TITLE: Why $|x|$ is not rational expression?
QUESTION [11 upvotes]: I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. One of my friends said that it is provable by use of differential, but I don't know about calculus. Is there any proof for this fact, without use of calculus?

REPLY [14 votes]: If there were polynomials $p$ and $q$ such that $|x|=\frac{p(x)}{q(x)}$ for all $x\in\Bbb R$, then $\frac{p(x)}{q(x)}$ would be defined for all $x\in\Bbb R$, and $q(x)$ could never be $0$. Moreover, we’d have $p(x)=|x|q(x)$ for all $x\in\Bbb R$. If $x\ge 0$, this means that $p(x)=xq(x)$, and if $x<0$ it means that $p(x)=-xq(x)$.
Let $r(x)=p(x)-xq(x)$; this is certainly a polynomial, and
$$r(x)=\begin{cases}
0,&\text{if }x\ge 0\\
-2xq(x),&\text{if }x<0\;.
\end{cases}$$
I’m going to assume that you know that a polynomial of degree $n\ge 1$ has at most $n$ real roots, though you’ve probably never seen a proof. Our supposed polynomial $r(x)$ evidently has infinitely many real roots, since each non-negative real is a root, so it must be the constant function $r(x)\equiv 0$. But then $-2xq(x)=0$ for each $x<0$, and it follows that $q(x)=0$ for each $x<0$. We saw at the beginning that this is impossible: $q(x)$ can never be $0$. This contradiction shows that in fact no such polynomials $p$ and $q$ exist, and $|x|$ is not a rational function.