Equivalence of Definitions of Complex Number

Theorem
The following definitions of a complex number 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 (definition 2) $\left({x_1, 0}\right)$ with the real number $x_1$.

Specifically, we can define an isomorphism between the set of complex numbers (definition 2) 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 $i = \sqrt {-1}$.