Explaining elementary arithmetic in terms of group theory

It is possible to explain elementary arithmetic in terms of group theory?

Addition and subtraction seem to be fine using $(\mathbb{R},+)$ but when it comes to multiplication and division it does not seem to fit as we can multiply by zero in arithmetic, but $0$ is never an element in a set when we define a group with operation $*$.

I had hoped that by explaining what a group was, it would then be possible to say "look, see how all of this (arithmetic $+,-,*,/$) can just be explained in terms of inverses, identity elements, associativity and commutativity".

Are there other abstractions that allow for this?

• I think what you mean is that $\mathbb Q$ is a field. And yes all properties of elementary arithmetic are explained by the field axioms. – lhf Jul 24 '15 at 18:58
• You may be looking for monoids, which are essentially to groups what rings are to fields: the requirement for inverses is dropped. – fkraiem Jul 24 '15 at 18:59
• For example $(\mathbf{R},\times)$ is monoid, but not a group (as is any ring under its multiplicative law). – fkraiem Jul 24 '15 at 19:07

A group has one operation. Ordinary arithmetic has two - addition and multiplication.

If you know all about groups you know all about addition (and subtraction). If your numbers are real or rational the group axioms tell you all about multiplication (and division) of the numbers that aren't zero. (If your numbers are just the integers then you don't have inverses for multiplication.)

Just knowing about groups can't get you all of ordinary arithmetic - you need one more thing: the axiom that connects addition and multiplication. It's the distributive law: $$a \times (b + c) = (a \times b) + (b \times c).$$

With this information you can in fact prove that $0$ times anything is $0$.

The abstraction that you're looking for is a field: two group operations like $+$ and $\times$ connected by the distributive law.

In addition to Ethan's excellent answer, I want to point out that the more of "elementary arithmetic" in $\mathbb R$ you seek to explain, the more specific an abstraction you need.

Patterns among inequalities, such as $a<b\implies a+c<b+c$, are expressed by saying that $\mathbb R$ is an ordered field.

If you want to understand why long division works and why every number has a decimal representation, you need even more: that $\mathbb R$ is an Archimedean field.

Finally, if you want to understand why every decimal represents a real number, you need the completeness axiom. At this point, we have become so specific that $\mathbb R$ is the only example!