts   , 1 h k  , i.e., there are common factors between h and k, contradictory to our assumption. Hence x must be an irrational number or an integer. Deduction : It follows as a corollary that the equation 2 2 0 x   has only an irrational solution. Alternative proof: To see that 2 cannot be expressed as a ratio of integers let us assume the contrary – let 2 mn  in lowest terms where m , n are integers i.e., if there be any common factors between m and n, let it be struck out in the beginning. Then we have 2 2 2 m n  , thus 2 m is even. only a square of even number is even and that of odd number is odd. Thus m is even , say 2 m k  for some integer k. Then 2 2 2 2 2 4 2 2 m k n n k     . Then 2 n becomes even and it follows that n is even. Then there is a common factor 2 between m and n. But we have excluded this possibility in the beginning and it is a contradiction; proving that 2 is not a rational number. Exercise12 If x is not n-th power of some integer, then n x is irrational number. (This straight follows from the above exercise as a corollary. §14:The Napierian number e : The formula for amount A with interest for a principal P with rate of interest r % per annum at the end of t years is given by Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 23 t 100 r 1 P A       , or , nt n 100r 1 P A       if interest is payable at every 1/n th fraction of a year. Obviously, the smaller the interval of payment of interest ( i.e., the larger the n) at the end of one year, when t = 1. If we change the rate of interest to 100% of principal, i.e., set, r = 100, we get an expression, n n1 1 P A       . If we further set P = 1, and presume that the interest is payable every moment, (i.e., n is infinitely large, we denote it by symbol lim   n ) , we arrive at Napierian number e. n lim n n1 1 e         which is approximately 2.71 and a transcendental number of immense importance, in Algebra, Trigonometry, Complex Analysis, Theory of Equations et al . We can arrive at its value to any desired place of decimals with the help of binomial theorem for rational index as below : lim lim 2 3 4 lim 1 ( 1) ( 1)( 2) ( 1)( 2)( 3) 1 1 ....... 2!. 3!. 4!. 1 1 1 1 2 1 1 2 3 1 1 1 1 1 1 1 1 ............ 2! 3! 4! .. n n n n n n n n n n n n n n e n n n n n n n n n n n Or                                                                           1 1 1 . 1 1 ...................... 2! 3! 4! e       by neglecting 1/n and its powers as n is supposed to be very large number, as large as one pleases. We can equate it to any positive number just by multiplying it with itself appropriate number of times . See natural logarithms Reverse processes as we see have extended the frontiers of our knowledge if we accept to remove the restrictions met with, logically but with open mind. There is another point of view in the extension of number system also. It is freedom to solve equations. Accepting solvability of a +x = b in the set of natural numbers for any a and b gave us the integers, accepting solvability of a .x = b in the set of integers gave us the set of rational numbers p/q . Accepting solvability of every quadratic equation with positive discriminant gave us the quadratic surds and accepting solvability of polynomials (finite power series) gave us the algebraic numbers. A real number is a cut separating two mutually exclusive parts of the rational number line. For example, 2 can be thought of as a cut separating all the rational numbers whose square is less than 2 from the set of rational numbers whose square is greater than 2. See the example and exercise below.  The concept of limit is thrust in here without introduction. It is not difficult to understand it even if formal presentation of limit is given in some chapter afterwards.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 24 The conception of ‘cut’ in the number line is due to Dedekind , which treats all numbers rational, irrational, transcendental , all on equal footing; not an extension of integers. Similarly, 0 and ∞ are discovered as extensions of the real number system. But the hyper-real number system treats them on equal footing as the real number system. in this system we do not loosely refer 10 as ∞ or 1 as 0, as division is an impossible operation, but call them infinitesimals and infinites which are reciprocals of each other. ( ‘infinities’ refer to “transfinite’ numbers discussed later on; and there is a concept like ‘ order’ of infinitesimal which we discuss later on) Example2 Take a set of positive rational numbers r, such that 2 2 r  . To show that the set has no largest number; i.e., we can always find an h > 0 ,such that  2 2 r h   however near r to 2 may be. Let us look for an h, 2 2 2 2 r rh h    . Since h is a small positive number , proper fraction , we can have 2 h h  . We can have 2 2 2 2 2 2 2 2 2 1 r r rh h r rh h h r          . Exercise13 There is no minimum of the set of positive rational numbers r such that 2 2 r  . Hint. You can always find a positive rational number k for every rational number r ,such that  2 2 r k   whenever 2 2 r  . Take 2 2 2 r k r   . Example14 Solve the equation 2 2 3 y x x    , if x and y are rational numbers. Since this is one equation in two variables it would have infinite number of solutions 2 2 2 2 2 2 Put , 3 , 3 , 3 1 , 3 2 , assuming p to be a rational number 1 2 2 y x p or x x x p or x x p x p or x x p px x or x p                    We can thus take p to be any rational number other than ½ . you can actually verify that y is a rational number for these values of x. Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 25 Both rationals and irrationals constitute the set of Real Numbers R . R is not countable (proved later on).(In fact, all the rational numbers from - to  can be squeezed into an small open interval of numbers ]2, 2.26[ or any small interval we please, and the rest of the ‘ Real number line’ is filled up with irrationals……;both ‘algebraic and ‘transcendental’( proved later on). Just as we can construct arbitrary sequences of infinite number of digits after a decimal point to arrive at some irrational number, so also we could make infinite series of powers of x with coefficients chosen according to some rule and choosing any value of x so that the series converges, we will arrive at a transcendental number. Thus we can visualize that there are far more irrational numbers than there are rational numbers and far more transcendental numbers than algebraic numbers ! Still we are unable to ‘write completely’ any irrational number or transcendental number only with 10 digits in the decimal system of representation; we have to take recourse to infinite series to write these numbers and to each such number we have to give a separate name like the ancient Romans naming the bigger and bigger natural numbers unendingly. The Hindu concept of place value could be able to write completely a natural number however big it may be, only with the help of 10 digits. It could also represent any rational number in a closed form p/q and could write this completely by a recurring decimal like efgh bcd . a Similar is the problem of finding solution in ‘closed forms’ for large numbers of differential equations we have to be satisfied only with series solutions. Numbers were primarily serving the basic purpose of counting. Rational numbers helped us with measuring also; a tailor could make three shirts for babies out of two meters of cloth. No difficulty in accepting numbers such as 2/3.Irrational numbers like 2, 3 etc. helped us measuring diagonals of squares and rectangles. So it is no wonder if we could not write completely an irrational number with the help of only 10 digits in a finite manner ! It hardly matters whether we can write a number in closed form or not. If it is not in closed form, or it is in the form of a sum of infinite converging series, just giving it a name like 2 or  or e etc. enables us to write is so called complete manner or in ‘closed form’. Exercise15 Show that 2 3  is an irrational number. Hint. If not, 1 3 2 2 3    is a ratio of rational numbers, hence rational. Now     1 2 2 3 3 2 2         , becomes a rational number, a contradiction. §15:The set of real numbers is not countable Consider the set of all numbers in {x : 0 ≤ x < 1} or [0, 1[ where any particular fraction a, is having decimal representation a = 0.a1a2a3a4…………….. or can be written as a non-terminating decimal. For example ½ can be written as 0.50000000000…. or 0.499999999…… (try to convert the latter two decimals into fraction form and their equality shall be evident.) . Let the set Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 26 {x : 0 ≤ x < 1} or [0, 1[ be countable and count its members in the following scheme: 1  x1 = 0.C11 C21 C31 C41……………., 2  x2 = 0.C12 C22 C32 C42…………….,, 3  x3 = 0.C13 C23 C33 C43,……………., 4  x4 = 0.C14 C24 C34 C44…………,and so on. Now we can construct a number xr = 0.q1q2q3q4………………….., such that non of the q’s is 0 nor 9 and for any I, qi  Cii . Now 0 < xr < 1 as none of the q’s is 0 or 9, and as such it belongs to [0, 1[. So it shall correspond to a natural number and can be represented as xr = 0.C1r C2r C3r C4r…………,Thus we have xr = 0.q1q2q3q4…………………..,= 0.C1r C2r C3r C4r…………, But these two decimal representations are identically same as none of the places contain 0 or 9 , thus excluding any ambiguity like 0.50000000000…. = 0.499999999…… We must have qs = Csr for any s ; so that qr = Crr – which is a contradiction, as we have assumed qi  Cii . In other words xr as per our construction , does not belong to {x : 0 ≤ x < 1} or [0, 1[, whereas xr is definitely a proper fraction. So the set {x : 0 ≤ x < 1} or [0, 1[ does not contain all proper fractions. This absurdity arises only because we have assumed it to be countable. Hence in fact the set is not countable. Now , this set is a proper subset of Q and not countable. Hence Q is not countable. By countability of a set , thus, we mean that a one-to-one correspondence between the said set and the set of natural numbers can be established. In other words, the members of the set can be attached with labels of natural numbers. From this point of view, it should not appear strange that there are as many integers or rational numbers as there are natural numbers ! Not in the least. No wonder that the set of integers or the set of rational numbers contain the set of natural numbers as a subset and still each of them are in one-to-one correspondence