REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 1 CHAPTER 1 : IDENTITY, DIVISION ,SYMMETRY §1:THE BEAUTY AND POWER OF, IDENTITY, DIVISION AND SYMMETRY You have a nature and a name, you have your tastes, choices, likes and dislikes, love and hate, relationships, characteristics and personality – the sum totality of this is your identity. It does not change, i.e., people still know and recognise you if you grow up in time, change houses or go abroad. You are still the same person with the same identity. Similarly your family has an identity, it does not change , for example, if represented by you or some other member of your family. In the same way the groups or communities or country you belong to, have identities for that matter. In mathematics you have come across the identity 0, celebrated as the identity of addition. You know it contains every number and its negative in pairs, for example 5 and – 5 . The identity 0 does not change any number with which it is added or subtracted from . You know 1 is an identity, the identity of multiplication, for example 5 and 1/5 both reside in 1 multiplied with each other. The 1 does not change any number it is multiplied with or divides. Such is the nature of identities. Any number such as 7 is called a constant as it does not vary. You will see this is also an identity in some sense. Note a class of integers which leave a remainder 1 when divided by 7, such as 8, 15, 22 etc. We can denote this class with a symbol [1]. Other such classes of integers are [3], [4] etc. leaving remainder 3 and 4 respectively . Take any two members, one in each class, say 10 from [3] and 25 from [4] and adding them gives 35, which belongs to the class [0] or [7]. This is no fallacy and we can call [7] or [0] each as an identity of addition in these seven classes of integers called congruent modulo 7. An identity does not change ; so also constants. You know (a + b)2 = a2 +2ab+ b2 is an identity. The formula holds no matter how a and b may change. An identity is an equation which holds ( = holds true) for any admissible value of the variable(s). For example, the equation xyy x y1 x1 holds for any value of x and y except 0. Also 2 2 y x y x y x is another example. Thus the identity stands on its own and does not depend upon the symbols though it is convenient to express them with the help of these symbols. Instead of identities that are evident at sight, like (a – b ) + (b – c ) +(c – a ) , we can assume or construct identities . If we assume (X– x )2 + (Y – y )2 +(Z – z)2 = 0 for all values of the variables X, Y, Z, x, y, and z ; or in other words if we assume it to be an identity,( call it a conditional identity) we immediately have the three equations X= x ,Y= y , Z = z. We dwell on this point again and again to prove beautiful results of use like equality of conjugate surds when given surds are equal; equality of complex numbers when given complex numbers are equal; proving that conjugate of a root of a polynomial is also another root of it; so on and so forth. An identity operation does not change or modify the argument (on which it operates) or the operand. For example, k = (k)2 ; an operation like squaring the square root of any number does not change that number. In other words it preserves the identity of the operand and it is therefore only that the operator is called so. Please note that the compound operation of first taking square root and then squaring is not same as first squaring and then taking the square REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 2 root; simply because, it does not result in identity. Taking square of a number and extracting the square root again are thus seen to be not the exactly reverse operations so as to result in an identity operation. We know reverse operation always takes us farther than we imagined. For example adding two natural numbers results in a natural number ; there is no restriction in adding and we cannot get out of the natural number system in the process of addition. The set natural number is thus said to be ‘closed’ under the process of addition . We would return to this point a little later. You see the power of identities in the next two examples. RESULT1; equating coefficients of similar powers in either side of an identity. One interesting thing about identities is that , If, t rx qx px d cx bx ax 2 3 2 3 is taken to be an identity, we must have, a = p, b = q, c = r , and d = t .(coefficients of corresponding powers of a particular variable from both sides shall be equal) This can be easily proved as under : Since the equation is true for any value of x, putting x = 0, we get, d = t and they obviously cancel out from both sides. Then we could assume values of x as 1, -1 and 2 say, and get three equations involving a -p, b -q, and c -r ; and on solving them , we can get, each of a – p etc., each = 0. Alternatively, after canceling out d and t from both sides, we can get another identity, or equation which holds for all values of x. Putting x = 0 again in this eqn., we get c = r . Repeating the process, we get the desired result. The utility of this result is indisputable as you are aware when apply this result to knotty problems. RESULT2 ; equating coefficients of similar trigonometric ratio’s in either side of an identity. If p cos x + q sin x + r = ( a cos x + b sin x + c ) + (b cos x -a sin x) + is taken to be an identity, (in other words, if a given expression in sin x and cos x is changed to another expression in sin x and cos x , for some desired convenience) then we must have the coefficients of sin x to be equal in both sides and so also the coefficients of cos x. We have, p cos x + q sin x + r = ( a cos x + b sin x + c ) + (b cos x -a sin x )+ for all x Or, p cos x + q sin x + r = (a + b) cos x + (b -a)sin x + (c + ) for all x, …………(a) Putting cos x = 1 throughout, ( so that sin x = 0), we get, p + r = (a + b) + (c + )………………………………………….……(b) Putting cos x = -1 throughout, ( so that sin x = 0), we get, -p + r = -(a + b) + (c + )……………………..……………….……(c) Adding (b) and (c) and dividing throughout by 2 , we get, REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 3 p = (a + b) and r = (c + ) ……………………..…………………...(d) Putting these values in (a) we get, q = b -a ……………………………….....