Equivalence of Definitions of Complex Number

Proof
Since:
 * $\tuple {x_1, 0} + \tuple {x_2, 0} = \tuple {x_1 + x_2, 0}$
 * $\tuple {x_1, 0} \tuple {x_2, 0} = \tuple {x_1 x_2, 0}$

we can identify a complex number (definition 2) $\tuple {x_1, 0}$ with the real number $x_1$.

Specifically, we can define an isomorphism between the set of complex numbers (definition 2) of the form $\tuple {x, 0}$ and the field of real numbers.

Now, we define $i = \tuple {0, 1}$.

Then:

Finally, we see that:

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