IB Maths AI HLComplex NumbersPaper 1 & 2Conjugates & division~8 min read
Operations with Complex Numbers
Complex numbers add, subtract and multiply just like ordinary algebra — with one extra rule, i² = −1. Division is the only operation that needs a trick: multiply top and bottom by the conjugate of the denominator to clear the i from below.
📘 What you need to know
Add / subtract by combining real parts and imaginary parts separately.
Multiply by expanding brackets, then replacing every i² with −1.
The powers of i repeat in a cycle of four: i, −1, −i, 1, …
The conjugate of z = a + bi is z* = a − bi — flip the sign of the imaginary part.
z + z* and zz* are always real; z − z* is always imaginary.
Divide by multiplying top and bottom by the conjugate of the denominator.
Adding, subtracting and multiplying
To add or subtract complex numbers, combine the real parts together and the imaginary parts together — nothing else changes. For example (5 − 3i) + (1 + 2i) = 6 − i.
To multiply, expand the brackets exactly as in algebra, then use i² = −1 to turn the i² term into a real number:
(a + bi)(c + di) = ac + adi + bci + bdi². Since i² = −1, the last term becomes −bd, so the result is (ac − bd) + (ad + bc)i.
Powers of i
Because i² = −1, every higher power of i collapses to one of just four values. After i4 the pattern starts again, so the powers of i form a sequence with period 4.
Multiplying by i moves one step around the cycle. To evaluate in, find the remainder when n is divided by 4: remainder 0→1, 1→i, 2→−1, 3→−i.
Fast method for in: divide the exponent by 4 and keep only the remainder. For i19, since 19 = 4×4 + 3, the remainder is 3, so i19 = i3 = −i.
The conjugate and division
The complex conjugate of z = a + bi is z* = a − bi — the same number with the sign of the imaginary part reversed. Conjugates have three useful properties worth memorising.
The complex conjugatez = a + bi ⇒ z* = a − biz + z* = 2a (real) zz* = a² + b² (real)z − z* is always purely imaginary
The product zz* being real is exactly what makes division work. To divide, write the quotient as a fraction and multiply top and bottom by the conjugate of the denominator — this clears the i from the bottom, just like rationalising a surd.
🧠Recipe — dividing two complex numbers
Write the division as a fraction, numerator over denominator.
Take the conjugate of the denominator — flip the sign of its imaginary part.
Multiply top and bottom by that conjugate.
Expand both products, replacing every i² with −1 — the denominator becomes a real number.
Split into real and imaginary parts to give the Cartesian form p + qi.
Worked examples
WE 1
Adding and subtracting
Given z1 = 5 − 3i and z2 = −2 + 8i, find (a) z1 + z2 and (b) z1 − z2.
(a) combine real parts, then imaginary parts(5 + (−2)) + (−3 + 8)i = 3 + 5i(b) subtract part by part(5 − (−2)) + (−3 − 8)i = 7 − 11i(a) 3 + 5i · (b) 7 − 11iwatch the double-negative in (b): 5 − (−2) = 7.
WE 2
Multiplying two complex numbers
Expand and simplify (4 − 3i)(2 + 5i), giving your answer in Cartesian form.
expand the brackets8 + 20i − 6i − 15i²use i² = −1, so −15i² = +158 + 14i + 1523 + 14ithe −15i² term flips to +15 — that sign change is the whole trick.
WE 3
Powers of i
Find the exact value of i19 + i32.
divide each exponent by 4, keep the remainder19 = 4×4 + 3 ⇒ i¹⁹ = i³ = −i32 = 4×8 + 0 ⇒ i³² = i⁰ = 1add the results1 − ionly the remainder matters — the cycle resets every 4 powers.
WE 4
Working with the conjugate
Let z = 7 − 4i. Write down z*, then find z + z* and zz*.
conjugate: flip the imaginary signz* = 7 + 4iz + z* — the imaginary parts cancel(7 + 7) + (−4 + 4)i = 14zz* = (7 − 4i)(7 + 4i) = 49 − 16i²= 49 + 16z* = 7 + 4i · z + z* = 14 · zz* = 65both z + z* and zz* land on real numbers — exactly as expected.
WE 5
Dividing complex numbers
Find (5 + 2i) ÷ (1 − 3i), giving your answer in Cartesian form.
Treat i like an algebra letter when expanding — only the final i² = −1 step is special.
For high powers, divide the exponent by 4 and use only the remainder.
The conjugate just flips the imaginary sign — the real part is untouched.
To divide, multiply by the conjugate of the denominator, never the numerator.
Your GDC can add, multiply and divide complex numbers — use it to check, but show the working for method marks.
âš Common mistakes
Forgetting i² = −1 — leaving an i² term in the answer instead of converting it.
Sign slips on i² terms — −15i² becomes +15, not −15.
Conjugating the numerator when dividing — it is the denominator‘s conjugate you need.
Using the wrong remainder for powers of i — i20 has remainder 0, giving 1, not i.
Stopping at one fraction — split (p + qi)/k into p/k + (q/k)i for full Cartesian form.
Next up: Complex Roots of Quadratics — when a quadratic’s discriminant is negative its roots are complex, and they always arrive as a conjugate pair. The conjugate you just met is the key to it.
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