Why does multiplication always have the associative and commutative property?

Question

We all know that Commutative property and Associative property of multiplication is always saved for the real and complex numbers. I know that if I recalculate it a million times, the result will be the same. But why does it work?

Is there an explanation for this, or is it just a coincidence that later made an axiom?

I am not an expert in mathematics, my level of knowledge is high school. Thank for you answer.

Answer 1

The standard system of axioms for arithmetic is the Peano axioms, which are defined in terms of the number $0$ and a 'successor' function: that is, a function that adds $1$ to a natural number, to give the next (or successive) natural number.

Using these axioms, associativity and commutativity of multiplication are theorems that can be proved. See On the commutative property of multiplication (domain of integers, possibly reals) or http://math.ucsd.edu/~nwallach/peano.pdf.

Answer 2

It is a theorem once we make a formal definition and can prove the property from the definition. Ultimately, these properties for complex numbers $\Bbb C$, are inherited from the properties for $\Bbb R$, which are inherited from them for $\Bbb Q$, then $\Bbb Z$ and in the end $\Bbb N$, whose properties we try to grasp with the Peano axioms.

The proofs of commutativity an associativity in $\Bbb N$ from a recursive definition using only the constant $0$ and the successor function $S$ a la $ n\cdot 0:=0$, $n\cdot Sm:=n\cdot m+n$ (and also $n+0:=n$, $n+Sm:=S(n+m)$) are somewhat technical (and perhaps surprisingly long if one really starts ab ovo) and it seems like a giant portion of good luck that we obtain such nice properties in the end.

So let's go back to a suitable motivation of multiplication: If we arrange pebbles in a rectangular grid of $n$ rows and $m$ columns, then the number of pebbles does not change if we look at the rectangle from a different point so that it appears as $m$ rows and $n$ columns. Hence if multiplication is to mimic the operation of "number of pebbles in a rectangle", then commutativity of multiplication is evident.

For associativity, consider a three-dimensional equivalent arranged in a cuboid grid $n$ long, $m$ wide, and $k$ high. We could rearrange the vertical columns in a line, then this line is $nm$ long and we obtain a (vertical) rectangle of $(n\cdot m)\cdot k$ pebbles. If we first rotate the cuboid to make $n$ the vertical extension, we arrive at $n\cdot (m\cdot k)$ pebbles. So in the end, via the "pebbles in a rectangle" definition of multiplication, commutativity and associativity are consequences of spatial symmetries and of the count of objects being invariant under movement in (abstract) three-dimensional space.