Imaginary & Complex Numbers

What is an Imaginary Number?

The imaginary unit is defined as:

i = √(−1), so i² = −1

This lets us take square roots of negative numbers:

√(−4) = 2i
√(−9) = 3i
√(−25) = 5i

Complex Numbers

A complex number has the form a + bi, where:

• a = real part (Re)
• b = imaginary part (Im)
• i = imaginary unit (i² = −1)

Examples:

3 + 4i   →  Re = 3,  Im = 4
5 − 2i   →  Re = 5,  Im = −2
7        →  Re = 7,  Im = 0  (purely real)
3i       →  Re = 0,  Im = 3  (purely imaginary)

Adding and Subtracting Complex Numbers

Combine real parts and imaginary parts separately:

(a + bi) + (c + di) = (a + c) + (b + d)i

Example: (3 + 4i) + (1 + 2i)

Real:      3 + 1 = 4
Imaginary: 4 + 2 = 6
Result:    4 + 6i

Example: (5 + 3i) − (2 + i)

Real:      5 − 2 = 3
Imaginary: 3 − 1 = 2
Result:    3 + 2i

Multiplying Complex Numbers

Use FOIL, remembering that i² = −1:

(a + bi)(c + di) = ac + adi + bci + bdi² = ac + (ad + bc)i + bd(−1) = (ac − bd) + (ad + bc)i

Example: (2 + 3i)(1 + 4i)

= 2·1 + 2·4i + 3i·1 + 3i·4i
= 2 + 8i + 3i + 12i²
= 2 + 11i + 12(−1)
= (2 − 12) + 11i
= −10 + 11i

The Conjugate

The conjugate of a + bi is a − bi (flip the sign of the imaginary part).

Key property: (a + bi)(a − bi) = a² + b² (always a real number!)

Example: (3 + 4i)(3 − 4i) = 3² + 4² = 9 + 16 = 25

Conjugates are used to divide complex numbers.

Magnitude (Modulus)

The magnitude (or modulus) of a + bi is:

|a + bi| = √(a² + b²)

This is the distance from the origin to the point (a, b) in the complex plane — the Pythagorean theorem!

Example: |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5

Common magnitudes (Pythagorean triples):

|3 + 4i|  = 5
|5 + 12i| = 13
|8 + 6i|  = 10
|7 + 24i| = 25

Powers of i

The powers of i cycle with period 4:

i¹ = i
i² = −1
i³ = −i
i⁴ = 1
i⁵ = i  (cycle repeats)

Trick: Divide the exponent by 4 and use the remainder.

• i¹⁰ → 10 ÷ 4 = remainder 2 → i² = −1
• i¹⁵ → 15 ÷ 4 = remainder 3 → i³ = −i

Practice