Question

Prove that the following are irrational numbers. (8)

i) 3 ii) 7​

Answer 1

Step-by-step explanation:

to be rational number every integer should have factor 2^n X 5^m

2^n X 5^m is not the factor of 3 and 7. therefore, 3 and 7 are irrational number

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Answer 2

ii.) let us assume that √7 be rational. then it must in the form of p / q [q ≠ 0] [p and q are co-prime] √7 = p / q => √7 x q = p squaring on both sides => 7q2= p2 ------> (1) p2 is divisible by 7 p is divisible by 7 p = 7c [c is a positive integer] [squaring on both sides ] p2 = 49 c2 --------- > (2) subsitute p2 in equ (1) we get 7q2 = 49 c2 q2 = 7c2 => q is divisble by 7 thus q and p have a common factor 7. there is a contradiction as our assumsion p & q are co prime but it has a common factor. so that √7 is an irrational.

i.) The number √3 is irrational ,it cannot be expressed as a ratio of integers a and b.

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