I was trying to prove $|x||y| = |x\cdot y|$ but do not have a clue to start. I have seen examples of |x|+|y| >= |x+y| but could not translate it to my problem. Please my fellow math geniuses, help a brother out. Much appreciated
$\begingroup$ You can break it into four cases. For example: If x Commented Oct 9, 2014 at 7:33 $\begingroup$ what happens if x>= 0 and y<0, this shall complete my proof. $\endgroup$ Commented Oct 9, 2014 at 7:39
$\begingroup$ @user182042 If $x\geq 0$, then $|x|=x$, and since $y Commented Oct 9, 2014 at 7:44
$\begingroup$ Or also: $|x|=\sqrt\quad\forall x\in\mathbb\Rightarrow |x|\cdot|y|=\sqrt\cdot\sqrt=\sqrt=|x\cdot y|$ $\endgroup$
Commented Oct 9, 2014 at 7:46It is obvious that if either $x=0$ or $y=0$, then $xy=0$ and $|xy|=0$, but also $|x||y|=0$, so the statement holds. Now, what if both $x$ and $y$ are nonzero?
If $x$ and $y$ are real numbers, then, as David Peterson wrote in his comment, you only have to separate $4$ cases:
In each case, proving that $|x||y|=|xy|$ is simple.
If $x$ and $y$ are real numbers, then the easiest way is to write $x=r_1e^