(e) So we get three independent equations in , , and from (d) and (e) which may be easily solved to find , , and in terms of p, q and r. ( the result has many uses in integration chapter) It seems at first sight that identities are trivially true and are of little utility. No. Just like 0 is a trivial nothing , but embodies a world of secrets like a black hole, identities contain a world of secrets inside them. We give below an example how the concept of identity is used to reveal the sum of an infinite series hidden in the symbols we define ,to get rid of a difficulty. The great scientist Albert Einstein perhaps said, if only you could know where exactly the problem lies, the solution lies there itself. In his times, half of the world believed there was the all pervading invisible ether and the other half believed there was no ether. He put an end to the controversy by telling everybody to raise only the questions which probably have answers, and discard all others. Did he take the cue from a small child he taught, who told, “there can be another way besides the Earth going around the Sun and the Sun going around the Earth ? ” If a point of difficulty is pinpointed, and is enclosed in a symbol, the solution lies nearby, just you have to manipulate with the adopted symbol in set procedures and arrive at the removal of difficulty. Do the duo of the difficulty and the solution constitute some sort of identity or invariable ? Maybe. Follow the examples: Example1 ; Quadratic Equations :an example in ‘difficulty identification’: First , take the quadratic equation, ax2 + bx + c = 0 with rational coefficients. How easy it would have been, only if the ‘bx’ term had not been there! So the difficulty seems to be the linear term and let us try to remove it. Now, put bx or x, inside a symbol t, such as t = x -h or x = t + h, proceed mechanically working with the equation so as to get rid of the linear term bx , and discard the t and h when x reveals its value and those extraneous symbols have nothing to do any more. Our equation becomes, a(t +h)2 +b(t +h) + c = 0 at2+ (2ah+b)t + ( ah2+bh +c)=0 If we choose, h = -b/2a, to make the second term 0, the equation reduces to , at2+ 0+(b2/4a-b2/2a+c)=0 ,(an eqn. in t having no first degree term in t) or, at2 = b2/4a – c t = a 2 ac 4 b2 x = a 2 b a 2 ac 4 b2 Now the poor things, h and t have been thrown out mercilessly once the difficulty is removed! The roots of quadratic equation t = f(x) = ax2 + bx + c = 0,REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 4 are a 2 ac 4 b b -2 and a 2 ac 4 b b -2 . The sum of the two roots is ba and the product of the roots ca as may be verified direct. We know in high school that this solution has been got by completing square method, either by dividing by a throughout or multiplying by 4a throughout. The ‘difficulty’ identified in this way is to make ax2 + bx , a part of a complete square (as this one is not a complete square, ) The roots are real if discriminant of the equation , ac 4 b2 is positive, i.e., b2 – 4ac 0 and imaginary otherwise( by imaginary we mean, the roots of a negative number, if at all we agree for its existence). They are rational if b2 – 4ac is a perfect square and irrational otherwise .Actually, the concepts of irrational numbers and imaginary numbers are gifts of the quadratic equations. The roots are equal if the determinant is 0. The expression under the square root is called discriminant because it determines the very nature of roots of the quadratic. If the roots are irrational or imaginary, one is the conjugate of the other, i.e, if one is p+ √q or p + iq, the other is p -√q or p – iq; ( This is evident from the structure of the roots , they are already in p+ √q and p -√q form. To prove it, assume m+ √n is another root, so their product (p+ √q)(m+ √n) = c/a, a rational number .This is possible only when m+ √n is a multiple of conjugate of p+ √q, say k(p -√q). Now, the sum of the roots is p + √q + k(p -√q) = p(1 + k) + (1 -k )q = -b/a. Solving this for q we see that either 1 – k = 0 or q = {-b/a – p(1 + k)}/(1 – k), a rational number, which is a contradiction. So 1 – k must be 0 or k = 1 or the other root must be p -√q . ). In another way, we observe that the sum of the roots is – b/a and the product of the roots is c/a; both rational numbers by assumption. Then , if the roots themselves do not occur in conjugate pairs, how can their sum and product be rational ? No matter if the reader does not understand imaginary numbers . We would return to the subjects, quadratic equation and imaginary numbers in detail in later chapters at appropriate places and this is merely one example how to apply the “Difficulty Pin-pointing Method”. Actually this is a method how second term in a polynomial equation could be removed. (See further the chapters on quadratic equations, theory of equations, Cardan’s solution of cubic equation etc.) Exercise1 : Remove the second term in the eqn. ax3 + bx2 + cx + d = 0 Hint : Put 3b x t a in the equation and get the cubic equation reduced to 3 0 t pt q …..(1) where 2 2 3 3 ac b p a and 3 2 3 2 9 27 27 b abc a d q a …………..(2)REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 5 Does this removal of square term help in solving the cubic equation ? It is not apparent on first sight. Any playing with (1) sooner or later leads back to the original cubic equation, back to square one. But Cardano decomposed the new variable t into t u v which changed equation (2) into 3 3 3 0 u v uv p u v q ……………………………….(3) Could we abruptly put 3uv p u v equal to 0. Why not ? Who deters us? ( If we are to put a symbol at the point of difficulty, and there are three unknown roots to search, Cardano might have been motivated to put two symbols instead of one!) Since the number of choices of u and v are infinite as t u v and since we are not actually bothered by the relation between u and v, if there be any; we incur no loss assuming a relation between u and v which would make3 0 3p uv p uv ……………(4) with a view just to simplify eqn (3) into 3 3 0 u v q ………………….(5). And lo, we have got two equations(4) and (5) in u and v and we can immediately solve them. Of course it is a quadratic equation 3 3 3 3 3 , 27 p u v q and u v in 3 u and 3 v ,i.e., 3 u and 3 v become the roots of the equation 3 2 0 27 p z qz and we get 2 3 3 2 4 27 q q p u and 2 3 3 2 4 27 q q p v ……………………(6) And we have not touched the holy cow! Now just retracing our contrivances u v t and 3b x t a , we land ashore quickly, 2 3 2 3 3 3 2 3 2 3 3 3 2 4 27 2 4 27 3 3 2 4 27 2 4 27 q q p q q p b u v t x a b q q p q q p x a Only a small assumption 2 3 0 4 27 q p gives us a real number solution of the cubic equation. We would know later on that any cubic equation has at least a real root. But don’t be swayed away with this REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 6 tempting, small and alluring example which reveals mathematics being so easy. Of course it is, the way you look at it is important. Example2 ; Infinite Sequences and Series : Another example in ‘difficulty identification’ or use of ‘equating of coefficients’ : to find out the sum of a series like the following; ) A .( .......... .......... terms n ).... 1 n ( n .. upto .......... 