Complex Numbers
Core concepts
Know the arithmetic of complex numbers in Cartesian form, the conjugate and its use in division, modulus and argument, the Argand diagram, polar (modulus–argument) form, and De Moivre's theorem for powers and roots. Understand that roots of a polynomial with real coefficients occur in conjugate pairs.
Traps
- The argument must be placed in the correct quadrant; a naive arctan(y/x) gives the wrong sign for some points.
- When taking nth roots, there are n distinct roots; omitting some loses marks. Space the arguments by 2π/n.
- Multiplying in polar form multiplies moduli and adds arguments; dividing subtracts arguments.
- For a real-coefficient polynomial, if a + bi is a root, so is a − bi; use this to factorise.
Worked example
Find the cube roots of 8. Write 8 = 8(cos 0 + i sin 0). Roots are 2(cos(2kπ/3) + i sin(2kπ/3)) for k = 0, 1, 2: 2, −1 + i√3, −1 − i√3. Check: each has modulus 2 and arguments spaced by 120°. Rule of thumb: always write the argument in a general form with + 2kπ before applying De Moivre.