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There is the proposition

Every integer $n$ can be represented uniquely as the product $2^km$ where $m$ is odd and $k\ge0$.

I have to use this to prove that $\sqrt{2}$ is irrational. As is it usual, assume the contrapositive

$$ \sqrt{2}=a/b, $$

where $a,b$ are integers and $b\neq 0$. Therefore

$$ a^2=2b^2. $$

Using the proposition, $b^2=2^jn$, with $n$ odd and $j\ge0$. Therefore $a^2$ is even (what the hell do I need this, since $a^2$ is clearly even, but I have to use the proposition...). Then I can continue in the usual way and finish the proof.

At this point, this proof seems very lame. Any ideas how to improve it?

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    $\begingroup$ Write $a$ and $b$ in that form then compare the parity of the exponents of $2$ in the resultant $2$-unique factorization of both sides $\endgroup$ – Bill Dubuque Jul 26 '17 at 2:55
  • $\begingroup$ If I do that I get, say $2^{2k}m^2=2^{2j+1}n^2$. I can determine that $n=m$. On the other hand, it says that $k\ge1$. Is this the punchline? $\endgroup$ – user2820579 Jul 26 '17 at 3:01
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    $\begingroup$ Hint: $m,n\,$ odd $\,\Rightarrow\, m^2$ and $n^2$ odd, so both sides are in said unique form, so $\,2k = 2j+1,\,$ contradiction. $\endgroup$ – Bill Dubuque Jul 26 '17 at 3:02
  • $\begingroup$ You are totally right. Thanks a lot! $\endgroup$ – user2820579 Jul 26 '17 at 3:04
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A easy way to show $\sqrt{2}$ is not a rational number:

Suppose $\sqrt{2}$ is rational. note that $\sqrt{2}>1$, so w.l.o.g. I can take $a,b\in \mathbb{N}$ such that $\sqrt{2}=\frac{a}{b}$

Then $a^2=2b^2$. Note that $a^2\equiv 0,1(\mod 3)$

Case 1:

If $a^2\equiv 1(\mod 3)$, then $b^2\equiv 1(\mod 3)\Rightarrow 2b^2\equiv 2(\mod 3)$, but $a^2=2b^2$, then $a^2\equiv 2(\mod 3)\Rightarrow 1\equiv 2(\mod 3)$, which is impossible.

Case 2:

If $a^2\equiv 0(\mod 3)$, then $b^2\equiv 0(\mod 3)$, then $a=3c$, $b=3d$ where $c,d \in \mathbb{N}$.

Now we again get $c^2=2d^2$, since $9c^2=2\times9d^2$. This lead to a loop. The loop is we can not go to case 1(since that case is impossible). Then we have stay in case 2, which will lead to $3\mid c$ and $3\mid d$ and this process will continue until we end up to $1=2$ which is again impossible.

So there does not exist $a,b\in \mathbb{N}$ such that $a^2=2b^2$. Hence $\sqrt{2}$ is not a rational number$.\space\space\space\blacksquare$

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  • $\begingroup$ Why do we have $a^2 \equiv 1, 0 (\text{mod 3})$? $\endgroup$ – Pablo S. Ocal Jul 26 '17 at 17:54
  • $\begingroup$ @galahad Since $a\equiv 0,1,2 (\mod 3)$, $a^2\equiv 0,1,4 (\mod 3)$. Now $4\equiv 1 (\mod 3)$., then $a^2\equiv 0,1 (\mod 3)$ $\endgroup$ – MAN-MADE Jul 26 '17 at 18:11
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Try applying the Proposition before squaring: consider $\sqrt{2} = a/b$ an irreducible fraction, write $a = 2^i n$ and $b = 2^j m$. We have multiple cases:

  • If $n=1$ or $m=1$, then respectively we must have $j=0$ or $i=0$ because of $a/b$ being irreducible, then (squaring the original equality) respectively $m^2=4^i/2$ or $n^2 = 2\cdot 4^j m^2$. Thus either $m^2$ is even or $n^2$ is even, but $m$ and $n$ are odd and the square of an odd number is an odd number, a contradiction.

Assume $n\neq 1 \neq m$. Since $a/b$ is reducible, we must have either $i=0$ or $j=0$.

  • If $i=0$ (notice that this contains the case $j = 0 = i$), squaring we obtain that:

$$2 = \frac{n^2}{4^j m^2} \text{ so } 2\cdot 4^j m^2 = n^2 \text{ is even, which is again a contradiction because $n$ is odd.}$$

  • If $j=0$ and $i\neq 0$, squaring we obtain that:

$$2 = \frac{4^i n^2}{m^2} \text{ so } m^2 = \frac{4^i n^2}{2} \text{ is even ($i>0$), which is again a contradiction because $n$ is odd.}$$

Notice how the fact that $n$ and $m$ can be $1$ forces you to consider extra cases (alternatively, you could say that the integers do not have multiplicative inverses and include the first case in the other two).

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