Why does a/b have to be in simplest form in the proof of irrationality for sqrt2

Question

The proof of the irrationality of $\sqrt{2}$ starts with the supposition that $\sqrt{2} = \frac ab$ where $a$ and $b$ are integers. I understand that, but why is it important that $\frac ab$ is expressed in simplest terms? I see that it is a major part of the contradiction, but why?

EDIT: I see that having the fraction be irreducible means it is unique. Why does it have to be unique though?

Answer 1

If they aren't in lowest terms, then it's possible that both $a^2$ and $b^2$ are even, while avoiding that possibility is the heart of the contradiction.

Answer 2

The importance of $\gcd(a,b)=1$ depends on the proof you are following.

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Assuming $\sqrt{2}=\frac{a}{b}$ with $a,b\in\mathbb{N}^+$ we get $$ a^2 = 2b^2 $$ but this is impossible since $\nu_2(a^2)$ is even while $\nu_2(2b^2)$ is odd.

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Here $\gcd(a,b)=1$ is irrelevant, for instance.
For any $n\in\mathbb{N}^+$, $\nu_2(n)$ is defined as $\max\{m\in\mathbb{N}:2^m\mid n\}$.

Answer 3

Depending how you want to do the proof, it doesn’t have to be in lowest terms. Just suppose $a$ and $b$ are both positive. If $\frac{a}{b}=\sqrt2$, then $a$ and $b$ are both even, by the usual argument. Say $a=2a’$ and $b=2b’$. Thus, $\sqrt2=\frac{a}{b}=\frac{a’}{b’}$, with $a’<a$ and $b’<b$.

Now, by the same argument, $a’$ and $b’$ are both even, so we reduce again, and obtain another, smaller pair. Continuing in this way, we obtain two infinitely descending sequences of positive integers, which is impossible. That’s your contradiction, and this method is called “infinite descent”.

Answer 4

It's usually just a time saver and an assurence that an infinite regress is a contradiction.

If $\frac ab$ is such that $(\frac ab)^2 =2$ then $a^2 = 2b^2$ so $2|a^2$ and so $2|a$. Let $a = 2a_1$ so $2a_1^2 = b^2$ so $2|b$. Let $b = 2b_1$ and $a_1^2 =2b_1^2$ so $a_1$ is divisible by $2$. Let $a_1 = 2a_2$. and so on ad infinitim.

Well... so what is wrong with ad infinitim? And how do we know that just because we got the same result $3$ times we will get it forever? And why is that a contradiction?

For that matter why is that all rational numbers actually have "lowest terms"?

Those all actually have answers and the pertain to the well-ordering principle and they are not trivial (although they aren't hard either).

We have a series of $a > a_1 > a_2 > ...$ so that $a_i = 2a_{i+1}$ as we as $b > b_1 > b_2 > ....$ so that $b_i = 2b_{i+1}$. But the induction principle in natural numbers we know these $a_i, b_i$ exist for all $i \in \mathbb N$ so that $a_i^2 = 2b_i^2$ and $b_i^2 = 2a_{i+1}^2$.

But by the well-ordering principle, we know that for such collections of $a_i$ and of $b_i$ that they must each have a least element.

And that is a contradiction.

But that involved a bit of a diversion that we feel simply distracts the student. Surely it's easier to just avoid the issue by saying "Oh, we can assume $\frac ab$ is in lowest terms".

But actually knowing that if $a,b \in \mathbb Z, b\ne 0$ that there must be some $a',b'$ so so that $\gcd(a',b') = 1$ and $\frac ab = \frac {a'}{b'}$ requires a similar well-ordering principle argument.