Complex Numbers and Complex Plane
What the exam tests
Complex Numbers and Complex Plane tests arithmetic, conjugates, modulus, polar form, De Moivre's theorem, and roots of polynomial equations. Kyoto problems often ask for a locus or for all complex solutions in a specified form.
Core rules
- i^2 = -1. For z = a + bi, the conjugate is z̄ = a - bi and the modulus is |z| = √(a^2 + b^2). Also z z̄ = |z|^2.
- Division: multiply numerator and denominator by the conjugate of the denominator.
- Polar form: z = r(cos θ + i sin θ), where r = |z| and θ is an argument. Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.
- De Moivre: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ).
- For a quadratic with real coefficients and negative discriminant, the roots are a conjugate pair.
Worked example
Write z = (1 + i)/(2 - i) in a + bi form. Multiply numerator and denominator by 2 + i: the numerator becomes (1 + i)(2 + i) = 2 + i + 2i + i^2 = 1 + 3i, and the denominator becomes 4 + 1 = 5. Thus z = 1/5 + (3/5)i. Its modulus is |1 + i|/|2 - i| = √2/√5 = √10/5.
Traps
The conjugate of a sum is the sum of conjugates, but do not conjugate only one term. Argument is not unique; choose the branch required by the problem. Do not treat i as a real variable when solving equations. Check that a division denominator is nonzero.