axiomatic sets 39 (complex numbers)

We shall define complex numbers from real numbers in set theory.

Let $(a,b)$ be a pair of two real numbers $a,b\in\mathbb{R}$.

Then, we decide four properties on $C=\left\{(a,b):a,b\in \mathbb{R} \right\}$ ;

(1)$(a,b)+(c,d)=(a+c,b+d)$ ,
(2)$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$ ,
(3)for any real number $k\in\mathbb{R}$ , $k(a,b)=(ka,kb)$ ,
(4)if $(a,b)\ne (0,0)$ , then $\frac{1}{(a,b)}=\left( \frac{a}{a^2+b^2},\frac{-b}{a^2+b^2} \right)$ ,

We call the pairs having above properties complex numbers $\mathbb{C}$ .

You must understand immediately $(0,1)\cdot (0,1)=(-1,0)$ ,
$(a,b)\cdot \frac{1}{(a,b)}=(1,0)$ , $(1,0)\cdot (a,0)=(a,0)$ ,
$(1,0)\cdot (0,b)=(0,b)$ , $(0,1)\cdot (a,0)=(0,a)$,
$(0,1)\cdot(0,a)=(-a,0)$ ,
and as we usually write $(0,1)=i$ , then $i^2=i\cdot i=(-1,0)$ .

And $(a,0)$ is fully corresponding to a real number $a$ .
Thus $(0,0)=0, (1,0)=1,(-1,0)=-1,\cdots$ and so on ,
and we also write $a+bi=(a,0)+(0,b)=(a,b)$

Complex numbers are the pair of two real numbers having some properties.
The set $\mathbb{C}$ does not include the set $\mathbb{R}$ formally
same as the relation of natural numbers and integers.
But the fact that $a$ is fully corresponding to $(a,0)$
makes us accept $\mathbb{R}\subset\mathbb{C}$ .