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

1.	Prove that 5 - √3 is irrational, given that √3 is irrational.
Answer» 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

Answer»

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

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