TITLE: Prove that zero multiplied by zero is equal to zero.
QUESTION [19 upvotes]: This is my proof:
So,  $0\cdot0=0$
And we know that
$a-a=0$
By substitution,
We have $(a-a)(a-a)=0$
Then by simplifying, $a^2-a^2+a^2-a^2=0$
and the we have $0-0=0$,
Therefore, $0=0$.
I am not sure about my answer. Will you please show me another way of proving it or some way to improve my answer? Thank you!

REPLY [2 votes]: If we take real numbers: 
$0\cdot0 = (0 + 0) \cdot 0\qquad$ ($0 = 0+0$ because $0$ is the additive zero element)
$\qquad= 0\cdot0 + 0\cdot0\qquad$ (distributive law, $(a+b)\cdot c = a\cdot c + b\cdot c$)
Since $x = x+a$ with $x = 0\cdot0$ and $a = 0\cdot0$, this makes $0\cdot0$ the additive zero.