with it.. Thus it follows that there is no harm if any set is equivalent to a proper subset of itself ! It is very much a reality stranger than fiction. (To have an idea of something like ‘total number’ of real numbers , just to satisfy curiosity of the reader, we may indicate that it is 2N where N is the ‘total number’ of natural numbers. And what 2n is, if n is a particular number, the total number of combinations obtained from n things by choosing 1,2,3…. upto n things at a time. See “Binomial coefficients” at the end of Binomial Theorem below ,in expansion of (1+1)n.) Note Since the rational numbers have a one to one correspondence with natural numbers, the set of rational numbers could well be used for counting. Bu the set of real numbers, the rational as well as the irrational numbers make room for all sorts of measurements. Extras 1) But the set of algebraic real numbers is countable! (try to visualize)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 27 2) Schroeder – Bernstein theorem : If A and B are two sets, both of which are numerically equivalent to ( one to one correspondence ) a subset of the other, they are numerically equivalent to each other. Example8:To prove that the number 0.01001000100001… = k, say, is an irrational number. To prove this, it is sufficient to see that it is not a non-terminating ‘recurring’ decimal. Evidently the number of zeroes between two successive 1’s go on increasing from step to step as such it cannot be a recurring decimal. Hence it is an irrational number . Now one can understand how could we construct or manufacture millions and millions of irrational numbers in this way. In fact the entire gamut of rational numbers is much fewer than the irrational numbers as they can be squeezed into a given small interval, however small we please. ( see one example below for proving this). Similarly the number of transcendental numbers is much more than that of algebraic irrational numbers. In this example, the given number k does not seem to be in “closed form” like the number 0.111111111……The latter can be written as a series in geometric progression , 91 10 1 1 10 1 .... .......... 101 101 101 10 1 4 3 2        which is in a closed form and is a rational number. k is not an algebraic number either. Can this number .... .......... 101 101 101 101 14 9 5 2     be an algebraic number ? Yes. All rational numbers are Algebraic numbers, as they are solutions of the polynomial 0 qx p   whose coefficients are rational numbers. We would return to the question in a chapter for series summations. Exercise16 Prove that any number having 0 at the decimal places of 10, 100, 1000, 10000……etc and no 0’s elsewhere is an irrational number. Hint. It can not be periodic, hence irrational. Exercise17 A set of real numbers is called a set of measure zero if it could be placed inside intervals whose total size may be made arbitrarily small. Show that a finite set of real numbers 1 2 , ......... n a a a is a set of measure 0. Take       1 1 2 2 , , , ,........., , n n a a a a a a             the desired intervals whose total length is n , which can be made arbitrarily small as we make arbitrarily small. Exercise18 Show that the set of natural numbers is a set of measure 0.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 28 Place the integer 1 in the interval 1 ,1 2 2           . Place the integer 2 in 2 ,2 4 4           . Proceeding in this way upto including n in the interval , 2 2 n n n n           , note that the sum of the lengths of the intervals is 2 3 ..................infinite terms = 2 2 2 2          ,evidently this can be made as small as we please , making sufficiently small. similarly it could be shown that the set of integers is a set of measure 0 (As it is in one to one correspondence with the set of natural numbers). (How do you appreciate the idea that the probability of a random number generator to return an integer is 0 ?) The entire set of rational numbers can be squeezed into small intervals of total length arbitrarily small. The rest of the number line is filled with irrational numbers. (This is only to visualize; for a proof refer to some elementary book on number theory or on irrationals). A natural generalization – any countable set of real numbers is of measure 0. The set of rational numbers Q is countable and hence is in one-to-one correspondence with the set of natural numbers N. Take a set M = { x : x = 1/n, for each n N}. Clearly this set is in one-to-one correspondence with N and therefore with Q , as N and Q are in one-to-one correspondence with each other. But M  ]0, 1]. Thus the set Q is mapped into ]0, 1]. This is not all. We can take still smaller intervals , like ]0, 1/k] where k is any pre-assigned large natural number . And we can take a subset P of ]0, 1/k] , such that P = { x: x = 1/(n +k) for each n  N}, which is in one to one correspondence with M and hence with Q. Thus we have mapped Q into P  ]0, 1/k] , an arbitrarily small interval as we pleased . instead of taking our arbitrarily small interval ]0, 1/k] , we can take the choice anywhere say at the number point 2 on the real line; i.e., we can choose our interval to be ]2, 2 + 1/k] instead of ]0, 1/k]. In that case take a set W instead of P such that W = { x: x = 2 + 1/(n +k) for each n  N}. But it is easily seen that the set W is in one-to-one correspondence with P (as per scheme 1/(n +k)  2 + 1/(n +k)) and that completes the proof. Try another proof , taking the arbitrarily small interval ] -,  [ instead of ]0, 1/k]. Hint : Take an interval ] – 1/r, 1/r [  ] -,  [ by taking an r >  . Then follow the lines of the above proof by taking the set of integers I instead of the set of natural numbers N, and complete the proof. Exercise19 An irrational number added to or subtracted from a rational number is irrational.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 29 Hint. If c = a + b, b only irrational, then c must be otherwise c – a = b ; the left side is rational and the right side irrational, a contradiction. Exercise20 Show that product of a non-zero rational and a n irrational number is irrational. §16:The Napierian number e is a irrational number. Suppose e is a rational number     1 1 1 1 1 1 1 ............................. ............ 2! 3! ! 1 ! 2 ! p e upto q q q q              Then,     ! ! ! ! ! ! ! ! ! ........... ............ 2! 3! ! 1 ! 2 ! p q q q q q q eq q q upto q q q q              or or , where     ! ! ............ 0 1 ! 2 ! q q R upto q q        Or,         2 1 1 1 1 1 1 1 1 1 ............ 1 ............ 1 1 1 2 1 2 3 1 1 1 1 1 1 R upto upto q q q q q q q q q q q q                                           Or, 1 R q  . Comparing this with 0 R  and       ! ! 1 1 1 ! ! ! ........... 1 ! ............ 2! 3! 1 ! 2 ! q q q p q q q upto q q                        ! ! 1 ! ! ! ........... 1 2! 3! q q q p q q R                 ! ! 1 ! ! ! ........... 1 2! 3! q q q p q q R              Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 30 We arrive at a contradiction that R is both a positive integer and a fraction less than 1. Thus we cannot assume p e q  or a rational number. It is interesting to know that a hyperbolic function is irrational for nonzero rational values of the argument, a standard result in number theory. The number ab is a transcendental number if a is an algebraic number not equal to 0 or 1, and if b is an irrational algebraic number. This was proved by Gelfond , a Mathematician from Russia in 1934, now known as Gilford’s theorem. (Gilford has in fact developed technique to study algebraic numbers) .Then Hermite proved that e is transcendental Similarly Lambert proved that if x is rational, tan x must be irrational Thus he proved from tan 1 4  that Л is irrational. This led Lederman to prove Л is transcendental with the help of Euler’s famous equation connecting 5 celebrated constants i π e+1=0, i π e=-1e, i, Л,1 and 0. From the last equation, , since -1 is algebraic, i Л has to be transcendental and since i is believed to be algebraic, Л must be transcendental! This does not mean ex is a transcendental number for all values of x. In fact ex may be put equal to 2 for some value of x ; but 2 is not a transcendental number. Transcendental numbers are roots of transcendental functions; not necessarily their values. Consider the converse .e.g., ex where e is an irrational number and x is any real number. We can make ex = N where N is any positive real number, may be rational even may be an integer. Actually ln N = x is defined in this manner. §17:Examples of transcendental numbers Apart from e and Л, we know very few transcendental numbers even though the gamut of transcendental is far more than algebraic numbers which is obviously far more than the rational numbers; the latter can be squeezed to any small interval in the real number line we please. One such number is Euler’s constant lim 1 1 1 1 ......................