Addition of Numbers

Since there many different kinds of numbers there are different kinds of additions. For every set of numbers we have to define an operation that satisfies the axioms of addition.

We have to start somewhere, and I shall assume that we know how to add (and subtract, actually also multiply) integers Z = {...,-2, -1, 0, 1, 2, ...}. But note that it's possible to derive all the properties of integer addition from a more basic, unary operation of passing to the next integer. The axioms for the next operation are known as the axioms of Peano (Giuseppe, 1858-1932). It's the same Peano who invented the space filling curves.

For rational numbers, addition is defined with

and we can verify that this operation satisfies axioms 1-4. First of all, recollect that representation p/q and (np)/(nq) define the same rational number. Consider then the number 0/q. By definition, 0/q+r/s=(rq)/(qs)=r/s. Similarly, r/s+0/q=r/s. So that 0/q takes the role of zero. It's next shown that numbers p/1 can be identified with integers. Therefore, there is no confusion between the "integer" and "rational" zeros. They are one and the same number. Associativity and commutativity are verified in a straightforward manner.

For real numbers addition can be extended "by continuity" which is not difficult but will sway us from the topic unnecessarily. Perhaps I'll return to this at a later date.

Complex numbers are pairs (x,y) of real numbers. Addition is defined componentwise:

With this definition (x,y)+(0,0)=(0,0)+(x,y)=(x,y) so that (0,0) takes the role of zero. Since real numbers x can be identified with pairs (x,0) there again couldn't possibly arise confusion between the "rational" and "complex" zeros. Both are denoted 0. Associativity (commutativity) of the complex addition is implied by associativity (commutativity) of the real addition. For example,