Question

Practice set 2.2
Show that 4 v2 is an irrational number.​

Answer 1

Step-by-step explanation:

Let 4√2 be a rational number.

A rational number can be written in the form of p/q.

4√2 = p/q

p = 4√2q

Squaring on both sides,

p²=16(2)q²

2 divides p² then 2 also divides p.

So, p is a multiple of 2.

 

p = 2a (a is any integer)

Put p=2a in p²=32q²

(2a)² = 32q²

4a² = 32q²

2a² = 16q²

2 divides q² then 2 also divides q.

Therefore,q is also a multiple of 2.

So, q = 2b

Both p and q have 2 as a common factor.

But this contradicts the fact that p and q are co primes.

So our supposition is false.

Therefore, 4√2 is an irrational number.

Hence proved.

Hope it helps