Prove that $\frac aa = 1$ when $a\neq 0$.

Question

We know the basic "a number $a$ divided by itself, i.e. $\frac aa$, is equal to $1.$ Except when $a=0$.

But why? How to prove this using axioms?

Answer 1

One axiom common to many systems is the multiplicative identity, $$x\cdot 1=x$$ Even in systems without a multiplicative inverse (such as integers), you can define "division" as $$\frac{a}{b}=c\iff b\cdot c=a$$ Note that this $c$ may not always exist for all pairs $a,b$ depending on your system. Working with these two assumptions, you can prove that since $x\cdot 1=x$ by the axiom, applying the definition of division with $a=x$, $b=x$, $c=1$ yields $$x\cdot 1=x\implies \frac{x}{x}=1$$ Additionally, note that $c$ may not be unique for each $a,b$ either - technically, under this definition you can show $\frac{0}{0}$ to be anything.