Are the propreties of arithmetic unproven?

Question

For example, the property which says that $$a(b+c)=ab+ac$$ This is very clear for integers, but is it actually provable for all real numbers (and complex maybe). Or the commutative property which says $$a*b=b*a$$ This one is very clear again if you only have 2 expressions in your multiplication, but once you get to $a*b*c*d*e$ it is not so intuitive anymore. So are these fundamental tools of arithmetic provable for all real numbers? Thanks.

Answer 1

The fact that, for example, $a\times b=b\times a$ is, as you say, intuitively clear for the natural numbers $0,1,2,3,\dots$. But even this is subject to proof. In the various formalizations of the theory of the natural numbers, we have a constant symbol $0$, a unary function symbol $S$ (for successor), and two binary function symbols $+$ and $\times$. There are various axioms that you can look into by searching for Peano aritmetic. The axioms include things like $x+Sy=S(x+y)$ and $x\times Sy=(x\times y)+x$. Typically, they do not include $x\times y=y\times x$, because this can be proved from more basic axioms, using mathematical induction.

Next we turn to the integers, that is, the natural numbers already discussed, plus their "negatives." It is not entirely obvious that addition and multiplication can be extended to these, keeping familiar properties such as $x\times(y+z)=(x\times y)+(x\times z)$. But they can. It takes a while.

After that come the rationals. Again, these must be carefully defined. We cannot simply say they are ratios of integers, since before the rationals are defined, things like $\frac{2}{3}$ have no meaning. So we must find another way of defining the rationals. This has been done, and addition and multiplication on them has been carefully defined. Then one proves that addition and multiplication on the rationals, as just defined, has the right properties.

The leap to the reals is a huge one. There are various definitions of the reals, most of which were devised around $1870$. The basic operations of algebra on these newly constructed objects were defined, and their properties proved.

The details are usually covered in a university course in a subject called analysis. To appreciate the complexity, you might turn to the Wikipedia article on the definition of the real numbers, looking in particular at the sections on Cauchy sequences and on Dedekind cuts.

Answer 2

The short answer is Yes, the integers, real numbers, and complex are what are called commutative rings (in fact they are Fields). Your concern is well-founded. For example, we know from linear algebra that matrix multiplication doesn't necessarily commute (i.e $ a*b \ne b*a).$

To deal with this, mathematicians use a concept called a group. Informally, a group is a set of elements from some set $A$ (not necessarily numbers) that is equipped with a binary operation $*: A \times A \to A$ (closed under *). Note that $*$ need not necessarily be multiplication. Also, there must be a neutral element (identity) and each element in the set must have a unique inverse(i.e $a*b=1 \to a^{-1}=b$ for a neutral element 1 in the set).

So for example, let $T$ be the group $<S=\{colors\}, mix( \cdot )>$, where $mix(a,b)=a+b=c$ for some a,b,c in S. This is the additive group of colors of light. It is easy to see that mixing any two colors is also a color, so that color is in S. All colors mixed together give white, the neutral element. Can you see what the inverse of each color would be? Can you see how it is unique? Special groups that have commutative operations are called Abelian groups. Is the group above abelian?

It turns out groups can be given more structure by assuming another operation along with $*$ and defining how these new operations interact with the elements. These are called rings.

A quick run-through is here:http://www.csee.umbc.edu/portal/help/theory/group_def.shtml