Question

Prove that `1/sqrt(2)` is an irrational number.

Answer 1

Given: We want to prove that the number `1/sqrt(2)` is irrational.

To Prove: `1/sqrt(2)` is irrational.

Proof:

Step 1: Assume, for the sake of contradiction, that `1/sqrt(2)` is rational.

That means it can be expressed in the form `1/sqrt(2) = a/b` where a and b are integers with b ≠ 0. 

Step 2: From the above,

⇒ `1/sqrt(2) = a/b`

Multiply both sides by `sqrt(2)`:

`1 = a/b sqrt(2)`

⇒ `sqrt(2) = b/a`

Note that a ≠ 0.

Step 3: Since a and b are integers, `b/a` is a rational number.

Therefore, this implies `sqrt(2)` is rational.

Step 4: But it is well-known and already proven that `sqrt(2)` is irrational.

This has been proven by contradiction, assuming `sqrt(2) = a/b` rational leads to the contradiction that both a and b share 2 as a common factor, which is impossible if the fraction is in the lowest terms.

Step 5: So the assumption that `1/sqrt(2)` is rational leads to a contradiction. 

Therefore, the initial assumption is false and `1/sqrt(2)` is an irrational number.