Question

Prove that 5 - 2√3 is an irrational number.​

Answer 1

Hii friend!!

assume that 5-2 root 3 is rational

5-2 root 3 = a/b , where a and b are integers .

-2 root 3 = a/b - 5

2 root 3 = 5 - ab

2 root 3 = 5b/b - a/b

root 3 = 5b - a / 2b

we know that a, b , 2 and 5 are integers and they are also rational

therefore root 3 will be rational

but we know that root 3 is irrational

there is a contradiction

so, 5 - 2 root 3 is an irrational number

Answer 2

Explanation:

Let us assume that 5 - 2 root 3 is a rational number and it can be written in the form of p/q where p & q are coprime and q is not equal to 0.

5-2root 3 = p/q => to prove this as irrational, 1st we must prove root 3 as irrational.

I think u know the method to prove root3 as irrational

After proving that the step continuous ;

5-2root3 = p/q

- 2 root 3 = p/q - 5

- 2 root 3 = p - 5q / q

Root 3 = p - 5q / - 2q

Root3 is irrational but p/q is rational number

Therefore, our assumption is wrong and by contradiction 5-2root3 is irrational

Hence 5-2root3 is irrational

Hope it helps you

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