Lesson 12.3: Complex Numbers

Property

Complex Number Standard Form:

A complex number is written as $a + bi$, where:

• $a$ is the real part
• $b$ is the imaginary part (coefficient of $i$)
• $i$ is the imaginary unit where $i^2 = -1$

Property

For any complex number $z = a + bi$ where $a, b \in \mathbb{R}$:

• $z$ is real if $b = 0$
• $z$ is purely imaginary if $a = 0$ and $b \neq 0$
• $z$ is complex (neither real nor purely imaginary) if $a \neq 0$ and $b \neq 0$

Examples

Property

Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form, $a+bi$.

Examples

• Simplify $(5 - 2i) + (3 + 8i)$: $(5 + 3) + (-2 + 8)i = 8 + 6i$.

• Simplify $(7 - 6i) - (4 - 2i)$: $7 - 6i - 4 + 2i = (7 - 4) + (-6 + 2)i = 3 - 4i$.