Real, Imaginary, and Complex Numbers

The imaginary unit $i$ is defined by $i^2 = -1$. A complex number is anything of the form $z = a + bi$, where $a$ is called the real part and $b$ the imaginary part.

• If $b = 0$, $z$ is just a real number, like $5$.
• If $a = 0$ and $b \neq 0$, $z$ is purely imaginary, like $3i$.
• Otherwise, like $2 + 7i$, it's a genuine complex number.

Every real number is also a complex number (with imaginary part zero) — the reals sit inside the complex numbers, not alongside them.