Question

Prove that 5 -- 2 v3 is an irrational number

​

Answer 1

Let us assume to the contrary that 5 - 2√3 is rational. Then there exist two positive integers such that,

=> 5 - 2√3 = a/b

=> -2√3 = 5 - a/b

=> -2√3 = 5b - a/b (taking LCM)

=> √3 = 5b - a/2b (making √3 the sub. and taking everything else on the right side)

Now, we know that a and b, 2 and 5 are integers and they're rational. Therefore, √3 is also rational.

But this contradicts the fact that √3 is irrational.

Therefore, 5 - 2√3 is irrational.

Answer 2

Refer to the attachment :)