This HL mini-lesson takes Number & Algebra to full depth: all the SL toolkit (error, standard form, sequences, financial maths, logs) plus the HL extensions — complex numbers (Cartesian and modulus–argument form, used to model AC circuits), solving systems of equations with matrices, and deeper amortisation.
AI HL flavour: complex numbers appear as phasors/impedance in electrical engineering, and matrices solve linear systems efficiently.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.
A complex number z = a + bi has real part a and imaginary part b, where i² = −1. It is a point (a, b) on the Argand diagram.
z = 3 + 4i.
|z| = √(3² + 4²) = √25 = 5
arg(z) = tan⁻¹(4 ⁄ 3) = 53.13°
In AC circuit analysis, each component has a complex impedance Z. Impedances in series add like complex numbers, and |Z| is the magnitude of the total opposition to current.
Z₁ = 3 + 4i (Ω), Z₂ = 5 − 2i (Ω).
Z = Z₁ + Z₂ = (3 + 5) + (4 − 2)i = 8 + 2i
|Z| = √(8² + 2²) = √68 = 8.25 Ω (2 d.p.)
The argument of Z is the phase angle between voltage and current — a genuine engineering use of complex numbers.
A linear system can be written Ax = b and solved with the inverse matrix (when det A ≠ 0): x = A⁻¹b. Your GDC does this directly.
2x + y = 5 and 3x + 4y = 10.
A = [[2, 1], [3, 4]], det A = 2×4 − 1×3 = 5 ≠ 0.
Solution: x = 2, y = 1
If det A = 0 the system has no unique solution (either none or infinitely many).
HL expects fluent use of the finance solver for loan amortisation and annuities. A loan L at periodic rate i over N periods needs equal repayments:
Loan L = $15 000 at 6%/yr compounded monthly (i = 0.005) over 5 years (N = 60).
PMT = 15000 × 0.005 ÷ (1 − 1.005⁻⁶⁰) = $289.99 per month
Tap an object, then the concept it belongs to.
Tap an item on the left, then its match on the right.
Add/subtract by combining real and imaginary parts; multiply using i² = −1.
(3 + 4i) + (5 − 2i) = 8 + 2i.
(1 + 2i)(3 + i) = 3 + i + 6i + 2i² = 3 + 7i − 2 = 1 + 7i
Compound interest is a geometric sequence (ratio 1 + i); an amortised loan is a geometric series of discounted payments. The finance solver just automates these.
Understanding the sequence behind each formula helps you check GDC answers for sense.
Complex numbers: z = a + bi; |z| = √(a²+b²), arg = tan⁻¹(b⁄a)
AC circuits: series impedances add as complex numbers
Matrices: Ax = b solved by x = A⁻¹b when det A ≠ 0
Financial: amortisation PMT = Li⁄(1 − (1+i)⁻ᴺ)
SL toolkit: error, standard form, sequences, logs all still assessed
GDC: use the complex, matrix and finance solvers fluently
You have worked through the whole topic. Press Finish to see your score.
You have covered HL Number & Algebra (with complex numbers and matrices) for AI. 🎉
Your stars: 0 / 0
Next: test yourself in the Evaluate stage, then lock it in with Verify.