Complex Numbers & Operations Cheat Sheet

Key Facts

• A complex number in standard form is $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.
• The imaginary unit is defined by $i = \sqrt{-1}$ and $i^2 = -1$.
• Powers of $i$ repeat in a cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$.
• To add or subtract complex numbers, combine real parts and imaginary parts: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• To multiply complex numbers, distribute and use $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• The complex conjugate of $z = a + bi$ is $\overline{z} = a - bi$.
• The modulus of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.
• To divide complex numbers, multiply by the conjugate of the denominator: $\frac{a + bi}{c + di} \cdot \frac{c - di}{c - di}$.

Vocabulary

Complex Number

A number that can be written as $a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$.

Imaginary Unit

The number $i$ defined by $i = \sqrt{-1}$, so $i^2 = -1$.

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

Modulus

The modulus $|z|$ is the distance from the origin to the point $z = a + bi$ on the Argand plane.