Complex Numbers – Why They Matter
Ever wondered why engineers talk about "i" when designing circuits? That tiny "i" is the gateway to a whole new number world that makes many real‑life problems solvable.
In simple words: a complex number is just a pair of ordinary numbers, written as a + bi. The i stands for the square root of –1, letting us handle equations like x² + 1 = 0 that have no real solutions.
What Are Complex Numbers?
Understanding the imaginary unit i
The word “imaginary” sounds made‑up, but mathematicians give it a precise meaning. The imaginary unit i is defined so that i² = –1. Think of it like a new direction on a map that lets you turn a “downhill” (negative) slope into a “sideways” move.
Rectangular (or Cartesian) form
When we write a complex number as a + bi, a is called the real part and b the imaginary part. It’s just like a point (a,b) on a flat grid, except the vertical axis is measured in multiples of i.
Polar form and why it matters
Sometimes it’s easier to think of a complex number as a length and an angle. The modulus (or magnitude) r tells how far the point is from the origin, while the argument (or angle) θ tells the direction. The polar expression looks like r(cosθ + i sinθ). This view is super handy for multiplication, division, and powers.
Operations with Complex Numbers
Addition & Subtraction
Just line up the real parts and the imaginary parts.
Example: (3 + 2i) + (1 – 4i) = (3+1) + (2‑4)i = 4 – 2i.
Multiplication
Use the distributive law (FOIL) and remember that i² = –1.
Example: (2 + 3i)(1 – i) = 2·1 + 2·(‑i) + 3i·1 + 3i·(‑i) = 2 – 2i + 3i – 3i² = 2 + i + 3 = 5 + i.
Division
Multiply numerator and denominator by the conjugate of the denominator. The conjugate of a + bi is a – bi; it flips the sign of the imaginary part.
Example: \frac{4 + 2i}{1 – i} \times \frac{1 + i}{1 + i} = \frac{(4+2i)(1+i)}{1² + 1²} = \frac{4 + 4i + 2i + 2i²}{2} = \frac{2 + 6i}{2} = 1 + 3i.
Conjugate and Modulus
The conjugate (a – bi) helps find the modulus: \|a + bi\| = \sqrt{a² + b²}. It’s like the distance from the origin on a city map.
Quick Comparison: Rectangular vs Polar
| Aspect | Rectangular (a+bi) | Polar (r∠θ) |
|---|---|---|
| Form | a + bi | r(cosθ + i sinθ) |
| Easy for | addition, subtraction | multiplication, division, powers |
| Key quantities | real part a, imaginary part b | modulus r, argument θ |
| Conversion | r = √(a²+b²), θ = tan⁻¹(b/a) | a = r cosθ, b = r sinθ |
Key Points to Remember
- i² = –1, the heart of every complex operation.
- Always keep real and imaginary parts separate when adding or subtracting.
- Use the conjugate to rationalise denominators during division.
- Modulus tells “how big” the number is; argument tells “which way” it points.
- Switch between rectangular and polar forms whenever it simplifies the problem.
📝 Likely Exam Questions
- Find the modulus and argument of 3 – 4i. Answer: Modulus = 5, Argument = tan⁻¹(‑4/3) ≈ –53.13° (or 306.87°).
- Express (1 + i)³ in a + bi form. Answer: (1 + i)³ = –2 + 2i.
- Divide (5 – 2i) by (1 + i) and write the result in a + bi form. Answer: = 1.5 – 3.5i.
- Convert 2(cos 45° + i sin 45°) to rectangular form. Answer: = √2 + √2 i.
- Show that the product of a complex number and its conjugate equals the square of its modulus. Answer: (a+bi)(a‑bi)=a²+b²=|a+bi|².