4 x 3 3 x 2 2 x 1 t n Difficulty here is that we do not know the sum of the series. But we think that it must involve n and its powers and some constant, of course, independent of n . So let us assume (like defining symbols as we often do) , ) B ....( .......... .......... .......... .......... .......... Dn Cn Bn A t 3 2 n The tr, the r-th term, or the general term for that matter, may be written as r(r + 1). (Had all of them been equal, we could have simply multiplied ‘n’ with any term to get the sum. That makes the sum one degree higher in ‘n’, than the degree of ‘n’ in tn . So it is safe to assume that, if n = 6, we don’t require more than 7 terms in the series for tn . So we have taken terms up to n3 in (B) when t n has no other power of n larger than 2. Similar must be the things for n+1 terms too; then,2 3 1 ( 1) ( 1) ( 1) ...............................................( ) n t A B n C n D n C 1 1 2 2 3 3 4.......... ..( 1)( 2).......................................( ) n t x x x upto n n D Subtracting(B) from (C), and (A) from (D) , we get, ) 1 n 3 n 3 ( D ) 1 n 2 ( C B ) 2 n )( 1 n ( t t t 2 1 n n 1 n ) E ( .......... .......... )......... D 3 ( n ) D 3 C 2 ( n ) D C B ( n n 3 2 2 2 Equating co-efficients of similar powers of n from both sides, we get, 3 2 B ... and ,... 1 C ,... 3 1 D 1 D 3 .. and ,... 3 D 3 C 2 , 2 D C B With these values and putting n = 1 (or any integer you like) in (B), which is also an identity, we get, 0 A .. Or ,.. 3 1 1 3 2 A 2 x 1 t t 1 1 3 ) 2 n )( 1 n ( n 3 n n 3 2 n t .. 3 n n 3n 2 t 2 n 3 2 n This is not certainly an universal technique for tackling any series whatsoever. But it gives a little bit of that feeling.REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 7 Review Exercise 2:Try for expression for sum of these series: 1) Sum of 1st n natural numbers, tn, tn =n, or in other words 1 2 n n n 2) Sum of squares of 1st n natural numbers, tn, tn =n2 or in other words 2 1 2 1 6 n n n n . 3) Sum of cubes of 1st n natural numbers, tn, tn =n3 or 2 2 3 1 4 n n n . Not only we use symbols to ‘avoid difficulties’, often we use them to take their advantage. Take the example of Trigonometric ratios; these symbols have been aimed at measurement of heights and distances, but have gone a long way in development of complex numbers, theory of equations and so on. In a similar manner Calculus is developed to deal with infinitesimal numbers (infinitely small numbers) by assigning symbols to them . The process or method is, to identify the ‘difficulty’ or pinpoint it, adopt some symbol to enclose the difficulty, then proceed with known and standard methods until the symbol reveals itself or until the difficulty vanishes otherwise and toss away the symbol mercilessly. Wait. We can throw the symbol out in a particular problem, or well keep the symbol for future use if it has general importance, like ( -1) = i or like log2 8 = 3 for 23 = 8 or sin – 1 ½ = 300 for sin 300 = ½ ., or a symbol for eliminant of equations such as matrices and determinants which find much use elsewhere. We shall return to the subject. Throughout the book series, we have adopted this method to develop a topic, adopting a symbol tacitly carrying its entire properties and characteristics; may it be Matrices and Determinants, Trigonometric functions, Inverse Circular Functions, Logarithms, Limits, Conic sections, Differential Coefficients, Integrals etc. etc. Due to this approach the topics appear in a manner how they were discovered and developed rather than a formal presentation. There are numerous examples throughout the book where we would be using this technique to rediscover and redevelop many topics from identities. Before that the reader may try some identities from high school some of which are given below. The reader can try as many of them as possible and at ease. Use the fact that if a = b put throughout the expression makes it 0, then a – b is a factor; similarly if a = -b put throughout the expression makes it 0, then a + b is a factor. 4) (a – b) + (b – c) + (c – a) = 0; 5) c(a – b) + a(b – c) + b(c – a) = 0 6) c(a – b)3 + a(b – c)3 + b(c – a)3 = (a + b + c)(a – b)(b – c)(c – a) 7) c(a – b)2 + a(b – c)2 + b(c – a)2 + 8abc = (a + b)(b + c)(c + a) 8) c4 (a2 – b2) + a4 (b2 – c2) + b4 (c2 – a2) = -(a – b)(b – c)(c – a)(a + b)(b + c)(c + a)REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 8 9) (b -c)3(b + c – 2a) + (c -a)3(c + a – 2b) + (c -a)3(c + a – 2b) = 0 10) (b -c)(b + c – 2a)3 + (c -a)(c + a – 2b)3 + (c -a)(c + a – 2b)3 = 0 11) (ab – c2)(ac – b2) + (bc – a2)(ba – c2) +(bc – a2)(ba – c2) = bc(bc – a2) + ca(ca – b2) + ca(ca – b2) 12) bc(b –c) + ca(c –a) + ab(a –b) = -(b –c)(c –a)(a –b) 13) a2 (b –c) + b2 (c –a) + c2 (a –b) = -(b –c)(c –a)(a –b) 14) a(b2 –c2) + b(c2 –a2) + c(a2 –b2) = -(b –c)(c –a)(a –b) 15) a3 (b –c) + b3 (c –a) + c3 (a –b) = -(b –c)(c –a)(a –b)(a + b + c) 16) a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – bc – ca – ab) 17) a3 + b3 + c3 – 3abc = (a + b + c)(b2 – ca + c2 – ab + a2 – bc) 18) a3 + b3 + c3 – 3abc = ½ (a + b + c)[(b – c)2 + (c – a)2 + (a – b)2] 19) (b – c)3 + (c – a)3 + (a – b) 3 – 3(b – c)(c – a)(a – b) = 0 20) (a + b + c)3 = a3 + 3a2b + 6abc 21) (a + b + c + d)3 = a3 + 3a2b + 6abc 22) bc(b +c) + ca(c +a) + ab(a +b) + 2abc = (b + c)(c + a)(a + b) 23) a2 (b +c) + b2 (c +a) + c2 (a +b) + 2abc = (b + c)(c + a)(a + b) 24) (b + c)(c + a)(a + b) + abc = (a + b + c)( bc + ca + ab) 25) (a + b + c) (-a + b + c) (a -b + c) (a + b -c) = 2b2 c2 + 2c2 a2 + 2a2 b2 – a4 – b4 – c4 26) c (a4 – b4) + a (b4 – c4) + b (c4 – a4) = (b –c)(c –a)(a –b)(a2 + b2 + c2 – bc – ca – ab) 27) (a + b)5 = a5 + b5 + 5ab(a + b)( a2 + ab + b2) 28) (a + b + c)5 = a5 + b5 + c5 + 5(a + b) (b + c)(c + a)( a2 + b2 + c2 + bc + ca + ab) 29) c2 (a3 – b3) + a2 (b3 – c3) + b2 (c3 – a3) = (ab + bc + ca)(a – b)(b – c)(c – a) 30) bc(c2 – b2) + ca(a2 – c2) + ab(b2 – a2) = (b – c)(c – a)(a – b)(a + b + c) 31) (b +c){(r + p)(x + y) – (p + q)(z + x)}+ (c +a){(p + q)(y + z) – (q + r)(x + y)} + (c +a){(p + q)(y + z) – (q + r)(x + y)} = 2[a(qz – ry) + b(rx – pz) + c(py – qx)] 32) (a – x)2{(b – y)2(c – z)2 -(b – z)2(c – y)2}+(a – x)2{(b – y)2(c – z)2 -(b – z)2(c – y)2} +(a – x)2{(b – y)2(c – z)2 -(b – z)2(c – y)2} = 2(b – c) (c – a) (a – b) (y – z) (z – x) (x – y) and so on and so forth. All Mathematical formulae read in high school are identities or conditional identities and we know their utility. As examples of some conditional identities put a + b + c = 0 in expressions above where a + b + c appears: the result may be taken as a conditional identity. If a + b + c = 0, prove the following : 33) 2bc = a2 – b2 – c2 34) 8a2b2c2 = (a2 – b2 – c2)( b2 – c2 – a2)( c2 – a2 –b2) 35) a3 + b3 + c3 = 3abc 36) 2(a4 + b4 + c4) = (a2 + b2 + c2)2 37) 3a2b2c2 – 2(bc + ca + ab)3 = a6 + b6 + c6 