Complex Numbers
Core concepts
Complex numbers extend the reals with i, where i² = -1. You must be able to add, subtract, multiply, and divide complex numbers, find the modulus |a + bi| = sqrt(a² + b²), use the conjugate a - bi to rationalise denominators, and represent numbers on the complex plane. Know the polar form r(cos θ + i sin θ) = r e^(iθ), multiplication as multiplying moduli and adding arguments, and De Moivre's theorem for powers and roots. Understand that every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle.
Traps to watch
- i raised to a power cycles with period 4: i, -1, -i, 1.
- Dividing by a complex number requires multiplying numerator and denominator by the conjugate.
- |z|² is not z² in general; z z̄ = |z|² is real.
- When computing arguments, check the quadrant; arctangent alone may be off by π.
- The quadratic formula always yields roots; a negative discriminant gives a conjugate pair.
Rule of thumb
To divide, multiply top and bottom by the conjugate of the denominator. To take powers, convert to polar form first.
Worked example
Compute (1 + i)/(2 - i). Multiply by (2 + i)/(2 + i): numerator = (1 + i)(2 + i) = 2 + i + 2i + i² = 1 + 3i. Denominator = 4 + 1 = 5. Result = (1 + 3i)/5. Check modulus: |1 + i| = sqrt(2), |2 - i| = sqrt(5), so the result has modulus sqrt(2/5), matching sqrt(1/25 + 9/25).