( ln ) 2 3 4 n n              = Another example is Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 31 One more example is Chapernowne's number, 0.12345678910111213141516171819202122232425... Morse-Thue's number, 0.01101001 ... Other transcendental numbers are 2 ln2, 2 etc. Hilbert’s number, ii= 0.207879576...etc., is surprisingly a real number. A sketch proof is provided after differentiation chapter. By countability of a set , thus, we mean that a one-to-one correspondence between the said set and the set of natural numbers can be established. In other words, the members of the set can be attached with labels of natural numbers. From this point of view, it should not appear strange that there are as many integers or rational numbers as there are natural numbers ! Not in the least. No wonder that the set of integers or the set of rational numbers contain the set of natural numbers as a subset and still each of them are in one-to-one correspondence with it.. Thus it follows that there is no harm if any set is equivalent to a proper subset of itself ! It is very much a reality stranger than fiction. §18:Transfinite numbers : cardinality of sets : Cardinality of a set is the number of elements or members it has . If the set is A = {a, b, c}, its cardinality is evidently 3. cardinality is a concept different from ordinality which has the meaning of sequential or serial number in which the member appears; e.g., in the above set ordinality of b is 2 or in other words, it is the ‘second’ element of the set. Of course sets are not distinguished in respect of ordinality of their members , i.e., the order in which the members of a set appear is not important. For example, if B = { c, a, b} , for all purpose we have A = B. If C be another set C = {1, 2, 3}, we can say C is equivalent to A (or B ), as there can be a one-to one correspondence between members of C and A( or B) like a2, b1, c3. if two sets are equivalent, they are of the same cardinality. Now if A = {a, b, c} be a finite set with cardinality 3, and the set (A), the power set of A, or the set of all subsets of A, (A)= {, {a}, {b}, {c}, {a, b},{a, c}, {b, c} , {a, b, c}} has definitely more members than A and its cardinality is definitely more than that of A. So cardinality of a power set of a finite set is more than cardinality of the set, or the power set is ‘bigger’ in ‘size’ than that of the finite set. Let us examine whether this statement can be extended to infinite sets as well. We see that the set of natural numbers is a proper subset of integers which is further a proper subset of rational numbers , but all these three sets are countable or equivalent to the set of natural numbers. As such the three sets are equivalent to each other, or are of the same size notwithstanding the fact that each of the first two is a proper subset of the next. The set of natural numbers has the same number of elements as the set of even positive integers, has the same number of elements as the set of even positive integers , have the same number of elements as the set of integers and even the set of rational numbers. But some of these sets are proper subsets of some others. So the subset relationship does not determine the size of the set. If a Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 32 finite set A has ‘a’ number of elements or members, we have its power ser (A) or the set of all its subsets which is having 2a number of elements. ( The number 2a may be understood to be (1 +!)a= aC0 + aC1 + aC2 + ….. aCn , all different combinations of a different things taking one at a time , taking two at a time, three at a time etc. which account for all the subsets of A by using binomial theorem). George Cantor of Germany who is one of the founders of set theory extended the concept to infinite sets to compare two infinite sets and examine ‘which one is bigger in size’. At this point it must be very clearly remembered that a finite set cannot be equivalent to or have one-to-one correspondence with one of its proper subsets. In fact, it is an alternative definition of an infinite set as one which is equivalent to one of its proper subsets. A real number is well represented by sum of a finite or convergent series of rational numbers. For example,  = 3.14159…….= (3+10 1 +100 4 +1000 1 + 10000 5 +100000 9 ….). or sum of (3, 10 1 ,100 4 , 1000 1 , 10000 5 ,100000 9 …….) . As such, every real number can be understood to be a combination of rational numbers. This in turn implies that the set of real numbers is a proper subset of power set of rational numbers or equivalently the set of real numbers is a proper subset of power set of natural numbers. So its ‘count’ or cardinality should be less than 20 where 0 stands for cardinality of natural numbers called aleph null after the Hebrew letter  pronounced as aleph; but at the same time its cardinality shall be more than 0 , the cardinality of rational numbers. Though cardinality of rational numbers or that of natural numbers is infinite, it is given this symbol by George Cantor ;which may be accepted with open mind as a special type of  and with patience to see what more possibilities it opens up. Cantor proved that there is no intermediate cardinality between 0 and 20 . in other words there is no other infinite set of different cardinality between the set of rational numbers and the set of real number. This makes the cardinality of real numbers 20 , although it is a proper subset of power set of rational numbers. The set of real numbers is a proper subset of power set of rational numbers as all combinations of rational numbers do not necessarily lead to real numbers; some of them are divergent series and do not lead to real numbers evidently. But still surprisingly and logically, the set of real numbers has the same cardinality as that of the power set of the set of rational numbers or natural numbers for that matter. The smallest transfinite number is thus 0. Example3 :Show that the set {n2} is equivalent to {n} = N. (Hint. Set up a one-to-one correspondence between the sets) Example4 : Show that the set of prime numbers is infinite and no general integer formula can represent the totality of prime numbers . Using the idea, figure out some idea about cardinality of the set of prime numbers.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 33 (Hint – assume the contrary, if the number of primes is finite, then their product + 1 is evidently a prime different from all of them. This is a contradiction) Example5: Arithmetic for cardinal numbers -assuming that the transfinite numbers can be added, and multiplied in conformity with the sense they stand for, show that 0 + 0 = 0 ( meaning thereby that the set of integers and rational numbers is equivalent to the set of natural numbers and such other examples) and 0 x 0 = 0( meaning thereby that Cartesian product of N x N is equivalent to N; the set of rational numbers is equivalent to a set with cardinal number 0 x 0 which is in turn equivalent to the set of rational numbers or equivalent to the set of natural numbers. . also show that c + 0 = c for any transfinite cardinal c ( meaning thereby that every infinite set has a subset equivalent to the natural numbers) . Transfinite numbers  = 20 denote the cardinality of the set of real numbers. The set of all real valued functions {f(x)} defined on the set of real numbers R = {x : x is a real number} can be visualized as a proper subset of (R) and must have a larger cardinality , the next bigger transfinite number 2 . Ordinarily ∞ or -∞ are not regarded and proper numbers. So we cannot write 1/∞ = 0 o r 1/0 = ∞; if we do so, to express the idea behind them, we call that number system extended number system. §19:Recapitulation of Complex numbers : A product obtained by solving quadratic equations Still we have been left with the restriction that square root of a negative number is not in R, for square of a positive number is positive and square of a negative number is also positive. When we preferred to throw away the restriction on square root of negative numbers, our number system was extended ,from a line to a whole plane instead . Then it was possible to solve the quadratic x2 + 1 = 0; in fact every equation of ‘n’ degree with rational coefficients could then be shown to have n roots. The √ -1 or ‘i’ is called an imaginary number, though it is very much existent and useful and the combination of real and imaginary numbers is called the field of complex numbers denoted by C. (When we simply chose to solve the equation x2 = -1). ‘i’ = √ -1 has got a meaning of turning a number by 900 anticlockwise when multiplied with it(for, multiplying twice by it, turns any number by 1800). A complex number is denoted in the form a + ib where a and b are real numbers and complex numbers obey usual properties of addition, subtraction, multiplication and division just as real numbers do. Just as every surd of the form a+b has a conjugate, a -b, (so that their product is a rational number a2 -b , where a and b are supposed to be rationals) every complex number admits of a conjugate, a -ib , so that its product with the conjugate produces a real number; (a + ib)(a – ib) = a2 +b2 , and ( a2 +b2 ),is the absolute value /modulus / If we regard squaring as not only a special case of multiplication but a separate process also, and regard ‘taking square root’ as its reverse process, we face the restriction that square roots of negative numbers cannot be extracted. But if we closely observe that we did not assume anything in the beginning to support this fact. So we can safely remove this restriction or simply chose to discard this restriction ( by introducing a symbol i, square root of a negative number , -1 .Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 34 norm /distance of the number point from the origin of coordinates. Also the sum a + ib +a – ib = 2a, again a real number. Most of the results of Calculus developed for R could be carried over to C. More about complex numbers, absolute values, triangle inequality etc are dealt in the section of vectors so that the comparison of complex numbers and vectors is evident. §20: Partially order relation, Total order, chains and lattices Another difference between real numbers and complex numbers is total order relation. (explained later on) i.e., given two real numbers a and b, either a > b, or a < b or a = b. The set of complex numbers has no such total order relation. For, given two different complex numbers a + ib and c + id, we cannot say which one is larger of smaller than the other, as they are line segments in different directions when the points (a, b) and (c, d) are joined to the origin (0, 0). Moreover their difference , which is the line segments joining the points (a, b) and (c, d)which cannot be said to be positive or negative. But we can have partial order relations in the set of complex numbers. For example, the number a + ib and 2a + 2ib are line segments in the same direction and definitely 2a + 2ib > a + ib , being comparable. In fact, all the numbers along this line i.e. real multiples of a + ib are in the same direction, they are comparable to each other and numbers in this line have an order relation which is called partial order relation and the numbers in this line are said to constitute a chain. Because of this difference between real and complex numbers, the equation x2 + 1=0 which has no solution in the field of real numbers, has a solution in the field of complex numbers. And because of this only, any polynomial or finite power series in x with real coefficients has at least a solution or root, real or complex. This statement is called fundamental theorem of Algebra and was first proved by C F Gauss, the towering genius in Mathematics. We would revert to the topic under theory of equations. Exercise21 Prove that a (Partial) order relation is reflexive, anti-symmetric and transitive taking the example of real and complex number sets. Let  symbolize a partially ordered relation; We may verify that i) x x  for any real number (reflexive), ii) x y and y x x y     ; this property is called antisymmetry. iii) x y and y z x z     (transitive) Some other examples of partially ordered sets : a) In the set of all positive integers, the relation if the relation m n  means m divides n, verify that it is a partial order relation. Note that there are so many pairs of numbers not comparable. b) If P(U) be the power set of some universal set U ( the set of all subsets of it) and if A B  means A is a subset of B, both being subsets of U, then it is a partial order relation.