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.