Complex Numbers
Add, subtract, multiply and divide complex numbers, raise them to a power or take their roots, step by step. Every answer comes in a + bi form and in polar form, and is drawn as arrows in the complex plane.
What is a Complex Number?
A complex number a + bi has a real part a and an imaginary part b, where i is a number with i² = −1. You can picture it as a point, or an arrow, in a plane: a to the right, b upwards.
Seen as arrows, the operations become geometry. Adding is putting arrows head to tail. Multiplying multiplies the lengths and adds the angles — multiplying by i simply turns an arrow a quarter turn. That is why complex numbers describe everything that rotates or oscillates, from alternating current to quantum waves.
In the Complex Plane, Step by Step
zStep-by-Step Solution
Key Relations
Divide by multiplying top and bottom by the conjugate c − di: the bottom becomes the real number c² + d².
In polar form multiplying is easy: r₁r₂ and θ₁ + θ₂.
Every nonzero number has exactly n different n-th roots, spread evenly around a circle.
Sources & Verification
- J. Stewart, L. Redlin & S. Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage, 2016 — Sec. 1.6 (Complex Numbers) and 8.3 (Polar Form; De Moivre’s Theorem).
- T. Needham, Visual Complex Analysis, Oxford University Press, 1997 — Ch. 1 (the geometry of complex arithmetic).
Frequently Asked Questions
How do you multiply complex numbers?
Multiply out the brackets and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 4i)(1 − 2i): 3 − 6i + 4i − 8i² = 3 + 8 − 2i = 11 − 2i. In the complex plane the lengths multiply (5 × √5) and the angles add.
How do you divide complex numbers?
Multiply the top and the bottom by the conjugate of the bottom (change the sign of its imaginary part). The bottom then becomes a real number c² + d². For (5 + 5i) ÷ (1 + 2i): multiply by (1 − 2i) to get (15 − 5i) ÷ 5 = 3 − i.
What is the modulus and argument of a complex number?
The modulus |z| = √(a² + b²) is the length of the arrow from 0 to a + bi; the argument is its angle with the positive real axis, atan2(b, a). For 3 + 4i: |z| = 5 and the angle is 53.13°. Together they give the polar form z = r(cos θ + i sin θ) = r·e^(iθ).
How do you raise a complex number to a power?
Use De Moivre's theorem: raise the length to the power n and multiply the angle by n. 1 + i has length √2 and angle 45°, so (1 + i)⁸ has length (√2)⁸ = 16 and angle 360°: it is exactly 16.
How do you find the roots of a complex number?
Take the n-th root of the length and divide the angle by n; adding full turns (360° · k) to the angle before dividing gives the other roots. There are always exactly n roots, evenly spread on a circle. The cube roots of 1 are 1 and −1/2 ± (√3/2)i, the corners of an equilateral triangle.
What is i, and why is i² = −1?
i is defined as a number whose square is −1. No real number does that, so i is drawn on a second axis, at right angles to the real numbers. Multiplying by i turns an arrow a quarter turn; doing it twice turns it half a turn, which is the same as multiplying by −1.