Introduction to Complex Numbers

Imaginary and complex numbers

Some equations (like x² + 1 = 0) have no real solutions. To solve them we define the imaginary unit:

i = √(−1),   so   i² = −1

A complex number has a real part and an imaginary part:

z = a + bi     (a = real part, b = imaginary part)

Powers of i

i¹ = i,  i² = −1,  i³ = −i,  i⁴ = 1

The pattern repeats every 4 powers.

Arithmetic with complex numbers

• Add/subtract: combine real and imaginary parts separately.

(3 + 2i) + (1 − 5i) = 4 − 3i.

• Multiply: expand like brackets and use i² = −1.

(2 + i)(3 − 2i) = 6 − 4i + 3i − 2i² = 6 − i + 2 = 8 − i.

The complex conjugate

The conjugate of z = a + bi is *z\ = a − bi** (change the sign of the imaginary part). Key uses:

• *z × z\ = a² + b²** (a real number).
• Dividing: multiply top and bottom by the conjugate of the denominator to make it real.

(1)/(2 + i) = (2 − i)/[(2 + i)(2 − i)] = (2 − i)/5.

Roots of polynomials

• Complex roots of a real polynomial always occur in conjugate pairs: if a + bi is a root, so is a − bi.
• A quadratic with negative discriminant has two complex conjugate roots (from the quadratic formula with √negative).

Worked example

Solve x² − 4x + 13 = 0.

• Discriminant = 16 − 52 = −36. x = [4 ± √(−36)] ÷ 2 = [4 ± 6i] ÷ 2 = 2 ± 3i (a conjugate pair). ✓

Common mistakes

• Forgetting i² = −1 when multiplying.
• Not using the conjugate to divide (leaving i in the denominator).
• Giving only one complex root — they come in conjugate pairs for real polynomials.

Exam tips

• Use i² = −1 throughout; simplify powers of i using the cycle of 4.
• To divide, multiply numerator and denominator by the conjugate.
• State both conjugate roots when solving real quadratics/polynomials.

Key facts to remember

• i = √(−1), i² = −1; complex number z = a + bi; powers of i cycle every 4.
• *Conjugate z\ = a − bi*; z·z\ = a² + b² (real); use conjugates to divide.
• Complex roots of real polynomials come in conjugate pairs.