← Complex numbers

9 Equations, arguments, and comparing components

A large share of the paper is not "compute this" but "an equation involving z, its conjugate, or an unknown real constant is true — find the constant." The technique is always to separate real and imaginary parts.

The method, every time

  • If two complex expressions are equal, their real parts are equal AND their imaginary parts are equal separately — that gives you TWO real equations from one complex one.
  • If z is real, Im(z) = 0. If z is purely imaginary, Re(z) = 0. Both are common conditions to set up an equation from a sentence like "show that ap = −bq".
  • For arg(z) = θ conditions: write z in the form a + bi (clearing any fraction with the conjugate first), then use tan θ = b/a — but check the quadrant, since arg = π/4 needs BOTH a and b positive, not just their ratio equal to 1.
  • When a problem states u/v = (something), it is usually faster to cross-multiply into u = v × (something) BEFORE separating real and imaginary parts, rather than rationalising the fraction first.
Worked example Comparing real and imaginary components to solve for two unknowns Find all possible values of g and h if (g + 2i)(3 + hi) = (10 − 4i)(3 − i). Expand the right side first, since it has no unknowns: (10−4i)(3−i) = 30 −10i −12i +4i² = 26 − 22i. Expand the left side: (g+2i)(3+hi) = 3g + ghi + 6i + 2hi² = (3g − 2h) + (gh + 6)i. Compare real parts: 3g − 2h = 26. Compare imaginary parts: gh + 6 = −22, so gh = −28. Substitute g = (26 + 2h)/3 into gh = −28: 2h² + 26h + 84 = 0, so h² + 13h + 42 = 0, giving (h+6)(h+7) = 0. ✓ h = −6, g = 14/3; or h = −7, g = 4 — both pairs are valid solutions. ✕

arg(z) = θ needs the actual quadrant checked, not just a ratio

Setting real part = imaginary part only proves arg(z) = π/4 OR arg(z) = 5π/4 (the same ratio, opposite quadrant). You must also check both parts are POSITIVE (for π/4) or both NEGATIVE (for 5π/4) before accepting a solution — an Excellence-level question can hinge on excluding a value that satisfies the equation but lands in the wrong quadrant. ✓

A reciprocal-symmetric equation is really a cubing identity in disguise

If x + 1/x = p, cubing both sides gives (x + 1/x)³ = p³. Expand the left side as x³ + 3x²(1/x) + 3x(1/x)² + 1/x³ = x³ + 1/x³ + 3(x + 1/x) = x³ + 1/x³ + 3p. So x³ + 1/x³ = p³ − 3p — a purely algebraic identity that never requires finding x itself.

A fast check before submitting an equate-components answer

  • Have I separated into a REAL equation and an IMAGINARY equation, rather than trying to solve one combined equation?
  • For an arg(z) = θ condition, have I checked the quadrant, not just the ratio of the two parts?

Practise just this