Arithmetic with complex numbers

What a complex number is

• Some equations, like $x^2 = -1$, have no real solution, because no real number squares to a negative. To solve them, mathematicians defined a new number:

$i = \sqrt{-1}, \qquad \text{so that } i^2 = -1$

• A complex number has the form $z = a + bi$, where $a$ and $b$ are real numbers.
  • $a$ is the real part, written $\text{Re}(z)$.
  • $b$ is the imaginary part, written $\text{Im}(z)$ (note $\text{Im}(z)$ is the number $b$, not $bi$).
• A real number is just a complex number with $b = 0$; a purely imaginary number has $a = 0$.

Adding, subtracting and multiplying

• Add or subtract by combining like parts — real with real, imaginary with imaginary.
• Multiply by expanding the brackets like normal algebra, then replace $i^2$ with $-1$.

$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

The conjugate

• The conjugate of $z = a + bi$ is $\bar{z} = a - bi$ — the sign of the imaginary part is flipped.
• Multiplying a complex number by its conjugate always gives a real number:

$(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2$

Dividing complex numbers

• To divide, multiply the top and bottom by the conjugate of the denominator. This makes the denominator real, so you can split into real and imaginary parts.