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 35 Some partial order relations possess a fourth property in addition to reflexivity, antisymmetry and transitivity; that is, any two members are comparable, or related. such as if a and b are two real numbers, and the symbol  has its usual meaning ( is less than or equal to) we can see that any two real numbers are comparable, e.g., any two set of real numbers a and b are comparable ,i.e., a and b are related as either a b or b a   . Such partially ordered relation is called a total order relation. The set is called totally ordered set or linearly ordered set or more often, a chain. Observe that in any partially ordered sets, there may be totally ordered subsets, or chains, the only point to note is, it is not one chain entirely. In a partially ordered set P, an element x is said to be maximal if x y x y    i.e., if it is  no other element in the set. For example in the class of subsets of U, U itself is the maximal element in each chain. In the set of integers, consider the partial order relation m n  means m divides n; in the subset {3, 5} . Any upper bound of {3, 5} may be 15, 30, 45, etc. divisible by both the numbers and evidently 15 is the least upper bound. In the set of rational numbers each of whose square is less than 2, has no least upper bound (verify). But in the set of real numbers each of whose square is less than 2, has a least upper bound, i.e. 2 . In the set of real numbers, if the least upper bound of a subset exists and is not infinite it is called supremum or maximum of the subset. Similar concepts are infimum or minimum. If A is any nonempty class of subsets of U, a lower bound of A is a subset which is contained in any set in A and the greatest lower bound is the intersection of all of its sets. Similarly the greatest lower bound of A is union of all of its sets. A powerful mathematical tool is what we call Zorn’s lemma or the axiom of choice. In a partially ordered set, if every chain has an upper bound, the set contains a maximal element. Why it is called the axiom of choice is, that it can be worded in a different way : given any nonempty class of sets all of which are nonempty, a set can be built up by taking one element precisely from every set. A little thought would reveal that Zorn’s lemma and the axiom of choice are one and the same concept. This logic is so fundamental that it could be taken as an axiom. A lattice is a partially ordered set in which each pair x and y have a greatest lower bound (denoted by x y  ) and least upper bound ( denoted by x y  ). If every nonempty subset of it has a maximal and minimal element, it is a complete lattice. §21:Ordered pair representation of numbers: extension of natural numbers to integers, rational numbers, surds, complex numbers, vectors and quaternions as numbers; An integer k can be represented by an ordered pair of natural numbers k = ( a, b ) the ordered pair denoting a – b, where a, b are natural numbers. The Arithmetic for these ordered pairs can be formulated , by stating how two integers are equal, how one is greater than or less than the other, Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 36 and how two integers are added or subtracted, and how they are multiplied. Since the integers are not closed with respect to division we need not consider division of these ordered pairs except when the dividend is an integral multiple of the divisor or when the divisor is a factor of the dividend. We can say two integers k = (a, b) and l = (c, d) are equal only when a + d = b + c . It follows that k has many such representations k = (a, b) = (p, q) for all values of p, and q satisfying a + q = b + p. It follows that the integer 0 can be represented by 0 = (r, r) for any natural number r. A negative integer can be represented by (u, v) where u < v . In this way we need not write a negative integer as a natural number with a minus sign and we need not write the symbol 0 also. Addition of two numbers (a, b) and (c, d) may be taken as ( a + c, b + d) and their subtraction may be written as (a -d, b -c). Their multiplication can be taken as (ac + bd, ad + bc). Division is met with restriction that division of any two natural numbers /integers does not necessarily lead to a natural number /an integer again. In a similar development, a rational number k can be represented by an ordered pair of integers k = (a, b) where (a, b) denotes a /b , when a, b are integers and b  0. We could say two rational numbers (a, b) and (c, d) are equal iff ac = bd. Similarly (a, b) > (c, d) iff ac > bd and (a, b) < (c, d) iff ac < bd . We can say (a, b) = 0 if a = 0. We can add, subtract, multiply and divide two numbers (a, b) and (c, d) like (ad + bc, bd), (ad -bc, bd), (ac, bd), ( if b  0  d ) and (ad, bc) ,(if b  0  c) respectively. The identity of addition may be taken as (0, a) if a  0, and the additive inverse of (a, b) may be taken as ( -a, b). the multiplicative identity may be taken as (a, a) for any non-zero integer a, and multiplicative inverse of any non-zero number (c, d) , c  0  d ,may be taken as (d, c).Thus the Arithmetic of rational numbers can be carried out without any notation of fractions. Quite analogously a complex number can be represented by an ordered pair of real numbers (a, b) . The ordered pairs (0, 0) and (1, 0) stand for additive and multiplicative identities respectively. Addition and subtraction of two complex numbers (a, b) and (c, d) may be given by (a  c, b  d) respectively. Two complex numbers (a, b) and (c, d) are equal only when a = c and b = d and vice versa. Multiplication of any two complex numbers (a, b) and (c, d) may be given by (ac – bd, bc + ad). Division of any two complex numbers (a, b) and (c, d) may be given by ( 2 2 d c bd ac   , 2 2 d c ad bc   ). Thus we may dispense with 1  or i for representation or Arithmetic of complex numbers. Similar notations for quadratic surds a + b may be developed and the sign  may be dispensed with. But the reader must have been aware by this time that the author has reached many dead ends in each of the treatments above and is only trying to take the reader for a ride. Admittedly yes. Dead ends are met with, in each of such attempts and we do not save much labour or thought apart from saving a symbol or notation; particularly in view of the fact that a different Arithmetic has to Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 37 be developed each time. So such ordered pair notations go well with theoretical study and prove less and less useful day by day. We never try to undermine one’s aspiration to be a ‘pure’ Mathematician; for some Mathematics discovered or developed today may be of use even after 300 years! The treatment given here is only with a purpose of illustrating how the notation of ordered pairs may be of some use and to illustrate that we must be better at ease with the symbols , (-1) or i, a/b and a – b etc. So again we assert the power of symbols and discard or make limited use of some symbols which do not promise great power. Such ordered pairs represent divisions in some sense. If a and b are natural numbers and (a, b) represents an integer a – b , it is a division of a in two unequal parts a – b and b. A positive integer can be represented as (a, 0) and a negative integer can be represented as (0, b). If the ordered pair (a, b) represents a rational number where a, b are integers, it is really a division. If the ordered pair (a, b) stands for the quadratic surd a + b, where a and b are rationals, it then shows separately the rational and the irrational parts. A rational number can be represented like (a, 0) and a pure quadratic surd can be represented as (0, b), so that we can write a + b = (a, 0) + (0, b). Similarly if (a, b) represents a complex number a + ib, the real numbers can be represented by (a, 0) and the purely imaginary numbers can be represented by (0, b) ;so that they are divisions or parts or components of a complex space. They are also called vectors in two dimensions and represent points in a plane of real and imaginary axes called Argand plane .An ordered pair like (a, (b, c)) of a real number and a vector or simply an ordered triplet (a, b, c) of real numbers is called a vector in three dimensions where a, b and c are its components in three arbitrarily chosen mutually perpendicular directions of x, y and z. In a particular set up of axes like this, two vectors (a, b, c) and (p, q, r) are said to be equal only when a = p, b = q and c = r. This is as long as we do not change the co-ordinate axes. If we choose a different set of axes again mutually perpendicular, the vector (a, b, c) shall be having another representation (u, v, w) = (a, b, c) where u  a, v  b and w  c; but u2+v2+w2 must be equal to a2+b2+c2 , called square of modulus or norm or length of the vector. There is another invariant , the direction of the vector which we would discuss in proper chapter for vectors. Thus that the vectors are directed line segments of free of bound types, well represented by ordered triplets of real numbers. A vector in three dimension (a, b, c) can well be represented by unit vectors ) 0 , 0 , 1 ( i ˆ , ) 0 , 1 , 0 ( j ˆ  , ) 1 , 0 , 0 ( k ˆ as k ˆc j ˆ b i ˆa   , where ) 0 , 0 , a ( i ˆa  , ) 0 , b , 0 ( j ˆ b  , ) c , 0 , 0 ( k ˆc  . Quaternions developed by Hamilton as division of two vectors ((a, b), (c, d)) or an ordered 4-tuples like (a, b, c, d) or written as a + bi + cj +dk in analogy with complex numbers and its Algebra in a manner similar to vectors in three or two dimensions. Hamilton was invited for the post of Professor at the very same university where he studied, even while he was in his graduation. We have already observed that this single entity notation like a + bi + cj +dk or like a + ib is more convenient for Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 38 