Ever wondered why engineers talk about numbers that have a “+ i” attached?
Complex numbers are just ordinary numbers paired with an imaginary friend called i, which squares to -1. Together they let us solve equations like x²+1=0 that ordinary numbers can’t handle.
What is a Complex Number?
A complex number looks like a + bi. Here a is the real part (just a regular number) and b is the imaginary part multiplied by i. Think of it as a point on a flat map: the horizontal axis shows the real part, the vertical axis shows the imaginary part. That map is called the Argand diagram, similar to a city grid where east‑west is real and north‑south is imaginary.
How to Add and Subtract Complex Numbers
Adding or subtracting is as easy as combining the real pieces and the imaginary pieces separately.
- Add: (a + bi) + (c + di) = (a + c) + (b + d)i
- Subtract: (a + bi) – (c + di) = (a – c) + (b – d)i
Example: Add 3 + 4i and -1 + 2i.
Real: 3 + (-1) = 2
Imaginary: 4 + 2 = 6
Result: 2 + 6i.
Multiplying and Dividing Complex Numbers
Multiplication uses the rule i² = -1.
Multiply: (a + bi)(c + di) = (ac – bd) + (ad + bc)i.
Example: (2 + 3i)(1 – 4i)
ac = 2·1 = 2
bd = 3·4 = 12 → 2 – 12 = -10 (real)
ad + bc = 2·(-4) + 3·1 = -8 + 3 = -5 (imaginary)
Result: -10 – 5i.
Division is easier if we turn the denominator into a real number by multiplying numerator and denominator by the conjugate (swap the sign of the imaginary part).
Divide: (a + bi) / (c + di) = [(a + bi)(c – di)] / (c² + d²).
Example: (3 + 2i) ÷ (1 – i)
Conjugate of denominator: 1 + i
Numerator × conjugate: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i – 2 = 1 + 5i
Denominator: 1² + (-1)² = 2
Result: (1/2) + (5/2)i.
Modulus and Argument – The Size and Direction
The modulus (also called magnitude) tells how far the point is from the origin, just like the length of a rope pulling a boat. It’s computed with the Pythagorean theorem:
r = √(a² + b²).
The argument (also called angle) tells the direction from the positive real axis, like the compass bearing of a ship. It’s the angle θ where tan θ = b/a, so θ = tan⁻¹(b/a). Always keep an eye on the quadrant to choose the right sign.
Example: For 3 + 4i, modulus r = √(3²+4²)=5, argument θ = tan⁻¹(4/3) ≈ 53.1°.
Rectangular ↔ Polar Form Conversion
Rectangular form is the a+bi we’ve been using. Polar form writes a complex number as r(cosθ + i sinθ) or the shorter r∠θ. It’s handy for multiplication, division, and powers because you just multiply moduli and add angles.
Conversion steps:
Example: Convert -1 + √3 i to polar.
r = √((-1)² + (√3)²) = √(1+3)=2
θ = tan⁻¹(√3 / -1) → second quadrant → 120° (or 2π/3 rad)
Polar form: 2(cos120° + i sin120°) or 2∠120°.
Quick Summary Table
| Operation | Rectangular (a+bi) | Polar (r∠θ) |
|---|---|---|
| Add / Subtract | Combine real parts, combine imaginary parts | Convert to rectangular first |
| Multiply | (ac‑bd)+(ad+bc)i | r₁r₂ ∠ (θ₁+θ₂) |
| Divide | [(a+bi)(c‑di)]/(c²+d²) | (r₁/r₂) ∠ (θ₁‑θ₂) |
| Modulus | √(a²+b²) | r |
| Argument | tan⁻¹(b/a) (adjust quadrant) | θ |
📝 Likely Exam Questions
- Find the modulus and argument of 5‑12i.
Modulus = √(5²+(-12)²)=13; Argument = tan⁻¹(-12/5) ≈ -67.38° (or 292.62°). - Write (1+ i)² in a+bi form.
(1+i)² = (1+i)(1+i)=1+2i+i²=1+2i‑1=2i. - Express -2+2√3 i in polar form.
r = √((-2)²+(2√3)²)=4; θ = tan⁻¹(2√3 / -2)=120°; Polar: 4∠120°. - Divide (3‑4i) by (1+ i) and give the answer in a+bi.
Conjugate of denominator: 1‑ i.
Numerator×conjugate: (3‑4i)(1‑ i)=3‑3i‑4i+4i²=3‑7i‑4=‑1‑7i.
Denominator: 1²+1²=2.
Result: (-1/2)‑(7/2)i. - Multiply (2∠30°) by (3∠45°) and write the result in polar form.
Moduli: 2×3=6; Angles: 30°+45°=75°; Result: 6∠75°.