Equivalence of Definitions of Complex Number

Theorem
The two forms of definition of a complex number:
 * The informal definition: the form $$a + b \imath$$ where $$\imath = \sqrt {-1}$$;
 * The formal definition: the ordered pair $$\left({x, y}\right)$$;

are equivalent.

Proof
Since:
 * $$\left({x_1, 0}\right) + \left({x_2, 0}\right) = \left({x_1 + x_2, 0}\right)$$
 * $$\left({x_1, 0}\right) \left({x_2, 0}\right) = \left({x_1 x_2, 0}\right)$$

we can identify a complex number $$\left({x_1, 0}\right)$$ with the real number $$x_1$$.

Specifically, we can define an isomorphism between the set of complex numbers of the form $$\left({x, 0}\right)$$ and the field of real numbers.

Now, we define $$\imath = \left({0, 1}\right)$$.

Then:

$$ $$

Finally, we see that:

$$ $$ $$ $$

Thus we can say that $$\imath = \sqrt {-1}$$.