Equivalence of Definitions of Complex Number

Theorem
The two forms of definition of a complex number:
 * The informal definition: the form $a + b i$ where $i = \sqrt {-1}$
 * 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 $i = \left({0, 1}\right)$.

Then:

Finally, we see that:

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

Comment
We have not proven that $i$ is the unique square root of $-1$.

In fact, $-i = \left({0,-1}\right)$ is also a square root.

However, $i$ can be defined as the principal square root of $-1$.