Real Numbers : Real Numbers Natural Numbers
Whole Numbers
Integers
Fractions
Terminating Decimals
Recurring or repeating decimals
Decimals that do not recur nor terminate Instructor : S.C.Benjamin
Email : sc.benjamin@yahoo.com
Real Numbers : Real Numbers All real numbers fall in two broad
categories :
Rational Numbers
Irrational Numbers
Irrational Numbers : Irrational Numbers Irrational numbers cannot be expressed in the
form p/q.
These are non-repeating non-recurring decimal
numbers.
Square roots of numbers that are not square
numbers.
Terminating Decimal Expansions of Rational Numbers : Terminating Decimal Expansions of Rational Numbers Let be a rational number such that prime factorisation of q is of the form , where m and n are non-negative integers ; then x has
a decimal expansion which terminates.
Non Terminating Repeating (recurring) Decimal Expansions of Rational Numbers : Non Terminating Repeating (recurring) Decimal Expansions of Rational Numbers Let be a rational number, such that the prime factorisation of q is not of the form , where m and n are non-negative integers ; then x has a decimal expansion which
is non-terminating repeatng (recurring).
Identify terminating and recurring decimal expansions : Identify terminating and recurring decimal expansions
Rational Numbers : Rational Numbers Every rational number can be expressed
in the form p/q, where p and q are integers A rational numbers can be expressed as a
terminating decimal or non-terminating
(recurring) decimal :
2.5 (terminating)
4.333… (non-terminating or recurring)
5.345345345…(recurring)
Prime Numbers : Prime Numbers A prime number is divisible by only two
numbers; itself and 1
2 , 3 , 5, 7 , 11, 13, 17 , 19, 23, …….
Name the only even Prime Number ? : Name the only even Prime Number ? Explain your answer
Composite Numbers : Composite Numbers A composite number is a whole number
greater than 1and is not a prime number
Fundamental Theorem of Arithmetic : Fundamental Theorem of Arithmetic Every composite number can be
expressed as a product of primes and
this expression is unique, apart from
the order in which the primes occur
Euclid’s Division Lemma : Euclid’s Division Lemma For positive integers a and b, there
exist unique integers q and r satisfying
the relation :
Slide 13 : Question The following real numbers have decimal
expansions as given below. In each case,
decide whether the number is rational or not.
If the number is rational, and of the form
p/q, what can you say about the prime factors
of q ?
Real Numbers : Real Numbers Euclid’s Division Lemma
Fundamental Theorem of Arithmetic
Theorem : Division of : Theorem : Division of Let p be a prime number. If p divides , then p divides a, where a is a positive integer.
Prove that following are irrationals : Prove that following are irrationals