38) a5 + b5 + c5 + 5abc(bc + ca + ab) 39) 7 c b a 5 c b a 2 c b a 7 7 7 5 5 5 2 2 2 40) c b a b a c a c b /3 3 /b a c a c b c b aREDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 9 §2:Symmetry , Anti-symmetry And Asymmetry Are some Aspects Of Beauty. A look at the previous examples of identities makes us think about symmetry anti-symmetry and cyclic symmetry of the expressions. Discoverer of electrical generator must have observed the change in electric field due to motion of chares, i.e., electric current causes a magnetic field; and it must have occurred to him that a change in magnetic field may generate electric current. Take the previous example of equalizing coefficients of similar terms from both sides of an identity equation. This reveals the beautiful feature of symmetry. Take the algebrical identities in the previous exercises. The hint is to put a = b in the expression. If the expression reduces to 0, then (a – b) is a factor. This is cyclic symmetry. Analyse any identity and you find some symmetry. Symmetry reveals the things that are not explicit. It also helps us to write expressions in brief. For example, a stands for a + b if two elements are taken; it stands for a + b + c or a + b + c + d if 3 or 4 elements are taken. Similarly a2 stands for a2 + b2 or a2 + b2 + c2 or a2 + b2 + c2 + d2 if 2 or 3 or 4 elements are taken. The expression a2 + b2 is symmetrical with respect to a and b in a sense that a and b can be replaced with each other without affecting the value of the expression. The feature may be termed bilateral symmetry. The expressions such as a2 + b2 + c2 or bc + ca + ab are bilaterally symmetrical, as any two of them can be interchanged without changing the value of the expression. In addition, the latter expressions are of cyclic symmetry; i.e., if a is replaced by b, b is replaced by c and c is replaced by a simultaneously, the expression is unchanged. To illustrate the method of applying symmetry concept in working out problems, consider factorizing the expression (a + b + c)5 -a5 -b5 -c5 . The value of the expression is unchanged if we put b in place of a , c in place of b and a in place of c. But the expression becomes 0 if b = -a throughout. Hence b + a or a + b must be a factor of it. Remember this is due to factor theorem read in high school. Similarly b + c and c + a must be factors of it. As such the expression contains (b + c)(c + a)(a + b) as a factor. The latter factor is of third degree whereas the expression to be factorised is of 5th degree. So it contains another factor of 2nd degree. As the expression is symmetric in a, b and c; so also all its factors must be symmetric cyclically. A general expression in three elements and in 2nd degree would be A(a2 + b2 + c2) + B(bc + ca + ab) where A and B have to be determined. So we have complete factorization as (a + b + c)5 -a5 -b5 -c5 = (b + c)(c + a)(a + b)[A (a2 + b2 + c2)+ B(bc + ca + ab)]REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 10 Now since this is an identity, the expression holds for any value of a, b and c. Putting each equal to 1 and each equal to 2 in turn we get, A + B = 10 and 5A + 2B = 35. Solving the equations, we get, the values of A and B , both equal to 5 and the complete factorization becomes, (a + b + c)5 -a5 -b5 -c5 =5 (b + c)(c + a)(a + b)(a2 + b2 + c2 + bc + ca + ab) An expression such as a2 (b –c) + b2 (c –a) + c2 (a –b) is changed to –[ a2 (b –c) + b2 (c –a) + c2 (a –b)], i.e., its own negative when any two of its variables are interchanged with each other. Such an expression is said to be alternating or anti-symmetric. More about symmetry and its uses shall be discussed from topic to topic later on; especially in transformation of graphs. Puzzle: In finding out factors of 3 3 3 3 a b c abc one observes that the expression reduces to 0 if a b c is put into it. But neither of the expressions , , a b b c c a is a factor of it. Explain how. ans. assuming stand alone expression like a b etc, does not make the expression 0, it becomes 0 only when all of a b c is assumed. So it indicates, neither of , , a b b c c a is a factor of the expression 3 3 3 3 a b c abc , but a factor of the expression involves some combination of all of , , a b b c c a at the same time! observe that 2 2 2 3 3 3 1 3 2 a b c abc a b c a b b c c a Some other aspects of beauty are continuity, completeness, compactness, connectedness, convergence and uniformity. Examples shall follow throughout the book. The concept of continuity of functions shall be discussed in Calculus in a later chapter. Convergence concept shall be discussed in the chapters for sequences a, series and limits. While watching a movie or drama we note a touching sequence of events and keep guessing what should happen at last. If the end comes of our expectation we feel continuity in the story line. If the end of the story keeps us guessing still , we must feel something lacking in the story, e.g., there may not be an end to the drama and it may not be said to be complete. What happens in this case is a sequence of chosen events leads to a limiting event, a point which is not included in the story; As such the sequence of events is not continuous and the story is not complete. The concepts as such, are better illustrated in Topology, which is a set of some subsets called open subsets with a structure – closed under arbitrary unions and finite intersections. The topic of topology is an attempt to provide a common platform to Algebra, Analysis, Differential equations, etc. etc. The principle of equivalence in mechanics , as propounded by Newton, states that the laws of mechanics are same with respect to all inertial frames; i.e. , they do not change if we change frames of reference with a new one moving at a constant velocity with the old one. AnREDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 11 illustration would be given at the appropriate place . It would be shown that acceleration of somebody measured in one frame of reference will just be same in be same as measured in a different frame of reference moving at a constant velocity from the initial frame of reference taken. It would be just child’s play and the reader even might have done the derivation is high school. Einstein derived the epoch making theory of relativity only from two assumptions; one : the principle of equivalence with only one word changed ; he wrote Physics in place of Mechanics. The second assumption is that the velocity of light in empty space does not depend on ( not added to nor subtracted from) the velocity of its source. The latter assumption is nothing but wise acceptance of failure to observe the expected result in the famous Michelson Morley experiment to measure absolute velocity of earth . ( Doesn’t it