Equivalence of Complex Number Definitions

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:

	\(\displaystyle \)	\(\displaystyle x + i y\)	\(=\)	\(\displaystyle \left({x, 0}\right) + \left({0, 1}\right) \left({y, 0}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x, y}\right)\)	\(\displaystyle \)	by the definition of complex addition and multiplication

Finally, we see that:

	\(\displaystyle \)	\(\displaystyle i^2\)	\(=\)	\(\displaystyle \left({0, 1}\right) \left({0, 1}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({0 \cdot 0 - 1 \cdot 1, 0\cdot 1 + 1 \cdot 0}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1, 0}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -1\)	\(\displaystyle \)

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

$\blacksquare$