Quick summary
This chapter introduces the imaginary unit i, complex numbers a + ib, their algebra (addition, multiplication, division, conjugate), square roots, the Argand diagram, modulus and argument, polar and exponential forms, De Moivre's theorem and the cube roots of unity.
Key points
- i = √−1, so i² = −1, i³ = −i, i⁴ = 1. Powers of i repeat in cycles of 4.
- A complex number is z = a + ib, where a = Re(z) and b = Im(z) are real. If b = 0, z is real; if a = 0 and b ≠ 0, z is purely imaginary.
- Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal.
- Conjugate of z = a + ib is z̄ = a − ib. z + z̄ = 2a and z z̄ = a² + b² (real).
- To divide, multiply numerator and denominator by the conjugate of the denominator.
- Complex numbers cannot be compared with < or >.
- In the Argand diagram, z = a + ib is the point (a, b).
- Modulus |z| = √(a² + b²) is the distance of z from the origin; argument θ = arg z is the angle with the positive real axis (tan θ = b/a, checking the quadrant).
- Polar form z = r(cos θ + i sin θ); exponential form z = re^(iθ).
- De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
- Cube roots of unity are 1, ω, ω², where ω = (−1 + i√3)/2. 1 + ω + ω² = 0 and ω³ = 1.
- Roots of ax² + bx + c = 0 with b² − 4ac < 0 are complex conjugates.
Important terms
- Complex number – number of the form a + ib with a, b real.
- Conjugate – a − ib for the number a + ib.
- Modulus – distance of the point from the origin, √(a² + b²).
- Argument (amplitude) – angle made with the positive X-axis.
- Argand diagram – plane in which complex numbers are shown as points.
Formulas
- i² = −1, i⁴ = 1 ; i^(4k+r) = i^r
- (a + ib)(c + id) = (ac − bd) + i(ad + bc)
- |z₁z₂| = |z₁||z₂| ; |z₁/z₂| = |z₁|/|z₂| ; arg(z₁z₂) = arg z₁ + arg z₂
- Polar form: z = r(cos θ + i sin θ), r = |z|
- Euler form: e^(iθ) = cos θ + i sin θ
- De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
- ω = (−1 + i√3)/2, ω² = (−1 − i√3)/2, 1 + ω + ω² = 0, ω³ = 1
5 important questions
Q1. Find the value of i¹⁰³ and i⁻³⁵.
103 = 4 × 25 + 3, so i¹⁰³ = i³ = −i. i⁻³⁵ = 1/i³⁵ = 1/i³ = 1/(−i) = i.
Q2. Express (3 + 2i)/(1 − i) in the form a + ib.
Multiply by (1 + i)/(1 + i): (3 + 2i)(1 + i)/(1 + 1) = (3 + 3i + 2i + 2i²)/2 = (1 + 5i)/2 = ½ + (5/2)i.
Q3. Find the modulus and argument of z = 1 + i√3 and write it in polar form.
r = √(1 + 3) = 2. tan θ = √3, θ in Quadrant I, so θ = π/3. z = 2(cos π/3 + i sin π/3).
Q4. Find the square root of 5 + 12i.
Let √(5 + 12i) = x + iy. Then x² − y² = 5 and 2xy = 12. Also x² + y² = |5 + 12i| = 13. So x² = 9, y² = 4; since xy > 0, roots are ±(3 + 2i).
Q5. If ω is a complex cube root of unity, show that (1 − ω + ω²)(1 + ω − ω²) = 4.
1 + ω² = −ω, so 1 − ω + ω² = −2ω. 1 + ω = −ω², so 1 + ω − ω² = −2ω². Product = 4ω³ = 4.
Exam tip
For the argument, first find the acute angle α = tan⁻¹|b/a|, then adjust for the quadrant of (a, b).