Why we square while doing the proof of √2 is irrational?

Question

When we prove that $\sqrt 2$ is irrational by the method of contradiction, we assume $\sqrt 2$ is a rational number:

$\sqrt 2 = a/b$

Squaring both sides, $2 = a^2/b^2$.

Here is my question: is there another way to derive a contradiction without squaring?

Answer 1

We square to be able to use the properties of integers instead of fractions and irrational numbers. Let us see how this plays into the proof that $\sqrt{2}$ is irrational:

Proof by contradiction - Suppose $\sqrt{2}$ is rational and thus can be written as a fraction $\frac{a}{b}.$ We can square both sides to get $2 = \frac{a^2}{b^2}.$ Rearranging, we have that $a^2 = 2b^2.$ But we see that no perfect square could be double another perfect square because one of the integers $a$ or $b$ would have an odd power of $2.$ Thus, $\sqrt{2}$ is irrational.

As you can see, squaring allows us to use properties of integers to complete the task.

Answer 2

If you don't square, all you can write is

$$\sqrt2=\frac pq,$$ which leads you nowhere. Squaring offers much better opportunities.