Question

.show that
$5 - \sqrt{3}$
Is irrational
​

Answer 1

Answer:

Let us assume the given number be rational and we will write the given number in p/q form

⇒5−

3

=

q

p

⇒

3

=

q

5q−p

We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence our assumption that given number is rational is false

⇒5−

3

is irrational

Answer 2

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

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