# 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 *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

p/q + r/s = (ps + rq)/(qs)

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,

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_{1},y_{1}) + (x_{2},y_{2})=(x_{1} + y_{1},x_{2} + y_{2})

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_{1},y_{1}) + (x_{2},y_{2})=(x_{1} + y_{1},x_{2} + y_{2})=(x_{2} + y_{2},x_{1} + y_{1})=(x_{2},y_{2}) + (x_{1},y_{1})

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