I have seen the proof that $\sqrt2$ is irrational However I'm a bit confused. The proof goes something like this. first assume $m$ and $n$ are integers such that $$ \frac{m}{n} = \sqrt{2} $$ $$ \frac{m^{2}}{n^{2}} = 2 $$ $$m^{2} = 2 n^{2}$$ $\implies m^{2} $ is divisible by $2 \implies m$ is divisible by 2. Let $2q = m$. Then $$m^{2} = 4 q^{2} = 2n^{2}$$ $$ n^{2} = 2 q^{2}$$ This means that $n^{2}$ is divisible by $2 \implies n$ is divisible by $2$...... Hence $\sqrt{2}$ is irrational. $$$$If $ m^{2} $ is divisible by $2$ how do we know that $m$ is also divisible by $2$? If you say it's a prime number then I can argue that I didn't need to do all these steps. I could have just said that its factors don't multiply to $2$ hence $\sqrt{2}$ is irrational
$\endgroup$ 35 Answers
$\begingroup$Suppose that $m$ is odd, then $m=2k+1$, and thus $$m^2=(2k+1)^2=2(2k^2+k)+1$$ and thus $m^2$ is odd too. That prove that if $m^2$ is even then $m$ is even too.
You can generalize this result : If a prime number $p\mid m^s$ for $s\in\mathbb N$ then $p\mid m$. Indeed, let $m=p_1^{\alpha_1}...p_n^{\alpha_n}$ the decomposition of $m$ in prime factor. Then $$m^s=p_1^{s\alpha_1}...p_n^{s\alpha_n}$$ and thus all prime that divide $m^s$ also divide $m$.
$\endgroup$ $\begingroup$To prove that if $m^2$ is even then $m$ is even, we proceed by contradiction. So suppose that $m$ is odd but $m^2$ is even. Then $m=2k+1$, so $m^2=4k^2 + 4k+1$. In particular, $m^2=4k(k+1)+1$ is one more than an even number, which is a contradiction.
$\endgroup$ $\begingroup$As the answer above put it, you can get it by contradiction.
If you want to deal with the square root of a general prime $p$, you can use the fundamental theorem of arithmetic: if $m^2$ is divisible by $p$, then it must have a prime factor of $p$, and all prime factors of $m^2$ are also prime factors of $p$.
To answer your last point: it is possible for a number to be a prime and still a product of two rational numbers $(3 = \tfrac{2}{1}\frac{3}{2})$. The square roots are irrational, because you impose the condition that it must be the product of two equal numbers.
$\endgroup$ $\begingroup$$\frac{m^2}{n^2}=2\implies m^2=2n^2\implies 2\mid m\implies 2\mid n \implies 2^2\mid m\implies 2^3\mid n$ etc... so that $m$ and $n$ are each divisible by arbitrarily many $2$'s, an absurdity...
Or, assume $\frac mn$ is reduced, then you get a contradiction when they are both divisible by $2$...
$\endgroup$ $\begingroup$Your entire proof is based on the assumption that m and n are integers. Till this assumption holds true. There is no fallacy in the argument.
But, Lets say m and n can be anything. That is, we don't assume anything for m andn. Then, your statement
m^2 is divisible by 2 => m is divisible by 2
no longer holds true. for eg: m=k*sqrt(2). Then m is not divisible by 2. But m^2=2*k^2 is divisible by m.
Hence, by contradiction sqrt(2) is irrartional.
$\endgroup$ 2
