Question

Prove that 5 - √3 is irrational, given that √3 is irrational​

Answer 1

Answer:

5-root3 is irrational

Step-by-step explanation:

let us assume to the contrary that 5-root3 is rational

=>5-root3=a/b     (where a and b are co primes and b not equal to zero)

=>root3=5b-a/b

Here a and b are integers, so 5-a/b is rational , hence root3 is also rational.

But this contradicts our fact that root 3 is irrational.

This contradiction has arisen bcoz of our incorrect assumption that 5-root3 is rational.

hence 5-root3 is irrational

Answer 2

Step-by-step explanation:

Let us assume that 5 - √3 is a rational

We can find co prime a & b ( b≠ 0 )such that

5 - √3 = a/b

Therefore 5 - a/b = √3

So we get 5b -a/b = √3

Since a & b are integers, we get 5b -a/b is rational,

and so √3 is rational.

But √3 is an irrational number

Let us assume that 5 - √3 is a rational

We can find co prime a & b ( b≠ 0 )such that

∴ 5 - √3 = √3 = a/b

Therefore 5 - a/b = √3

So we get 5b -a/b = √3 Since a & b are integers, we get 5b -a/b is rational, and so √3 is rational.

But √3 is an irrational number Which contradicts our statement ∴ 5 - √3 is irrational