REAL NUMBERS

Decimal Expansion Of Rational numbersEuclid’s Division AlgorithmFundamental Theorem Of ArithmeticPROOFS BASED ON IRRATIONAL NUMBERS

2

REAL NUMBERS

Proofs Based On Irrational Numbers

Q) Prove that √7 is irrational.

SOLUTION:

Let’s say that √7 is actually a rational number.

Since √7 is a rational number we can express √7 as

√7 =

where a & b belong to integers and have no common factor other than 1 and b 0. So , since √7 =

Squaring on both the sides we get,

7 =

Or, 7b2 = a2 ------------------------------ (1)

Proofs based on irrational numbers

Chapter : Real Numbers Website: www.letstute.com

a2 is divisible by 7. (7b2 is divisible by 7)

a is divisible by 7. (7 is prime & divides a2, 7 divides a)

Let, a = 7 c ( where c is some integer)

Substituting a = 7c in (1), we get

7b2 = (7c)2

7b2 = 49 c2

b2 = 7 c2

b2 is divisible by 7. (7 c2 is divisible by 7)

b is divisible by 7. (7 is prime & divides b2, 7 divides b)

Proofs based on irrational numbers

a and b are divisible by 7. Chapter : Real Numbers Website: www.letstute.com

7 is a common factor of a and b, but this contradicts the fact that a and b have no common factor other than 1.

This contradiction has arisen because of our incorrect assumption that √7 is rational.

Hence, √7 is an irrational number.

Proofs based on irrational numbers

Chapter : Real Numbers Website: www.letstute.com

Chapter : Real Numbers Website: www.letstute.com

IF ‘p’ (PRIME NUMBER) DIVIDES ‘’ THEN ‘p’ DIVIDES ‘a’ WHERE ‘a’ IS POSITIVE INTEGER.

RATIONAL NUMBER + RATIONAL NUMBER = RATIONAL NUMBER.

RATIONAL NUMBER RATIONAL NUMBER = RATIONAL NUMBER.

= MAY OR MAY NOT BE A RATIONAL NUMBER.

ALWAYS TRY TO PROVE THESE TYPES OF PROBLEMS BY THE CONTRADICTION METHOD.

APPLY THE DEFINATION OF RATIONAL NUMBERS.

Please visit www.letstute.com to take a

test.

Chapter : Real Numbers Website: www.letstute.com