Introduction to Complex Numbers

The Imaginary Unit

The imaginary unit $i$ is defined as: $i = \sqrt{-1}$

Therefore: $i^2 = -1$

Complex Numbers

A complex number has the form: $a + bi$

where:

• $a$ = real part
• $b$ = imaginary part
• $i$ = imaginary unit

Example: $3 + 4i$

Powers of $i$

Pattern repeats every 4:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$
• $i^5 = i$ (pattern repeats)

Adding and Subtracting

Combine like terms (real with real, imaginary with imaginary):

$(3 + 4i) + (2 - i) = (3 + 2) + (4i - i) = 5 + 3i$

Multiplying

Use FOIL and remember $i^2 = -1$:

$(2 + 3i)(1 + 4i)$ $= 2 + 8i + 3i + 12i^2$ $= 2 + 11i + 12(-1)$ $= 2 + 11i - 12$ $= -10 + 11i$

Complex Conjugates

The conjugate of $a + bi$ is $a - bi$.

Property: $(a + bi)(a - bi) = a^2 + b^2$ (always real!)