What it shows
A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. On the Argand plane it is the point (a, b), or the arrow from 0 to that point. Its length is the modulus |z| = √(a² + b²) and its angle from the positive real axis is the argument arg z. Seen this way, the algebra turns into geometry:
- z + w is the fourth corner of the parallelogram on z and w (like adding vectors).
- z × w stretches z by |w| and turns it by arg w: lengths multiply, angles add.
- z̄, the conjugate a − bi, is the reflection of z in the real axis.
- i × z turns z a quarter turn anticlockwise, which is why i² = −1 (two quarter turns point backwards).
It follows complex numbers and quadratic equations in Class 11 maths. The unit circle page explains the angles and the cos and sin inside the polar form; symbolic algebra simplifies complex expressions; and the graphing calculator draws any parabola.
How to use it
- Pick an operation: z + w, z − w, z × w, z ÷ w, the conjugate z̄, i × z, or zⁿ.
- Drag the blue point z and the orange point w on the plane, or type their parts in the boxes. Snap to halves keeps the numbers tidy.
- The green arrow is the answer. Under the picture, the working line fills in your numbers, and the facts give each number’s modulus, argument and the polar form of the answer.
- For zⁿ, slide n from 0 to 8 to see the powers spiral out (|z| > 1) or in (|z| < 1).
- In Complex roots of a quadratic, slide a, b and c. The left picture is the parabola; the right one shows the two roots on the Argand plane.
Worked example
Take z = 3 + i and w = 1 + 2i. Then z + w = 4 + 3i. For the product, (3 + i)(1 + 2i) = 3 + 6i + i + 2i² = 1 + 7i. Check with the geometry: |z| = √10 ≈ 3.1623 and |w| = √5 ≈ 2.2361 multiply to 7.0711 = |1 + 7i|, and arg z ≈ 18.43° plus arg w ≈ 63.43° gives 81.87° = arg(1 + 7i). Dividing, (3 + i) ÷ (1 + 2i) = (3 + i)(1 − 2i) / 5 = 1 − i.
For x² + 2x + 5 = 0, the discriminant is 4 − 20 = −16, so the parabola never touches the x-axis and the roots are (−2 ± 4i) / 2 = −1 ± 2i, a conjugate pair.
Things to notice
- The argument shown is the principal value, between −180° and 180° (−π and π). When the angles of a product add past 180°, subtract 360°.
- Real quadratics always have complex roots in conjugate pairs: mirror images in the real axis.
- Powers of i repeat every four: i, −1, −i, 1.
Check yourself
Five questions with fresh numbers: a modulus, the real part of a product, a power of i, an argument, and the roots of a quadratic. Nothing is saved.
FAQ
Why is the argument of a negative real number 180° and not −180°? The principal argument is taken in (−π, π], which includes π but not −π.
Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid.
Where the numbers come from
- Standard complex number algebra: i² = −1, the Argand plane, modulus, principal argument in (−π, π], polar form, and the quadratic formula with a negative discriminant. Written for fibub; no textbook text, figures or exercises are used.
- Checked against Python's complex numbers and cmath.phase (sums, products, quotients, powers, modulus and argument) and numpy.roots (roots of five quadratics).
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.