I'm relearning mathematics by reading through What is Mathematics?. It begins by explaining the natural numbers, introducing the following property:
$a.0=0$
To make $a$ the subject, I would divide both sides by zero, leaving:
$a=\frac{0}{0}$
I'm aware, but unsure why, that division by zero isn't allowed. I also look at that equation and read it to mean that any natural number $a$ is given by $\frac{0}{0}$, which can't be right. Moreover, cutting a non-existant cake into no pieces makes my head spin.
Is it correct to say that the mistake is the division by zero, or is there more to this than meets the eye?