carrying out calculations rather that ordered(a, b, c, d) or (a, b) notation. An ordered n-tuples of real numbers may be studied for similar rules of calculations . Quaternions find place in theory of relativity in 4-dimensions and n-tuples find use in mechanics of n-particles with 3n degrees of freedom with or without constraints. Particularly a statistics for a gas ensemble was developed by Maxwell when he was 16 years old only paving a way for kinetic theory of gasses. Two gas particles or point masses which are free to move independently except when they are colliding with each other may be thought of as having 6 degrees of freedom or may be thought of as a single particle with motion in 6 dimensions and the concept is extended to dynamics of particles and kinetic theory of gasses. The dynamics of rigid body reduces to the dynamics of particles in this framework with constraints, or fixed distances among them. §22:Absolute value of a number and triangle inequality Absolute value of a real number is distance of the number point from the origin. Thus absolute value of 5 is 5 and absolute value of – 5 is also 5. The absolute value of ‘a’ is denoted by |a|. Absolute value of a complex number a + ib is denoted by | a + ib | and as it is distance from the origin is OP = 2 2 a b  . If a complex number a + ib is represented by a point P in the complex plane, and another number c + id is represented by the point Q, then verify that the point R represented by their sum a +c + i( b + d) is the diagonal OR of the parallelogram whose adjacent sides are OP and OQ. This is similar to the law of addition of vectors to be explained afterwards. Also OQ = PR = 2 2 c d  . Then OR =     2 2 a c b d    . In the triangle POR the sum of the sides OP PR OR OP +OQ OR     . As sum of two sides of a triangle is greater than the third. Other related things shall be discussed in appropriate places. Now at least it is clear that the absolute values of real as well as complex numbers obey triangle inequality, sum of two sides of a triangle is greater than the third. Parallelogram law of addition of complex numbers. O P Q R X a b c d b + d a + c YSeries : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 39 Example9 : To prove that |x| = x + 100 is possible if x is a real number ! But |x| = x – 100 is impossible. By |x| we mean absolute value of a real number x , i.e., |x| = x if x ≥ 0 and |x| = -x if x ≤ 0 . If x ≥ 0, then |x| = x and so |x| cannot be equal to x + 100 ; as such |x| = x + 100 has no solution. If x < 0, then |x| = -x so that |x| = x + 100  -x = x + 100  2x = -100 i.e., x = -50. Similarly if x ≥ 0, then |x| = x and so |x| cannot be equal to x – 100 ;thus |x| = x – 100 has no solution for this case. But if x < 0, then |x| = -x so that |x| = x -100  -x = x – 100  2x = 100 or x = 50. But this is not tenable as we have assumed x < 0 in the beginning of this case. Example10 Simplify 3 4 2 x   We have 3 4 2 1 3 4 2 4 1 3 2 4 1 2 x x x x                Exercise22 Show that the other diagonal PQ of the parallelogram in the above figure gives the absolute value of difference of two complex numbers a + ib and c + id. Exercise23 With the help of the fact that difference of two sides of a triangle is less than the third, show that OP OQ PQ   Try to prove and remember Exercise24 a) Now, for both real and complex numbers p, q, we have p q p q    and p q p q    ; Or, p q p q p q      . b) If ‘a’ and ‘b’ are real numbers, ab a b  c) If ‘a’ and ‘b’ are real numbers, and 0 b  , then a ab b  Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 40 d) If “a’ is a real number and ‘b’ is a positive real numbers and a b  , then b a b    , i.e. a lies between – b and b. Exercise25 a b a b    if a and b have same sign. Exercise26 With the help of the fact that difference of two sides of a triangle is less than the third, show that OP OQ PQ   Find the interval of definition of x if   2 2 6 5 6 5 x x x x       . ans. 2 3 x   Hint. a a   if a<0 exercise27 Find the domain of x if 1 1 0 1 1 x x x x       . Ans. 1, , 1 x or x    Hint. a a  only if a>0. Exercise28 Solve 2 3 10 0 x x    Ans. 5 3 x    Hint.    5 3 0 x x    implies the factors are of opposite signs. It suffices if we say one of the factors is negative and the other is positive. Example10  2 3 4 2 3 2 1 5 x x x           Example11 sin sin 1 x x   This would be valid only if sin 0 x  . In that case, sin sin x x   . Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 41 Thus 0 1 sin sin 1 2sin 1 sin 30 2 x x x x x          Exercise30 Solve 2 3 2 0 x x    Hint. Solve in two parts separately , one for x > 0, and another for x < 0. There is no real solution for the latter case. Otherwise since 2 2 x x  , write the equation as 2 3 2 0 x x    and solve for x §23:Continued fractions; a product of solving quadratic equations Ratio of two positive integers is always a proper fraction or a ‘terminating ‘ CF. Are there other CFs. Then it must be something other than ratio of two positive integers ; it is as simple as that. Take for example the solution of the equation 1 2 2 5 1 0 , 5 1, , 5 x x or x x or x x        We have been able to avoid squares , the point of difficulty, in a way. So it should lead to a solution. Using the last equation as a formula and applying it on itself repeatedly, ( as if we have got value of the x at the left side of the last eqn.) we have, 1 1 1 1 5 5 5 5 1 1 1 5 5 1 1 1 5 1 1 1 1 ..................... x x x x                Or x = [5;5555………………..] in our standard notation for CF ( if all parts like 1 5 r  have 1 in the numerator, , a semicolon is placed after the integer part and the successive denominators are written next to one another in order of their appearance).Though we cannot make arithmetic operations direct on the CFs as such, quadratic surds can be represented as recurring CFs ( analogous to recurring decimals), to be content with being able to write a quadratic surd in a ‘closed form’. This seems to be the solution or value of x, as the x in the right side is supposed to be diminishing in its place value after each step and may safely be supposed to have no effect after a very large number of steps. Is it a solution? I wonder . (Do you wonder too ?) Then let us do one thing, find out the two real solutions of the given eqn. , which are 5 25 4 2   .Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 42 We should verify surprisingly that at least one of the roots, i.e., 5 25 4 2   is represented by the CF, [5;5555………………..] .More surprise is in stock when we discover that the other root is also represented by the same CF, only with its integer part different ! This has got to be. Because we have found a solution of the quadratic by iterating 1 5 x x   , this should have included all (both in this case) solutions. The quadratic equation is nothing but only a representation of two numbers ( the roots ) in the form of a formula, the formula hiding the roots (remember that x is only a symbol given to the number or the root) . The formula ( the Quadratic eqn. ) only contains x and hides the two numbers inside it. Now, 1 5 x x   is only another way of writing the ‘formula’ for the two numbers (roots) and the x in it’s left side must stand for both the numbers ! But this little creature [0;5555………………..] shall represent the fraction part of the other root, only if we agree to represent negative CFs by positive fractional part and their integer part in negative, by one digit higher. Calculate the other root 5 25 4 2   and verify whether its CF fractional part is [0;5555………………..], in this way. Now it is fully evident that the recurring continued fractions are quadratic surds, which are evidently solutions of quadratic equations. This means at least two things, Quadratic surds can be represented by recurring CFs and if we could be as familiar with CFs as with commensurate fractions then the above is an easy way of solving quadratic equations. By and by we shall see more and more of the proposition we just made. It is now clear why in ancient times, people called simple fractions as ‘ commensurate ‘ fractions. For, ratio of two positive integers a and b must be terminating CF when it is converted to CF as there must be a GCD of a and b, no matter if it is 1 also. Then would it right to be said ‘ non-terminating CFs are ratio of two positive integers who have no GCF, not even 1 ? I am afraid it is better not to make such statement at all; it sounds logical, but looks weird ! Could we extend the shadow logic (pseudo logic) as farther as saying that, complex numbers with imaginary part are ratio of two positive integers whose GCD is negative ? Don’ tax your brain; it is too much weird . Not for a little bit of fun at great cost ! This is just an introduction to continued fractions. Perhaps the ancient Indians developed high precision Astronomy with use of continued fractions. Take the example of sundial, to observe time with the help of a vertical stick in the centre of a calibrated circle. Time was calculated by measuring ratio of the stick to its shadow, with counting finger widths, making correction and further corrections in a manner as if finding CFs of the ratio. Ancient Indian treatises of Astronomy, such dividing integers,, taking their integer value of the quotient, then making corrections by dividing the quotient by the remainder ,adding the integer value in parts of one-sixtieth units, and carrying forward so on as further corrections. Perhaps for this method, they did not need irrational numbers formally. Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 43 §24:Continued fractions; a further vain attempt to write numbers in closed form: Any positive rational number x = p/q where p > q and there are no common divisors of p and q except 1, can be written as a sum, x = [x] + (x), where [x] denotes the integer part of x (the largest integer not exceeding x), and (x) denotes the difference x -[x], called the fractional part of x. If x is an integer the fractional part (x) is zero. If x is not an integer the fractional part (x) is positive and less than 1, so it can be written as 1/y for some real y > 1. The number y in turn can be written as the sum [y] + (y), giving       y y 1 x x    If (y) = 0, then x = [x] + 1/[y], a rational number, and the continued fraction terminates. This process is just like finding out GCD of p and q which is assumed to be 1 as there are no other common factors between them as assumed. Thus, for any rational number the continued fraction terminates at the end of some finite number of steps like this. Termination of the process speaks nothing more than the fact that the greatest common divisor of p and q is 1 if one remembers how did we arrive at GCD in pre-high school. But in some case when (y) > 0, then (y) = 1/z for some z > 1, and             z z 1 y 1 x z1 y 1 x x        If this process does not stop after a finite number of steps, and the sequence of such successive fractions converges to a limit, the number x is evidently not a rational number as it cannot be expressed as p/q where p and q are integers. For, if they are any two integers, either they have a natural number GCD or their GCD is 1 and the above process which is same as the process of finding GCD must end. Hence, if this process does not end, the number represented by the continued fraction is an irrational number. ( this line of argument is not certainly a proof; for it still remains to prove that the continued fraction is convergent and it does represent a number at least. It is no difficulty to understand that if the infinite convergent fraction is at all convergent, then it must represent a real number which is not rational ). More important is the reverse statement; that an irrational number can be represented by an unending continued fraction .( a preliminary proof of the statement follows in case when the number is a quadratic surd). If the irrational number is ‘a’ and if we denote the successive integral parts by a1, a2, a3, a4 etc, we write ..... ab ab ab a a 4 4 3 3 2 2 1      in short, for ....... ab a b a b a a 4 4 3 3 2 2 1      and 1 a , 22 1 ab a  , 33 2 2 1 33 2 2 1 ab a b a ab ab a      etc. are called 1st, 2nd, 3rd … n-th convergents of the number a. They are easily seen to be successive approximations to the number a. We consider the cases when each an is positive integer and each bn = 1 for sake of simplicity. For, it would be seen that it makes no difference to the Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 44 number represented by the continued fraction if all the numerators and denominators are multiplied or divided by a constant integer ( even by -1 as a special case),except the first convergent a1). We consider 11 1 1 qp 1 a a   , 2 2 1 2 1 22 a 1 a a a1 a qp     , 1 a a a ) 1 a a ( a a1 a 1 a a1 a 1 a qp 2 3 1 2 1 3 3 2 1 3 2 1 33           etc. and observe that the numerator of the third convergent can be formed by multiplying the numerator of the second convergent by the third quotient and adding the numerator of the first convergent. Similarly the denominator of the third convergent may be formed by multiplying the third quotient to the denominator of the second convergent t and adding the denominator of the first convergent (1 in this case) to it. In this sort of representation, a number shall have a unique expansion as a continued fraction. For, if possible, let a particular number be expressed as a contd. fraction in two different ways as ..... a 1 a 1 a 1 a a 4 3 2 1      = ..... b 1 b1 b 1 b 4 3 2 1     Now a1 or b1 is the positive integer part of ‘a’ or it is the largest positive integer that can be derived from ‘a’. Thus it cannot have two values. And we have a1 = b1 . Subtracting them from both sides and dividing 1 by the equal results we get ..... a 1 a 1 a 1 a a a 1 5 4 3 2 1       = 1 5 4 3 2 b b 1 ..... b1 b 1 b1 b       Repetition of the same argument forces a2 = b2. continuing in this manner, we prove the unique representation. But if a number is expressed as a continued fraction as ..... ab ab ab a a 4 4 3 3 2 2 1      , instead of all numerators as 1, there is no unique representation in this manner as we can multiply or divide all the numerators and denominators by a particular integer excepting a1. If we denote the successive convergents by nn qp , we can generalize this rule to a formula like pn = an pn-1 + pn-2 and qn = an qn-1 + qn-2 ………(a) And the formula evidently holds for n = 3. Now the (n+1)st convergent is obtained from the n-th convergent by replacing an with 1 n n a1 a   , so that the (n+1)st convergent 1 n 1 n qp  may be written as Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 45 1 n n 1 n 1 n n 1 n 1 n 2 n 1 n n 1 n 1 n 2 n 1 n n 1 n 2 n 1 n 1 n n 2 n 1 n 1 n n q q a p p a q ) q q a ( a p ) p p a ( a q q a1 a p p a1 a                                          which is in the same form as (a), when n is replaced with n + 1. Thus by induction , the formula (a) hold for all values of n. Now pnqn-1-pn-1qn = (anpn-1+ pn-2)qn-1 – pn-1(anqn-1+ qn-2) = (–1)(pn-1qn-2 – pn-2qn-1) = (–1) (–1) (pn-2qn-3 – pn-3qn-2) following the same steps again. = …………………………further continuing the same process = (–1)n-2 (p2q1 – p1q2) = (–1)n-2 [(a1a2 + 1) – a1a2] = (–1)n-2(1) = (–1)n-2(–1)2 = (–1)n As such pnqn-1-pn-1qn = (–1)n ……………………………………..(b) The result also holds when the contd. fraction is < 1, taking a1 = 0 in this case. The result further gives us a method to verify correctness of each successive step of calculation in converting something into a continued fraction. Two important conclusions result from (b) above. One – this gives a recursive formula to calculate successive convergents or verify correctness of such calculations, and two – it states that the successive numerators and denominators are ever increasing positive integers. It immediately follows that if pn and qn had any common divisors say d , then d must divide (– 1)n, which forces d = 1. Thus pn and qn are already in their lowest terms….(c) 1 n n 1 n n 1 n n n 1 n nn 1 n 1 n q q 1 q q q p q p qp qp          ……………………………………………(d)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 46 which shows that the difference between two successive convergents = 1 n nq q 1  Let ....... a1 a 1 a 1 a x      . We can write x1 a x   with no loss of generality and on simplification, we immediately see that the number x is a root of a quadratic equation x2 – ax – 1 = 0. Similarly a number represented by a continued fraction ..... a 1 a 1 a 1 a x 2 1 2 1      or ....... a 1 a 1 a 1 a x 2 1 2 1      may be written as x1 a 1 a x 2 1    ; i.e. x can be thought of as a root of a quadratic equation, a2x2 – a1a2x – a1 = 0. In the same manner, suppose ........ a1 a 1 a 1 a 1 a 1 a x 3 2 1 3 2 1       , which can be written as x1 a 1 a 1 a x 3 2 1      x a x a a 1 x a a x 2 3 2 3 1      1 x a x a a a x a a a ) x a x a a ( x 3 1 2 1 3 2 1 2 3 2         the number x is again a root of some quadratic equation. Thus it is shown that if the continued fraction representation of a number is only a block of elements repeated and there is no non-repetitive part in the beginning from the point of repetition, the further repetitions may be replaced by the number itself and thus the number is seen to be a root of a quadratic equation, It is necessarily a quadratic surd, since it is not a rational number. If the number has a non-repetitive part , the convergent containing the non-repetitive part may be separated and what remains is a repetitive or cyclic continued fraction. The former part is a rational number and the latter part is a quadratic surd. Thus an infinite continued fraction in which a block of elements are repeated with possibly exception of a few elements in the beginning, represents a quadratic surd. (a simple proof follows a little later)Thus quadratic surds at least can be represented ‘in a closed form’ of continued fractions ( a proof follows later on); although all algebraic numbers or for transcendental numbers cannot be. Actually to expect to express every number in closed form is not at all realistic and it must be more natural to expect to the contrary . Any attempt to express all numbers in closed forms must fail, as numbers were basically invented to count rather than to measure. It is only later on we put the Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 47 numbers to use in measurement. Naturally all measurements cannot be carried out only with the help of counting. Still it is a very big achievement that at least the natural numbers and even rational numbers could be expressed in closed forms , thanks to the concept of place value of the Hindus. We already know that algebraic numbers are in closed forms as they are obtained by finite number of algebraic operations on a rational number. Some transcendental numbers whom we have given some name like sin 2, e2 etc. are also in closed forms. Once we are habituated in converging series expressions for numbers or functions there would be no more obsession with ‘closed forms’ and we would be completely at ease with series representations in carrying out algebraic operations on them. Whether in closed form or not, a real number can always be expressed as an unending decimal number. if it is an integer, the decimal places are all 0’s; if it is a rational number, the decimal places after a finite number of places are all 0’s or a block of digits repeating indefinitely, if it is a surd or algebraic irrational number, the sequence of truncated decimals converge to a limit in closed form and if the number is transcendental, the sequence converges to a limit, which cannot be expressed in general in a closed form unless it is separately named ( like  or e). §25:To convert a surd to a continued fraction ; It would be interesting to note how a surd 3 4 19  can be converted into a continued fraction. Care should be taken to see that a quadratic surd is of type 1 1 r b N  where its value is greater than one and N – b12 is divisible by r1; if not, first of all the surd may be converted to one of this type. The integer part must be that of 3 4 19  or 2, as integral part of 3 4 16  is 2. Its fractional part may be written as 3 4 19  -2 or 3 2 19  or 2 1931 . So 3 4 19  = 2 + 2 1931 ………………..……………….…(1) Now 2 193 can be written as 4 19 ) 2 19 ( 3  or 5 2 19  . Its integral part is 1 and fractional part is 5 2 19  -1 = 5 3 19  , which can be written as ) 3 19 ( 5 9 19   or 3 192 or 2 3 191 . Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 48 So 3 4 19  = 2 + 2 1931 =2 + 5 2 191 =2 + 2 3 191 1 1   …………….……………..(2) Now 2 3 19  can be written as 3 + ( 2 3 19  -3 ) or 3 + 2 3 19  . Then 2 3 19  can be written as ) 3 19 ( 2 9 19   or 3 195 or 5 3 191 . So 3 4 19  = 2 + 2 1931 =2 + 5 2 191 =2 + 2 3 191 1 1   =2 + 5 3 191 3 1 1 1    ……..(3) Now 5 3 19  has integral part 1 and fractional part 5 3 19  -1 or 5 2 19  or 2 1951 . Now 2 195 can be written as 4 19 ) 2 19 ( 5  or 3 2 19  . So 3 4 19  = 2 + 2 3 191 1 1   =2 + 5 3 191 3 1 1 1    =2+ 3 2 191 1 1 3 1 1 1     ………….