MATH1961 · Mathematics 1a (advanced)
Complex Numbers
A complex number is z = x + iy with x, y real and i² = −1. The reals sit inside ℂ as the y = 0 line; what you gain is that every polynomial now factors (the Fundamental Theorem of Algebra). Arithmetic is just real algebra with the single rule i² = −1 applied at the end, and the workhorse identity is |z|² = z·z̄ — division is ‘multiply by the conjugate’. The geometry comes from the Argand diagram: plot z as a point, and it acquires a length (modulus) and a direction (argument). Polar and exponential form with Euler’s formula make multiplication a matter of multiplying lengths and adding angles, de Moivre’s theorem (provable by induction) powers it, and the roots of unity sit at equally-spaced points on the unit circle.
What this chapter covers
- 01Arithmetic, the conjugate & modulus — dividing by multiplying by the conjugate
- 02The Argand diagram — modulus and argument replace x and y
- 03Polar & exponential form, and Euler’s formula
- 04de Moivre’s theorem (with the induction proof) & powers
- 05Roots of unity and the unit-circle / trig connection
Worked example: divide complex numbers using the conjugate
- +1Multiply top and bottom by the conjugate of the denominator, which is 1 + i. This makes the denominator real.
- +1Denominator: (1 − i)(1 + i) = 1 − i² = 1 − (−1) = 2 = |1 − i|².
- +1Numerator and finish: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i, so the result is (1 + 5i)/2 = 1/2 + (5/2)i.
Key terms
- Complex number
- z = x + iy with x, y real and i² = −1; x = Re z and y = Im z. The reals are the y = 0 line inside ℂ. The gain over ℝ is that every polynomial factors completely — the Fundamental Theorem of Algebra — so a degree-n polynomial has n roots in ℂ.
- Conjugate and modulus
- The conjugate of z = x + iy is z̄ = x − iy, and the modulus is |z| = √(x² + y²). The key identity is z·z̄ = |z|² (a real number), which is why dividing by z means multiplying by z̄. Useful rules: conjugation distributes over + and ×, and |zw| = |z||w|.
- Argand diagram
- The plane in which z = x + iy is plotted at the point (x, y), the real part horizontal and imaginary part vertical. It gives z a length (modulus |z|) and a direction (argument arg z), turning complex arithmetic into geometry.
- Polar / exponential form
- z = r(cosθ + i sinθ) = r eiθ by Euler’s formula, where r = |z| and θ = arg z. In this form multiplication multiplies the moduli and adds the arguments, which makes powers and roots simple.
- de Moivre's theorem
- (cosθ + i sinθ)n = cos(nθ) + i sin(nθ), provable by induction on n. It powers complex numbers in polar form, derives multiple-angle trig identities, and gives the n distinct n-th roots of unity, equally spaced around the unit circle.
Complex Numbers FAQ
How do I divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. The denominator (c + di)(c − di) becomes the real number c² + d² = |c + di|², and the rest is bookkeeping. This works because z·z̄ = |z|² is always real — the single most useful identity in the chapter.
Why is writing i = √−1 dangerous?
Because the real square-root identity √a·√b = √(ab) fails for negatives: √−1 · √−1 would give i·i = −1, not √((−1)(−1)) = 1. Define i by the rule i² = −1 and never push real square-root identities onto negative numbers — that is a classic marked error.
What does multiplication look like on the Argand diagram?
In polar form z = r eiθ, multiplying two complex numbers multiplies their moduli and adds their arguments. So multiplication is a scaling-and-rotation, which is exactly why polar/exponential form makes powers (de Moivre) and roots so clean compared with multiplying out x + iy brackets.
How do I find the n-th roots of a complex number?
Write the number in polar form r eiθ, take the real n-th root of the modulus, and divide the argument by n — then add multiples of 2π/n to get all n roots, equally spaced on a circle. The n-th roots of unity are the special case r = 1, θ = 0, sitting at the vertices of a regular n-gon on the unit circle.
Exam move
Anchor everything on two ideas: z·z̄ = |z|² for arithmetic (it turns every division and many ‘show that…’ questions into a one-line conjugate argument) and the Argand picture for geometry. Get fluent moving between rectangular x + iy and polar r eiθ form, because multiplication, powers and roots are trivial in polar form and clumsy in rectangular. Learn de Moivre with its induction proof (the Advanced unit wants the proof, not just the formula), and practise roots of unity as equally-spaced points so you can both compute them and draw them. Watch the √−1 trap.