seem that the theory of relativity was derived from the very antithesis of relativity ?) The conservation laws in Physics like conservation of mass, conservation of energy, conservation of momentum, conservation of angular momentum, conservation of spin etc. tell us that the totalities of quantities like mass, momentum, energy, spin etc before an event like collision or explosion remains unchanged after the collision or explosion etc., as the event may be. So total mass, momentum, energy etc. are invariants .The concept is similar to the concept of identity. So the totals of these quantities in a system do not change i.e., they are symmetrical about a point of event like collision or explosion etc. Total mass or energy or momentum in a system of particles before a collision taking place in the system is conserved after the process of collision. Similar is the case after a chemical reaction is completed. The new theory of relativity has combined the laws of conservation of mass and conservation of energy into one law, conservation of mass and energy together, by showing equivalence of mass and energy. Not a single instance has been observed violating the these principles of conservation. We would give an expression to illustrate the principle of conservation of momentum and conservation of energy at appropriate place .The principle of equivalence in theory of relativity for those who refuse to wait until then. Let us measure acceleration a of some object while we stand still on the ground. Acceleration is t v v a 0 1 , where v1 , v0 are its final and initial velocities and t is time taken for this change of velocity. If we observe the same from object from a train with velocity u, do not observe the initial and final velocities, but observe the initial and final relative velocities instead , v1 – u and v0 – u . Now acceleration observed from the train is a t v v t ) u v ( ) u v ( 0 1 0 1 , again, which proves the proposition. One can change all these symbols except for time to vectors and prove the proposition in case of vector velocities too. Suppose two bodies of masses m1 and m2 with velocities u1 and u2 respectively collide and their velocities get changed to v1 and v2 respectively. By Newton’s third law the force exerted by one body on the other should be equal and opposite. Let them be F and – F respectively. Equivalently, t ) v (u m t ) u (v m 2 2 2 1 1 1 where t is the brief time for which the two bodies are in contact while colliding, or , m1v1 + m2v2 = m1u1 + m2u2 ; i.e., the total momentum before collision is the total momentum after collision , as expected.( vectors may replace scalars throughout, if you please)REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 12 has led to Lorenz transformation of coordinates which in turn, has led to the result of equivalence of mass and energy. Some of the invariants in transformation of coordinates we shall discuss later on in the chapter for transformation of graphs. The starting point in solving problems involving equations of motion is these conservation laws which give the differential equations of motion in a particular situation; one equation if got from law of conservation of momentum, another is got from law of conservation of energy; the latter are then solved by applying standard Mathematical techniques to get the equations of motion. By differential equation we mean an equation involving physical quantities such as velocity etc. and their rates of change; the latter called differential coefficients. A differential equation is solved when we get equations involving the physical quantities only and not involving their differential coefficients. Do the phrases “The principle of equivalence”, “The conservation laws in Physics” and “ Constants and invariants in Physics or in Mathematics” sound like the concept of identity we just discussed. Decide for yourself. A special mention may be made of Heisenberg’s uncertainty principle which enunciates that the product of errors of measurements of complementary physical quantities like position and momentum shall be at least equal to Plank’s constant; h p . s ; This is a completely theoretical fact having nothing to do with precision of measuring instruments. If we set s = 0 to know s or position completely precisely, it requires p to become infinite; i.e., the momentum p shall have infinite error in its measurement and thus cannot be determined at all. This is nothing but symmetry; just in the same sense as 4/1 and ¼ are symmetrical. Symmetrical conjugates combine with each other to result in identity ! The way they combine or associate with each other may be different i.e. (+4) adds up with ( -4 ) to result in 0, the identity of addition and 4/1 and ¼ have to be multiplied with each other to result in multiplication identity 1. Break open the identity and you get symmetrical parts; again fit the parts together, you get the identity. The beauty of the statement lies in the philosophy of quantum theory stating that no physical quantity can ever be measured deterministically but only probabilities can be expected. When the notion of exactitude of measurements is lost, the philosophy of cause and effect, the basic philosophy of all experimental sciences is put at stake. By exactitude of physical laws we mean the current event is the result of the previous event and the cause of the next. It turns out that a bigger particle has some chance of tunneling through a smaller particle or simply you could just pass right through a wall for that matter. The concept of continuum of cause and effect still thrives, albeit in terms of probabilities though not in terms of exactitude as was in the good old days of classical physics. Behind these conservation laws or invariance, there is a grand principle of nature. Economy of action is the grand principle behind. Events would proceed in the direction in which the total energy of the system would be the least. Newton was a great integrator. He integrated the works of Kepler’s, Hooke’s and Copernicus and declared the beginning of the Similarly , from the principle of conservation of energy, we can derive an expression of kinetic energy of a body of mass m, the energy, W say, possessed by it by virtue of its velocity. Surely it would be equal to the work done by it against a force opposing its motion until it comes to rest. If the body travels a distance s in the process, then we have, the work done W = Fs = mas = ½ m 2as = ½ m1v2 which is the expression of kinetic energy we sought after. REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 13 era of scientific progress giving science a foundation; his three laws , appearing almost axiomatic. Another great integration or synthesis was achieved with the advent of a new era