Complex numbers were invented to solve equations like x squared equals negative one, which has no real solution. We introduce the imaginary unit i, where i squared equals negative one. Any complex number can be written as z equals a plus b i, where a is the real part and b is the imaginary part. For example, 3 plus 2i has real part 3 and imaginary part 2.
The complex plane, also called the Argand diagram, provides a geometric way to visualize complex numbers. The horizontal axis represents real numbers, while the vertical axis represents imaginary numbers. Each complex number z equals a plus b i corresponds to the point with coordinates a comma b. For example, 3 plus 2i is plotted at point 3 comma 2, negative 1 plus 4i at negative 1 comma 4, and 2 minus 3i at 2 comma negative 3.
Complex numbers can be added, subtracted, multiplied, and divided. Addition is straightforward: add real parts together and imaginary parts together. For example, 3 plus 2i plus 1 minus 4i equals 4 minus 2i. Geometrically, addition works like vector addition in the complex plane. Multiplication uses the distributive property and the fact that i squared equals negative 1. For instance, 2 plus 3i times 1 plus 2i equals 2 plus 4i plus 3i plus 6i squared, which simplifies to negative 4 plus 7i.
Complex numbers can also be expressed in polar form. Instead of a plus b i, we write r times cosine theta plus i sine theta, or more compactly, r e to the i theta. Here r is the modulus, equal to the square root of a squared plus b squared, and theta is the argument, equal to arctan of b over a. For example, 3 plus 4i has modulus 5 and argument arctan of 4 over 3. Polar form makes multiplication and division much easier to visualize and compute.
De Moivre's theorem states that r cosine theta plus i sine theta to the power n equals r to the n times cosine n theta plus i sine n theta. This makes computing powers much easier. For example, 1 plus i to the eighth power: first convert to polar form as root 2 times cosine 45 degrees plus i sine 45 degrees, then apply the theorem to get 16. For roots, there are always n distinct nth roots. The cube roots of 8 are located at angles 0, 120, and 240 degrees, giving us 2, negative 1 plus i root 3, and negative 1 minus i root 3.