(4) Now 3 2 19  has integral part 2 and fractional part 3 2 19  -2= 3 4 19  = ) 4 19 ( 3 16 19   = 4 191Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 49 So 3 4 19  = 2+ 3 2 191 1 1 3 1 1 1     = 2+ 4 191 2 1 1 1 3 1 1 1      ……………………….(5) Now 4 19  = 8 +( 4 19  -8) = 8 + 4 19  =8 + 4 19 16 19   = 8 + 4 193 =8+ 3 4 191 .The last denominator 3 4 19  indicates that the process will be repeated henceforth indefinitely. Example11: Try to convert 9 8 37  straight into a continued fraction and face the difficulty met. Then convert this in the form 1 1 r b N  by adjusting the numerals such that 1 1 r b N  > 1, N – b12 divisible by r1 and see that the difficulty is removed. Example12: Convert 19 into continued fraction by first converting it to the standard form 1 1 r b N  > 1, N – b12 divisible by r1 . Example13: Convert 19 -4 into continued fraction by first converting it to the standard form 1 1 r b N  > 1, N – b12 divisible by r1 . §26:To convert a surd 1 1 r b N  , where b1 < N, b1 and r1 are positive integers and 2 1 b N  is divisible by r1; (A surd of the type BA for A > B, also falls in the above category for B 0 AB BA   and N is also included in the same category where A = N and B = 1. Again if 2 1 b N  is not divisible by r1, we can always add and subtract some suitable number c to b1 and get the condition satisfied )Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 50 If a1 be the integer part of 1 1 r b N  , where b1 < N, b1 and r1 are positive integers; then . 1 a r b N a 1 1 1 1     So . r r a b N r a 1 1 1 1 1 1     ; i.e., . r b r a N b r a 1 1 1 1 1 1 1      ……….…(1) Let 2 1 1 1 b b r a   so that b2 is a +ve integer and . r b N b 1 2 2    ……(2) (subtracting b1 from three sides). Now 1 1 r b N  can be written as 1 2 1 1 r b r a N   = 1 2 1 r b N a   = ) b N ( r ) b N )( b N ( a 2 1 2 2 1     = ) b N ( r b N a 2 1 2 2 1    = 2 2 1 b Nr a   = 2 2 1 r b N1 a   ………………………………………………(3) where we have put 2 1 2 2 r r b N   .If we could prove b2 and r2 to be positive integers and b2 < N, we could write, 2 2 r b N  as 3 3 2 r b N1 a   or 1 1 r b N  = 2 2 1 r b N1 a   = 3 3 2 1 r b N1 a 1 a    …………………….(4) and proceed with formation of the required continued fraction. (This may be made by process of induction, assuming it true for bn and proving for bn+1.) Assume to the contrary, b2  0, then . r b N 1 2   (from (2)), which makes N < r1 . This, alongwith b1 < N gives b1 < r1 0 and thus we arrive at a contradiction. Further b2 = a1r1 – b1 implies that b2 is a positive integer…………………………….…………………………………………..(5)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 51 Again, we have 1 1 r b N  = ) b N ( r b N a 2 1 2 2 1    > a1 ( a1 being integral part of it) , so that N – b22 > 0. So 2 1 2 2 r r b N   > 0. ) b a 2 r a ( r b N ) b r a ( N b N 1 1 1 2 1 1 2 1 2 1 1 1 2 2         It is at this point we understand that we require N – b12 should be divisible by r1; it is to make to make N – b22 divisible by r1. We have already proved 2 1 2 2 r r b N   > 0. So it follows that r2 is a positive integer as desired. And the process at (4) may possibly be repeated. On similar reasoning we can arrive at n 1 n r b N  =an + n 1 n r b N   = an + ) b N ( r b N 1 n n 2 1 n     = an + 1 n 1 n b Nr    ……………………….(6) for some value of n ( as we have just derived it for n = 1), where 1 n n 2 1 n r rb N     ………………………………………………………………………(7) where an < n n r b N  < an + 1 , and bn+1 = anrn – bn .........................………………….(8) indefinitely again and again, for any positive integer n , just as we have already seen it to be true for n = 1, 2, 3… In view of rn, and rn+1 being positive integers, and rnrn+1 = N – bn2, bn must be less than N . Thus bn, which is an integer, cannot have more than b1 different values, for b1 is the largest integer less than N .From (8), rn = (bn + bn+1)/an < bn + bn+1, (as, an is a positive integer, i.e.,  1),  N + N =2 N .Now rnbn  2 N N = 2N. In the same reasoning, rn is an integer and less than 2 N . The largest integer less than 2 N is 2b1. so rn cannot have more than 2b1 number of values. Thus n n r b N  , cannot have more than 2b1xb1 or 2b12 number of different values; all combinations Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 52 q p r b N  for all different values of p and q. But fractions like this, (called complete quotients) shall come again and again as the process is unending. Hence they would recur. §27:To convert a series ..... .......... u1 u1 u1 3 2 1    to a continued fraction ; ( An irrational or transcendental number which is not a surd is a series like this one where u1,u2 etc are powers of 1/10.) We convert the converging infinite series as above with the help of a simple device ,i.e., by putting r r 1 r r v u 1 u1 u1      vr = 1 r r 2 ru u u    ………………………………………………….(1) We have, v1 = 2 1 2 1u u u  .so that 2 1 2 1 1 2 1 2 1 1 1 1 2 1 u u u u1 u u u u 1 v u 1 u1 u1          ……………..(2) The last one in the notation of continued fraction in short. Now, 2 2 1 2 1 1 2 2 1 3 2 1 v u u u u 1 v u 1 u1 u1 u1 u1          writing u2 +v2 in place of u2 in (2) where v2 = 3 2 2 2u u u  = 3 2 2 2 2 1 2 1 1 u u u u u u u 1     = 3 2 2 2 2 1 2 1 1 u u u u u u u1     ……………………(3) in continued fraction notation. Hence in general, ..... .......... u1 u1 u1 3 2 1    = n 1 n 2 1 n 3 2 2 2 2 1 2 1 1 u u u .......... u u u u u u u1        Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 53 Thus an algebraic number which is not a quadratic surd, or a transcendental number which is nothing but a converging series of powers of 1/10, like that given above, can well be converted into a continued fraction , to any step as far as we know its value correct up to that decimal place. §28:Better approximations to irrational/transcendental numbers with continued fractions: Transcendental numbers can be represented by unending continued fractions . if one truncates the continued fraction at any step a rational number m/n is arrived at, on simplification and this would be the best approximation to the transcendental number . no other rational number having denominator less than n shall be a better approximation to the transcendental number. For example the transcendental number , the ratio of circumference of any circle to its diameter  =3.14159 …… can be approximated by 3, 10 31 , 100 314 , 1000 3141 , 10000 31415 , 100000 314159 etc. But the denominators are larger and larger ever, to get approximation to only one step further in the decimal scale. The truncated continued fractions at second , third and fourth steps 22/7 = 3.142857 . . ., 333/106 =3.141509….and 355/113 = 3.1415920. . .respectively give better approximations equivalent to 3rd, fourth and 5th step in the decimal scale of approximations and yet with much smaller denominators 7 , 106 and 113 compared to 100, 1000 and 10000 respectively. Continued fractions are a point of interest for this charming utility for approximating transcendental numbers rather than giving closed forms to quadratic surds.. Let us examine this aspect of approximation in a little detail. With a view to explaining how continued fractions are better approximations to transcendental numbers we need a few general results about continued fractions and we proceed to derive those below . §29:The continued fraction always lies between any two successive convergents : It we write the continued fraction as x and x = ....... a 1 a 1 a 3 2 1    , certainly x > a1. Then 2 1 a1 a  > ....... a 1 a 1 a 3 2 1    = x, as the former has a smaller denominator. So 2 1 a1 a  > x. Now 3 2 1 a1 a 1 a   < .......... a 1 a 1 a 3 2 1    = x ,as .......... a 1 3  < 3 a1 , So, 3 2 1 a1 a 1 a   < x. This process can be indefinitely continued observing that each successive partial quotient an+1 is obtained by writing n n a1 a  in place of an. Thus it is observed that odd the convergents of the continued fraction are all Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 54 less than the number represented by it and the even convergents are all greater than it. The statement also hold good even when the contd. fraction is less than 1; for in that case the first convergent is 0 and the rest of the statement follows. §30:Each successive convergent is a better approximation to the contd. fraction than its predecessor. From (d) above, we expect this result . instead of the difference between two successive convergents, we need to find out the difference between any convergent and the number itself , represented by the continued fraction. To arrive at the result, we have to pass through a concept – complete quotient ....... a 1 a 1 a 2 n 1 n n      in contrast to partial quotient an at any stage n . if we denote this by k, we can write 2 n 1 n 2 n 1 n q kq p kp x       ………………………………………(e) for the number represented by the continued fraction just in the same analogy as we write n-th partial convergent as 2 n 1 n n 2 n 1 n n nn q q a p p a qp       ………………………………….(f) in accordance with (a) above, where an is replaced by k. Evidently k > 1 as all an’ are > 1. Now                 nn 1 n 1 n qp x qp x =        1 n 1 n nn qp qp = 1 n n n 1 n 1 n n q q q p q p     = 1 n n 1 n q q ) 1 (   (by (a)); Whose absolute value is 1 n nq q 1  which is less than 1 So that 1 n 1 n qp  is numerically nearer to x than nn qp ,no matter their difference from x may be positive or negative. Further , as                1 n 1 n nn qp x qp x =         nn 1 n 1 n qp qp = n 1 n 1 n n n 1 n q q q p q p     = n 1 n n q q ) 1 (   ; and its absolute Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 55 value is 1 n nq q 1  which is > 1 n nq q 1  ( as qn-1 < qn < qn+1), which indicates that nn qp gets nearer and nearer to x as n increases. pn = an pn-1 + pn-2 and qn = an qn-1 + qn-2 Further we have,   2 n 2 n qp x 2 n 2 n 2 n 1 n 2 n 1 n qp q kq p kp        = 2 n 2 n 1 n 2 n 2 n 1 n 2 n 2 n 2 n 2 n 1 n q ) q kq ( p q kq p q p q kp                = 2 n 2 n 1 n 1 n 2 n 2 n 1 n q ) q kq ( ) q p q p ( k          = 2 n 2 n 1 n 1 n q ) q kq ( ) 1 ( k      = 2 n 2 n 1 n 1 n q ) k /q q ( ) 1 (       , where k > 1 ( as it is a positive integer >an ; so that 2 n 2 n qp x   = 2 n 2 n 1 n q ) k /q q ( 1     Now, 2 n 2 n qp x   = 2 n 2 n 1 n q ) k /q q ( 1      2 n 1 n q q 1   by decreasing the denominator, neglecting a term, which is further  2 2 n q1 , by further decreasing the denominator as qn-1 > qn-2. Again, 2 n 2 n qp x   = 2 n 2 n 1 n q ) k /q q ( 1      2 n 2 n 1 n q ) q q ( 1     ,( by increasing the denominator, by taking qn-2 in place of qn-2 /k )  1 n 1 n 1 n q ) q q ( 1     = 2 1 n q 2 1  , taking qn-1 in place of qn-2 where qn-1  qn-2. Combining the results, 2 1 n q 2 1   2 n 2 n qp x    2 2 n q1 and similarly, 2 1 n q 2 1   nn qp x   2n q1 Hence the error in taking nn qp in place of x , lies between 2 1 n q 2 1  and 2n q1 …………..