of Quantum Mechanics, with its harbingers, Max Plank, Neil Bohr and others. Big controversies about particle nature of radiation and wave nature of matter were settled by giving matter and energy a common platform , the wave equation due to Schrödinger. Radiation was accepted to be effected in quanta, or ‘in packets’ instead of in continuous fashion. Classical Mechanics predicted that a charged particle shall continuously radiate energy resulting in shrinking of its orbit and shall fall into the nucleus within very short time and stable atoms were incomprehensible as such. Neil Bohr used the ‘quantum’ idea of Max Plank and told to the world to accept that the electrons simply do not radiate while being in their orbits and only radiate in lump sums while jumping from orbit to orbit. The structures of orbits became complicated to comprehend of course, but the dual nature of radiation and matter were explained. Classical mechanics which beautifully predicted celestial phenomena, could be viewed as a limiting case of Quantum Mechanics and the two domains were unified. Another great integration or synthesis was brought about , at about the same time by Albert Einstein for explanation of behaviour of matter approaching the speed of light and the concept of matter, energy, space and time had to be redefined. He brought about the Special Theory of Relativity , the single most outstanding discovery and philosophy of the twenty first century, which identified the century, together with Quantum Mechanics. The simple idea behind was, that the speed light does not depend on the speed of its source. Mechanics at lower velocities was viewed as a limiting case of the theory of relativity. Necessary Mathematics was already at hand , developed by Maxwell in his electrodynamics theory and the simple idea of constancy of velocity of light in empty space was acceptance of failure of the famous Michelson & Morley experiment aimed to use difference of velocities of light in moving back and forth to measure absolute velocity of earth. These great integrators like Newton and Einstein were motivated by the idea of sort of identity, symmetry and invariance etc. What was fundamental belief driving behind these principles of invariance and symmetry – it was the principle of economy – the way of Nature’s doing things. It was as back as the days of Ptolemy and Aristotle when observations were made regarding angles of incidence and angles of refraction when path of light changed mediums; but the relationship between them eluded until some three hundred years until the days of Fresnel who correlated them in the laws of refraction. About a hundred years later, Format gave the law a firm footing; the law was deduced from the belief that light travels through the path in which minimum time is taken. It is only through this principle of economy that Hamilton ‘deduced’ the Newton’s laws again. Euler went further through the problems of maximization or minimization and created a new discipline in Mathematics which is known as Calculus of Variations Optimization theory today. The development of these topics are giving rise to Fuzzy Logic and Neural Networks progressing in the direction of Artificial Intelligence and Industrial Mathematics at large. Now any idiot can swear that if the grand unification theory is going to be complete, it would be only through courtesy of belief in nature’s principle of economy – from which the relative external features like symmetry and identity emerge. The scientific world today is striving towards grand unification theory of the four fundamental forces of nature or looking for a common origin of the four forces; namely, electromagnetism, gravitation, the strong and the weak nuclear forces. It is the sense of beauty REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 14 only which enables integration of apparently contradicting theories such as the Classical and the Quantum Mechanics, the wave theory and corpuscle theories of light, the classical and the relativistic Mechanics and so on. It is the art and science of Mathematics only which formulates and guides such endeavors. This is the dream idea of Einstein and we are nearing the idea year by year.Newton has given us a comprehensive concept for force, mass, energy, velocity etc. and explained the gravitational force of universal application with the formula 2 r Mm G F for interaction between any two (point masses or spherical bodies of) masses M and m at a distance r from each other. So gravitational force has been recognized as a fundamental force in nature. This force is responsible to hold together solar systems and nebulae as well as pieces of any matter together to have some shape or size. it goes without saying that solids have definite size and shape only because of small intermolecular distance between them; liquids have only definite ‘size’ or volume as intermolecular distance is more than that in the solid state and gasses have no definite size nor shape but for the large intermolecular distances. Similar is the Coulomb’s law of electrostatic attraction or repulsion, describing the force between two point charges Q and q at a distance r apart from each other to be 2 r Qq k F . Electric currents or moving charges showed magnetization as natural magnets and as such, all magnetism was ultimately attributed to electric currents. Faraday imagined that , if a motion of electric charges gives rise to magnetic poles in effect, a motion of magnetic poles must produce some electric current and it was very much true. This led to the fact that electricity and magnetism are not two different entities , but are manifestation of one thing, electromagnetism which also included the Coulomb’s law for magnetic force 2 2 1rm m F between two magnetic poles 1 m and 2 m at a distance r apart.. So electromagnetic forces are fundamental forces responsible for binding electrons to the nucleus to form atoms and are sufficient to explain away chemical properties of matter. Though the electrostatic attraction and gravitational forces both vary inversely as square of distances, the magnitude of the latter is 1038 times the former. EXAMPLE 3: Calculate the forces between a proton and an electron in an hydrogen atom given r – 10 0.53 x 10 m, charge of one electron as well as that of proton being e = 19 1.6022 x 10 Coulomb. in magnitude; mass of electron m = – 31 9.1093 x 10 kg, mass of proton M being some 1836 times that of electron; G = 11 6.673 x 10 Nm2/kg2, k = (1/40) = – 19 9.1 x 10 Nm2/Coul2. So much of data is absolutely not necessary to compare the forces; neither the distance of electron from nucleus, nor its mass and charge both; only e/m is sufficient for the purpose , it being 11 1.76 x 10 Coul/kg. A single electromagnetic