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Complex Numbers
Edexcel A Level Further Mathematics – Core Pure Revision Notes
The Basics
A complex number has the form z = a + bi, where a and b are real numbers and i =
−1, so i^2 = −1.
a = Re(z) is the real part. b = Im(z) is the imaginary part (note: b is real, not bi). The complex conjugate of z = a + bi is z∗^ = a − bi.
Adding/subtracting: Treat real and imaginary parts separately. (3 + 2i) + (1 − 5 i) = 4 − 3 i
Multiplying: Expand like a bracket, replacing i^2 = −1. (3 + 2i)(1 − 5 i) = 3 − 15 i + 2i − 10 i^2 = 3 − 13 i + 10 = 13 − 13 i
Dividing: Multiply numerator and denominator by the conjugate of the denominator.
3 + 2i 1 − 5 i
(3 + 2i)(1 + 5i) (1 − 5 i)(1 + 5i)
3 + 15i + 2i + 10i^2 1 + 25
−7 + 17i 26
i
Modulus and Argument
Think of a complex number as a point (a, b) on the Argand diagram (where the x-axis is the real axis and y-axis is the imaginary axis).
Modulus and Argument Definitions
Modulus: |z| =
a^2 + b^2 (distance from origin to the point) Argument: arg(z) = θ where tan θ = b a (angle from positive real axis) The principal argument is always in the range (−π, π]. Be careful with quadrant: use the Argand diagram to place the point correctly before computing θ.
Key properties:
|z 1 z 2 | = |z 1 ||z 2 | arg(z 1 z 2 ) = arg(z 1 ) + arg(z 2 ) (mod 2π)
z 1 z 2
|z 1 | |z 2 |
arg
z 1 z 2
= arg(z 1 ) − arg(z 2 )
|z∗| = |z|, arg(z∗) = − arg(z) z · z∗^ = |z|^2 (always a real, non-negative number)
Modulus-Argument Form and Euler’s Form
Instead of z = a + bi, you can write a complex number using its distance from the origin and its angle:
z = r(cos θ + i sin θ) where r = |z|, θ = arg(z)
This is called modulus-argument (polar) form. Even more compactly, using Euler’s formula eiθ^ = cos θ + i sin θ:
z = reiθ
Converting Between Forms
Cartesian → mod-arg: r =
a^2 + b^2 , θ = arctan(b/a) (adjusted for quadrant). Mod-arg → Cartesian: a = r cos θ, b = r sin θ.
Multiplying in mod-arg form: Multiply the moduli and add the arguments.
r 1 eiθ^1 × r 2 eiθ^2 = r 1 r 2 ei(θ^1 +θ^2 )
De Moivre’s Theorem
This is one of the most powerful results in the whole topic.
De Moivre’s Theorem
[r(cos θ + i sin θ)]n^ = rn(cos nθ + i sin nθ) Or equivalently:
reiθ
n = rneinθ This works for any integer (or rational) n.
Applications of De Moivre
- Proving trigonometric identities:
Since (c + is)n^ = cos nθ + i sin nθ (where c = cos θ, s = sin θ), expand the left side using the binomial theorem and equate real and imaginary parts.
Example: Prove cos 3θ = 4 cos^3 θ − 3 cos θ.
(cos θ + i sin θ)^3 = cos^3 θ + 3 cos^2 θ(i sin θ) + 3 cos θ(i sin θ)^2 + (i sin θ)^3
Real part: cos^3 θ − 3 cos θ sin^2 θ = cos^3 θ − 3 cos θ(1 − cos^2 θ) = 4 cos^3 θ − 3 cos θ ✓
- Expressing powers of cos θ and sin θ:
Using z = eiθ, we get z + z−^1 = 2 cos θ and z − z−^1 = 2i sin θ.
So: 2 cos θ = z + z−^1 , 2 i sin θ = z − z−^1 , (2 cos θ)n^ = (z + z−^1 )n.
