A complex number extends the real number line with a second, perpendicular axis for multiples of i, the imaginary unit where i² = −1. This guide walks through the four arithmetic operations, why the conjugate makes division possible, how modulus and argument describe a complex number's polar form, and how De Moivre's theorem turns powers and roots into simple angle arithmetic.
Why i exists, and what a + bi means
Real numbers alone cannot solve an equation like x² = −1, because squaring any real number gives a non-negative result. Mathematicians resolved this by defining a new number, i, with the property i² = −1. A complex number a + bi then combines an ordinary real part a with an imaginary part b, a multiple of i.
Geometrically, every complex number corresponds to a point (a, b) on a two-dimensional plane — the real part along the horizontal axis, the imaginary part along the vertical axis. This is called the Argand plane (or complex plane), and it turns algebra on complex numbers into geometry: addition becomes vector addition, and multiplication becomes a scaling-plus-rotation.
Arithmetic: add, subtract, multiply, divide
Addition and subtraction work component-wise — combine the real parts, then the imaginary parts, with no interaction between the two. Multiplication is FOIL-expansion followed by simplifying with i² = −1: the cross terms become the new imaginary part, and the i·i term flips sign into the real part.
Division is the one operation that needs a trick: you cannot rationalize a complex denominator the way you would a square root, so instead you multiply top and bottom by the denominator's conjugate. Since z·conj(z) = |z|², the denominator becomes a plain real number, and the division reduces to two ordinary real divisions. Dividing by 0 + 0i is undefined for the same reason dividing by 0 is undefined on the real line — this calculator flags it explicitly rather than showing a broken result.
Modulus, argument, and polar form
Every complex number has two useful descriptions besides a + bi. The modulus |z| = √(a² + b²) is its distance from the origin — literally the Pythagorean theorem applied to the point (a, b). The argument is the angle that point makes with the positive real axis, computed with atan2(b, a) so every quadrant (and the axes) resolve correctly instead of dividing by zero.
Together, modulus and argument give the polar form r∠θ — an equivalent way to write the same number. Polar form is often more convenient than a + bi: multiplying two complex numbers in polar form just multiplies their moduli and adds their arguments, which is far simpler than the FOIL expansion in rectangular form.
De Moivre's theorem: powers and roots made simple
Repeatedly multiplying a complex number by itself would normally mean repeated FOIL expansions. De Moivre's theorem shortcuts this: in polar form, z^n = r^n(cos nθ + i sin nθ) — raise the modulus to the n-th power and multiply the argument by n. Squaring 1 + i (modulus √2, argument 45°) this way gives modulus 2 and argument 90°, i.e. 2i, matching direct expansion.
Finding an n-th root reverses the process, with a twist: because angles repeat every 2π (360°), there are exactly n distinct roots, evenly spaced around a circle of radius r^(1/n). Each one satisfies root^n = z. This is why, for example, the equation x³ = 1 has three complex solutions, not just one — the real root 1, plus two complex roots symmetric across the real axis.