Complex number calculator
Type z in the form a + bi. Choose what to find: the modulus and argument, a sum, product or quotient with w, a power, or the nth roots. Each answer is plotted on an Argand diagram.
- |z|
- 1.41
- arg z
- 2.36
Show the working
|z| = √(−1.00² + 1.00²) = 1.41
arg z: z is in the second quadrant of the Argand diagram; tan−1(1.00 ÷ 1.00) = 0.785 is the angle to the real axis, so
arg z = 2.36 (the principal argument, between −π and π)
z = 1.41(cos 2.36 + i sin 2.36) = 1.41e2.36i
Conjugate z* = −1.00 − 1.00i.
Answers are rounded to 3 significant figures, which is what IB papers ask for unless a question says otherwise.
How to do it by hand
- Modulus: |a + bi| = √(a² + b²).
- Argument: the angle from the positive real axis. Sketch the point first to see its quadrant, then use tan⁻¹(|b| ÷ |a|) and adjust; the principal argument is between −π and π.
- Multiply or divide in polar form by multiplying or dividing the moduli and adding or subtracting the arguments. De Moivre: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
- The n nth roots all have modulus r^(1/n) and arguments (θ + 2πk) ÷ n for k = 0, 1, …, n − 1.
Worked example
Find the cube roots of 8i, giving each in the form a + bi.
Solution
z = 8.00(cos 1.57 + i sin 1.57)
Each cube root has modulus 8.001/3 = 2.00 and argument (θ + 2πm) ÷ 3 for m = 0, 1, …, 2: they are equally spaced round a circle.
w0 = 1.73 + 1.00i
w1 = −1.73 + 1.00i
w2 = −2.00i
Answer: 3 roots: 1.73 + 1.00i, −1.73 + 1.00i, −2.00i
Every number in this example is worked out by the same code as the tool above, and the code is tested against independent results from Python's SciPy library.
Where you need it
| AA SL | Not in the syllabus |
|---|---|
| AA HL | Most complex-number questions are on Paper 1 (no calculator); use the GDC on Papers 2 and 3 and to check your algebra. |
| AI SL | Not in the syllabus |
| AI HL | Arithmetic, modulus–argument form and complex numbers in sinusoidal models. |
On your calculator
In the exam you use your own calculator, so practise it too. Our guides give the exact keystrokes for the Casio fx-CG50, TI-Nspire CX II and TI-84 Plus CE, with a worked example to check against:
Common mistakes
- Taking tan⁻¹(b ÷ a) without checking the quadrant: −1 − i has argument −3π/4, not π/4.
- Leaving the calculator in degrees when the question wants radians (or the other way round).
- Giving only one root of zⁿ = w: there are always n of them, equally spaced round a circle.
- Writing a decimal when the question asks for an exact modulus or argument such as √2 or π/4.
Notes and practice
Questions students ask
What is the principal argument?
The argument in the interval −π < θ ≤ π (−180° to 180°). Courses and calculators normally use it, so −π/2 rather than 3π/2.
How do I divide complex numbers by hand?
Multiply the top and the bottom by the conjugate of the bottom. The bottom becomes the real number c² + d², and you can split the answer into real and imaginary parts.
Why do the nth roots form a regular polygon?
They all have the same modulus and their arguments differ by 2π ÷ n, so they sit equally spaced on a circle centred at the origin.