Understanding the proof of "$\sqrt{2}$ is irrational" by contradiction.

Question

I have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here )

This is the snapshot of it:

The proof starts with assuming that $\sqrt{2}$ can be written as a ratio of two integers and then that this fraction can be reduced to its lowest terms i.e. $\sqrt{2}=\dfrac ab$, where gcd(a,b)=1 . Then at last we reach at the contradiction that gcd$(a,b)\neq1$. Then they say that because of this contradiction $\sqrt{2}$ cannot be a rational number.

What I do not understand is that how the contradiction proves that $\sqrt{2}$ cannot be a rational number. The contradiction only proves that $\sqrt{2}$ cannot be written as the ratio of two coprime numbers. But can't we write $\sqrt{2}$ as the ratio of two non-coprime numbers?

Let us consider two statements, X and Y as:

X: $\sqrt{2}$ cannot be written as the ratio of two coprime numbers.

Y: $\sqrt{2}$ cannot be written as the ratio of two non-coprime numbers.

The contradiction proves only the statement X not the statement Y.

I guess that we can prove statement Y from X as: Let us suppose that $\sqrt{2}$ can be written as the ratio of two non-coprime numbers, i.e. $\sqrt{2}=\dfrac RS$, where $R$ and $S$ are mutually non-coprime. But every rational number can be written as a fraction in lowest terms. So let's say $\dfrac RS$ in its lowest terms is $\dfrac rs$, but this means that $\sqrt{2}$ is also equal to $\dfrac rs$, where $r$ and $s$ are coprime. This eventually contradicts the statement X, hence by contradiction $\sqrt{2}$ cannot be written as the ratio of two non-coprime numbers, or the statement Y is true.

Question:

1. Did I prove the statement Y from X correctly ?

2. Why does the book directly mention "$\sqrt{2}$ is irrational" without justifying statement Y? Is the justification too trivial to be mentioned?

3. Is there any other way than mine(proof by contradiction) to deduce Y from X?

I only want to clarify these three doubts, nothing else.

Answer 1

What is a rational number? The definition I use is that it's a number that can be written as a ratio of two integers. If your book doesn't say explicitly that the integers can be coprime, it's because the notion of a reduced fraction is fundamental to working with fractions. You're not allowed to work with fractions until you believe that they can be written in a unique reduced form, and that your numerator and denominator will be coprime.

If you wanted to, you can prove this with the fundamental theorem of arithmetic. Suppose your rational number $q$ can be written as the ratio of integers $\frac{a}{b}$. Then by the fundamental theorem of arithmetic, $a$ and $b$ have unique factorizations into primes. So factorize them, and the factors that appear in both will cancel. Then you have a coprime numerator and denominator.

In the proof that you're trying, I don't see the need for contradiction. Statement X and Y imply each other. If you have a ratio of non-coprime numbers, reduce it. If you have a ratio of coprime numbers, multiply them both by 2 and now you have a ratio of non-coprime numbers. (The statements you have written are the contrapositive of this.)

Answer 2

Let $\sqrt2 = a/b$ where $a$ and $b$ are not co-prime, that is they have a common divisor greater than or equal to $2$, so let $d = gcd(a, b)$, $a’=a/d$, $b’=b/d$. Now $a’/b’ = a/b = \sqrt2$, and $a’$, $b’$ are co-prime.