Why is $|x|/x^2=1/|x|$

Question

This may be a stupid question but I was looking over a proof and one of the steps simplifies $|x|/x^2=1/|x|$ and I was wondering what the rigorous justification of that is. Is it because $x^2$ is essentially the same as $|x|^2$?

Answer 1

Yes!

For intuition, split into $2$ cases, $x$ positive and negative

Answer 2

If $x\in\mathbb R$, then by definition of absolute value:

• If $x>0$, then $|x|=x$, which implies $$|x|^2=x^2$$

• If $x<0$, then $|x|=-x$, which implies $$|x|^2=(-x)^2=x^2$$

So, you can deduce that $$|x|^2=x^2$$ holds for all $x\in\mathbb R.$

Thus,

$$\frac{|x|}{x^2}=\frac{|x|}{|x|^2}=\frac{1}{|x|}.$$