force can be understood to act upon a charged particle, whether stationary or moving, accounting for electrostatic and electromagnetic forces. This is REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 15 Lorentz force given by F = qE + qv x B where q is the charge with velocity v, E is the electrostatic field, and B is the magnetic induction at the point where the charge lies instantaneously. What makes protons stay together inside the nucleus when they should have repelled each other away ? It is the strong nuclear force which should be great enough to overcome the electrostatic repulsion and in fact it is of the order of 1038 times the gravitational force between two protons . or of the order of 100 times the electromagnetic force so that it overcomes repulsive Coulomb forces between protons in the nucleus. But it is short ranged and almost ineffective beyond a distance of 15 10 m which is in the order of nuclear radius. Beyond the nucleus the electromagnetic force reigns supreme , but it dies out fast beyond atomic or molecular radius though theoretically it acts up to infinite distance. Beyond the atomic radius of 10-10m only gravitational force is visible as there are no charged particles to masquerade it. We see it acts through great distances as known from its formula. The strong nuclear force is said to act upon quarks and gluons of which the protons and neutrons are made up of, and thus accounts for formation of nucleus and disintegration of nucleus too, for that matter. Radiation of -particles has been successfully explained in terms of this force. An -particle is equal to Helium nucleus consisting of two protons and two neutrons. Crudely speaking neutrons are necessary to bind protons together in the nucleus on the face of electrostatic repulsion and more and more of them is necessary as atomic number is increased, until the nucleus becomes unstable and becomes radioactive to emit an -particle to become stable. For protons and neutrons cannot be shed off unless they are in an unit of two plus two of each, i.e., an -particle. More correct picture is of course, the one involving quarks and gluons, believed to be ultimate particle of every matter . Protons and neutrons are believed to be made up of 3 quarks each of different ‘colour’. Colour is a property of quarks and gluons analogous to charge , are of three types, ‘red’ ,‘blue’ and ‘green’ and quarks with different colour attract each other and quarks of same colour repel ach other. Gluons carry a colour and an anti-colour charge.. A proton consists of three quarks of each different colour. Within a distance of less than 10-15m (a femtometteror nuclear radius the strong force is repulsive and keeps the nucleons (protons and neutrons) apart. Within a distance of 1 to 2 femto-meters the strong force is attractive force overcomes the electrostatic repulsion being of the order of 100 times the electromagnetic force Quarks and leptons were believed to be ultimate building blocks of all matter. The leptons, which include the electron, do not “feel” the strong force. However, quarks and leptons both experience a second nuclear force, the weak force. This force, which is responsible for certain types of radioactivity classed together as beta decay, is feeble in comparison with electromagnetism , of the order of 10-15 compared to electromagnetic forces and act in very short range of 10-15m, within nuclear radius. In -decay, the nucleus emits an electron with high energy and an antineutrino of no rest mass and a neutron is converted into a proton in the process. Similar is the process of electron capture by a nucleus from the innermost shell and similar is the process of positive electron or positron emission ; in all the three cases the week nuclear force is involved. REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 16 Much is to be done in the area of unification of forces or to have closer look for symmetries and similarities though much has been done. To remind of history, it was Sir Isaac Newton around 1687 who ‘unified’ celestial forces with ‘terrestrial’ forces, i.e., made it evident that the acceleration due to gravity is nothing but a manifestation of universal gravitation. All magnetism was traced back to moving electric charges , thanks to the works of Hans Christian Oersted and Michel Faraday , around 1825. It is J. Clark Maxwell who showed that visible light was an ‘electromagnetic’ radiation; which paved the way to understand that heat waves, ultraviolet, microwave, x-rays, -rays, radio waves , cosmic rays all are electromagnetic radiations. Such unification brought about great confusion about wave nature of particles and particle nature of waves and confusion about ether, the thinnest wall in the empty space to lean on. So developed the quantum theory to unify waves and particles; a number of scientists were involved , over many decades and ultimately the concept of ether was discarded. Mathematics required it to be of very high density and nobody was prepared to swallow the idea along the gut anyway. Nuclear physics and radio activity brought newer phenomena to observation and again we were in the mesh of particles, particles and more particles. Force, which was not due to contact, or action at a distance, whether gravitational, electromagnetic, strong or week nuclear forces, were seen to be understood with ‘carrier particles’ of forces; most successfully electromagnetic forces were explained, with exchange of ‘photon’ as carrier of electromagnetic forces; just as a play ball influencing the behaviour of a pack of playing children. ( When two children run toward the ball to catch it, they seem to attract each other). Albert Einstein’s general theory of relativity postulated ‘graviton’ , photon-like particle as carrier of gravitational force although quantum theory for it is yet to be fully developed. ‘Gluon’ was postulated to be carrier of strong nuclear force or quarks exchanged gluons on interaction under influence of strong nuclear forces. W-particles and Z-particles were thought to be the exchange particles for week nuclear forces though big particles they were indeed, the former with + or – charges and the latter neutral. A single ‘electro-week theory’ was postulated by Sheldon Glashow, Abdus Salam, Steven Weinberg around 1980 in order to look for a common origin of electromagnetic and week nuclear forces and it worked very well. It was verified experimentally by Cario Rubia and Simon Vander Meer around 1984 but the grand unification is still a far cry. But the efforts in this direction are not discouraging ; Abdus Salam had said that the unification is expected if proton decays at all, and it was found to be decaying, with experiments carried out in the deepest mines