Complex Number Operations Reference Cheat Sheet

Key Facts

• A complex number in standard form is written as $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.
• The imaginary unit satisfies $i^2 = -1$, so powers repeat in the cycle $i, -1, -i, 1$.
• To add complex numbers, combine like parts: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• To subtract complex numbers, distribute the negative sign: $(a + bi) - (c + di) = (a - c) + (b - d)i$.
• To multiply complex numbers, use distribution and replace $i^2$ with $-1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• The complex conjugate of $z = a + bi$ is $\overline{z} = a - bi$, and $z\overline{z} = a^2 + b^2$.
• To divide complex numbers, multiply by the conjugate of the denominator: $\frac{a + bi}{c + di} \cdot \frac{c - di}{c - di}$.
• The modulus of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$, and the polar form is $z = r(\cos \theta + i\sin \theta)$.

Vocabulary

Complex number

A number of the form $a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$.

Real part

The real part of $z = a + bi$ is $a$, written as $\operatorname{Re}(z) = a$.

Imaginary part

The imaginary part of $z = a + bi$ is $b$, written as $\operatorname{Im}(z) = b$.

Complex conjugate

The conjugate of $a + bi$ is $a - bi$, which changes the sign of the imaginary part.

Modulus

The modulus of $z = a + bi$ is the distance from the origin to the point $(a, b)$, given by $|z| = \sqrt{a^2 + b^2}$.

Argument

The argument of a complex number is the angle $\theta$ it makes with the positive real axis in the complex plane.