Complex numbers as pairs of reals

Summary

The complex numbers can be constructed as ordered pairs of real numbers. On $\mathbb{R}^2$, define addition by $(a, b) + (c, d) = (a + c, b + d)$ and multiplication by $(a, b) \cdot (c, d) = (a c - b d, a d + b c)$. With these operations, $\mathbb{R}^2$ is a field. It is the field $\mathbb{C}$ of complex numbers. The pair $(a, 0)$ is the real number $a$, and the pair $(0, 1)$ is the imaginary unit $i$. Then $(a, b) = a + b i$, and the map $a + b i \mapsto (a, b)$ is a bijection $\mathbb{C} \to \mathbb{R}^2$.

Remarks

This construction shows that the complex numbers exist, without the assumption of a mysterious square root of $-1$. The bijection with $\mathbb{R}^2$ respects addition and multiplication by real numbers. It is the base of the complex plane. Another construction is the quotient ring $\mathbb{R}[X]/(X^2 + 1)$.

Explanations

L1 human

A complex number is just a pair of real numbers with a special rule for multiplication. The rule is made so that $(0, 1) \cdot (0, 1) = (-1, 0)$, which says $i^2 = -1$.