e.g., the Kolar gold fields in India. A significant improvement has been achieved in 2004 towards The Grand Unification Theory or The Theory of Everything and Nobel prize for Physics has been shared by three physicists; for the first time for theoretical research. §3:Super-symmetry and anti-symmetry : Particles have broadly been divided into two types, fermions having spin quantum number ½ or – ½, and bosons are the other type, having spin numbers 1.The symmetry we have referred in connection with conservation of momentum etc. means that total momentum after a collision (say) is equal to total momentum before collision. A similar ‘super-symmetry’ is displayed in Three people shared the 2004 Nobel prize for Physics for a major development in the Grand Unification Theory (GUT), or Theory Of Everything (TOE), or Quantum Chromodynamics (QC), by successfully including gravitation into the common fold of the field theories.REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 17 subatomic level. Fermions can be transformed into bosons without changing the structure of the underlying theory of the particles and their interactions and vice versa ( i.e., back again to fermions). Super-symmetry provides a connection between the known elementary particles of matter (quarks and leptons, which are all fermions) and the messenger particles that convey the fundamental forces (all bosons). Thus it shows that one type of particle is in effect a different facet of the other type. Super-symmetry reduces the number of basic types of particle from two to one. This feature has in fact encouraged to look for a grand unification theory. Another advantage of super-symmetry is that it requires a particular fermions must have a supersymmmetri conjugate boson particle somewhere. This has doubled the number of types of fundamental particles known and compelled scientists to look for an anti-particle of every particle discovered or postulated. Super-string theory of 1970’s which assumes strings as ultimate extended particles to explain all the four fundamental forces has been supported by concept of super-symmetry, although it has problems to sort out. It works in a field of 10 dimensions, , 6 of which are assumed to be truncated to very short distances within which quarks have a free play. This is rather strange and asymmetrical considerations to explain the most recent ‘theory of everything’ TOE. But any asymmetry, as it were, promises of more subtle symmetry. For example, Higgs suggested ‘ breaking of symmetry’ for explaining week nuclear reactions where ‘parity’ did not seem to be conserved and this led to the concept of super-symmetry in course of development. In Mathematics, we come across non-commutative algebraic structures or operations more often than not. Not only it is tolerated with open mind, but is sincerely accepted as it gives philosophical insight and is meant for theoretical development after all. An example of ‘breaking’ symmetry: If , are the two roots of quadratic equation 2 0 ax bx c , then , b c a a . The expressions are symmetric as , can be interchanged and the sum and product are still the same. If one tries to solve the quadratic equation from the sum of product of the roots, one again lands on the same quadratic equation. Same is the case with n-th degree equations, only the n relations between the roots and coefficients does not help substantially in getting the roots. But when we get from 2 2 2 24 4 b ac a . The sum and difference of the roots gives us the roots immediately. This is called ‘ breaking of symmetry’. Galois carried further the argument and discovered that such process cannot be possible for 5th degree equation and beyond, although an equation of n-th degree must possess n roots. The roots only could be numerically calculated. Breaking symmetry although did not produce the roots, it produced valuable knowledge! To explain symmetry from a different angle , just imagine what would have been the situation if both the roots of the quadratic 2 0 ax bx c been equal. Then the expression must have been a whole square ! And what if the two roots are not equal in general; then we should try to REDISCOVER MATHEMATICS FROM 0 AND 1 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 1 : Identity, Division ,Symmetry in Mathematics ©copyright protected Page 18 take out a whole square out of it i.e. 2 2 2 2 2 2 2 b b b x x c a a a removing the remaining terms whatsoever to the other side of the equation. And this is the whole square method taught in high school. If the roots are not equal, they must differ from the average of the two roots by the same amount and this is in fact so. The average of the roots is 2ba and the actual roots differ from it by the same amount , namely 2 4 2 b ac a . If (3 , 5) is an unsymmetrical division of 8 , then they must differ from the symmetrical division of 8 i.e., 4 by the same amount 1 . §4:Division: Symmetry may be considered as a division of identity. Subtraction may be viewed as a division into unequal parts and can be represented by ordered pair (a, b) standing for a – b. A rational number is a division and can be represented by an ordered pair. An irrational number is a continued fraction, again a division. A complex number can be represented by an ordered pair, so it is some sort of division. So vectors and matrices are. Quaternions were discovered as vector division. The sequences and series may be thought of as some sort of division . e.g, 2 3 4 1 1 ............ 1 x x x x x . The entire Calculus may be viewed as some sort of division, limits of division. In a locus in Coordinate Geometry the point varies but the scheme, or the equation represents the entire locus; So it is a concept of entirety ,similar to the concept of identity, invariants, constants fixed points etc. A function f(x) =0 may be written as x = φ(x); the roots of the former are fixed points of the later. In short, the writer believes Mathematics and Physics are visible with the telescope or microscope of identity–division–symmetry . The difficulty and its removal together constitute a division of knowledge. Symmetry is a division into symmetrical parts of a whole. Symmetry and identity are only different views of looking at the same thing. By saying momentum is conserved in a collision is same thing as saying total momentum before collision is equal to the total momentum after collision. It is the invariance laws or conservation laws of momentum and energy produce differential equations of motion which in turn, yield the equations of motion when solved. Thus identity – division – symmetry is the only formula for ultimate knowledge of Nature. Thus 'identity' and 'symmetry' are intimately connected with each other and the connection is 'division'