………(g)Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 56 From the reverse point of view, this result enables us to form a continued fraction to a number of steps with an error less than a pre-assigned number 1/v if desired . Obviously we have to calculate nn qp to n steps as long as qn2 is less than v and stop when qn2  v. ……………..(h) Further, the error in taking nn qp in place of x < 1 n nq q 1  = ) q q a ( q 1 1 n n 1 n n    < 2n 1 n q a 1 . Thus the error is small enough as soon as the succeeding convergent an+1 is large enough. This gives a further short cut to calculation of number of steps, for we can stop as soon as an+1qn2  v. So a good approximation is any convergent which immediately precedes a large quotient. It is now evident at least in principle, how the Aryans excelled in Astronomical calculations and predicted eclipses etc. highly accurately, even not using Calculus in its form as we see it today. The Aryans must have been very good at continued fractions solely depending upon integers instead of adding large number of terms in infinite convergent sequences with the help of Calculus. So ‘closed forms’ or ‘no closed forms’ – continued fractions are a great help in any calculation to any desired degree of accuracy. Continued fractions play a dominant role in theory of numbers too and are important as such. However, we would have to close the topic with a few examples .They have big application in theory of computing also But the reader is referred to the end of this chapter where for the illustrative problems , where some other aspects of approximations are also figured. § 31: Fibonacci numbers and Golden Ratio Look at the sequence of integers 0, 1,1,2,3,5,8,13,21,34……….. and so on. Any number in the series is the sum of two immediately preceding number. Such a sequence is called Fibonacci sequence named after Leonardo of Pisa, although most scholars agree that it was of Indian origin. The sequence can be arrived at , by a model of a couple of rabbits, one male and the other female which further produce a pair. If the scheme continues beyond time then the number of pairs present at the end of each step is the sequence of Fibonacci numbers. Alternatively, it is also known as bee ancestry code. Let a beehive is stated by a male drone and female bee couple, and let the scheme of reproduction be such: If an egg laid by unmated female hatches let it produce a drone bee of a male one. If a fertilized egg hatches let it produce a female bee. Thus a male has one ancestor  it would not be an exaggeration to say that the modern theory of functions and its structure is based on the inherent structure of numbers most conveniently known to us in decimal form. So also the theory of differential equations and their solutions generally in the form of convergent infinite series and rarely in closed forms and still very rarely in rational forms. The fact that continued fractions give us far better approximations than the method of addition of large number of terms of convergent infinite series, they might promise for a good alternative function theory and a good alternative structure of differential equations if proper computer programmers are designed and their theory is developed zealously. The civilization should remember that we have not yet landed on our immediate cosmic neighboring planet Mars, so, far enough we have to go.Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 57 immediately before it and a female has two. If the ancestry of a male drone is traced, he has one female parent, which in turn has two parents ( one male and the other female) so the couple together have 3 parents. Among the parents two have two parents each ( as they are female bees) and the drone has one parent. So together they have 5 parents. If the step is retraced, we arrive at precisely the numbers in Fibonacci sequence. The Fibonacci sequence is characterized by a recursive relation 1 2 , 2 n n n F F F n      and two seed values are 0 1 0, 1 F F   . The florets in a sunflower, the branches of a tree, arrangement of leaves on a stem, fruitlets of a pine apple, leaves in an uncurling fern, etc. are numerous examples of Fibonacci sequence. Though the above two situations is highly idealized, non the less, the Fibonacci sequences are abundant in Nature. The ratio of any term to its predecessor 1 /n n F F  is more and more approximate to the irrational number 1.6180339887   as more and more terms are taken, which is called golden ratio because of its abundance in nature .In olden days’ architectures like temple of Panthenon, in design of pyramids, in paintings such as a man in the pentagram of Leonardo da’ Vinci , naturally occurring such as the ratio of lengths of upper half of human body to the lower half or the ratio of human body to the upper half approximate the golden ratio. The ratio of length to breadth of A4 size of paper, that of A3 size of paper, the ratio of length of A3 size of paper to that of A4 size all are close to . Post cards obey the same rule. Even it is associated with structure of human genome as some modern scientists propose. Share brokers find is useful and have an analysis tool called Fibonacci retracement. Computer programmers use it in a search criteria called Fibonacci search and it is used usefully in coding decoding theory. For an interesting example see the best seller “The Da’ Vinci Code’ by Dan Brown. It is even traced in crystal structure, even in the subatomic environment, in the magnetic resonance of spins in cobalt niobdate crystals. Look at the pentagram below, the ratio of a solid unit portion of length show to that of a dashed unit is this ratio ; which is also same as which is same as the ratio of length of one long arm of the star to the m of lengths of one solid unit and one dashed unit. Look at the isosceles triangle below with sides 1  units and the base  units. This is called Golden triangle . Bisector of the base angle meet the other side dividing it in the golden ratio and the bisector itself is of length .Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 58 Look at the regular pentagon above with each side a and each diagonal b such that /a b   .mark evident symmetries in it and prove Ptolemy’s theorem 2 2 a b ab   . Assuming the limit of the sequence exists and non-zero, let   1 lim /n n n F F     , we may write 2 1 1 1 1 lim lim 1 1 lim 1 1 0 n n n n n n n n n n F F F F F F F                                               .(To prove that the limit exists, we have to employ standard tests about sequences and series. This quadratic equation enables us to solve for which may be found in standard processes to be 5 1 1.6180339887 2     The quadratic equation also gives us two interesting forms of golden ratio which stands for a b a a b     a) As a continued fraction:   2 2 1 1 1 1 1 0 1 1 1 1 1 1;1,1,1,1..... 1 1 1 1 1 1 1 1 1 1 1 ......                               Also interesting to note that   1 1 1 0;1,1,1,1..... 0.6180339887 1 1 1 1 1 ......             ,called the inverse or conjugate of . [Pentagon] a b φ φ φ 1 φ+1[Golden triangle] [Pentagram (star)]Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 59 The successive convergents of these two continued fractions , namely, 1,2, 3/2, 5/3, 8/5 …. Etc are ratio’s of terms of Fibonacci sequence to their previous terms. b) As an infinite root sequence of roots : 2 1 0 1 1 1 1 1 1 1 1 1 1 ........                         The golden ratio is also associated with painting, such as that of Da’ Vinci and good music , such as that of Pearl Drums or that of Debussy’s ‘reflection in water’ etc. to modern day sculpture and industrial design. A perfect favorite for anybody’s eyes and ears. That is an irrational number may be given a short proof as under : Let mn  , assuming the contrary, m, n being positive integers, m > n, and let the ratio be in lowest terms. Since 1 1 1 n m m n m n m n n n m n             . This shows that mn  can be further reduced ; therefore, it is not in lowest terms – evidently a contradiction. One more exhibit from the last argument – irrational numbers are those which cannot be expressed as commensurable ratio’s. In any case, assuming them so lands us in contradiction. A few other interesting properties of Fibonacci numbers and the golden ratio: 1) 2 0 1 n i n i F F      ( to prove, proceed from the right just using the definition). 2) 1 2 1 2 0 n i n i F F      (to prove,     1 1 2 1 2 2 0 2 2 1 0 0 n n i i n n i i i F F F F F F              3) 2 3 0 2 n i n n i F nF F        (Prove by induction) 4) 2 1 0 n i n n i F F F    Look at the figure:Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 60 The whole rectangle is 1 n n F F  which is easily seen to be sum of the squares 2 0 n i i F  . (Starting from Fn, Fn-1, Fn-2……..). 5)   2 2 2 1 1 1 1 n n n n n n F F F F F F         6) 1 1 2sin cos 2cos 10 2 10 5 ec        7) The successive powers of  obey Fibonacci recursion relation. 1 2 1 n n n n n F F            8) Every third number in the Fibonacci sequence is even. 9) Every n-th number in the sequence is divisible by n F 10)     1 1 2 , , 1 n n n n GCD F F GCD F F      ; any two of three successive terms are relatively prime. 12)     , , m n GCD m n GCD F F F  13) 9 6 11 n i n i n F F      ; sum of any successive 10 terms is divisible by 11. 14) Any positive number can be written as a sum of Fibonacci numbers using any term only once. Fn Fn+1 Fn Fn-1 Fn-1 Fn-2 Fn-3 Fn-4 Fn-5 [Successive Fibonacci Squares; The arc seen in the figure is a logarithmic spiral.] Fn+1 = Fn + Fn-1 Fn = Fn-1 + Fn-2 Fn-1 = Fn-2 + Fn-3 Fn-2 = Fn-3 + Fn-4 And so on….Series : REDISCOVER MATHEMATICS FROM 0 AND 1 Book : Calculus And Analytic Geometry Of 2D and 3D (Concepts and fundas for IITJEE and other competitive exams) PART 1:THE SWORD AND THE SHIELD Section 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions Chapter 4: The Story Of Numbers © copy right protected Page 61 15) n F is the closest integer to 5n  16) A positive integer k is a Fibonacci number iff one of the two expressions 2 2 5 4, 5 4 k k   is a perfect square.