Complex Numbers
i² = −1: the second dimension of numbers
Summary
Complex numbers have the form a + bi, where i² = −1. They are represented not on a line but on a plane: the Argand plane. Drag the two points and see the geometric meaning of addition, subtraction, multiplication and division.
In depth
Real numbers line up along a line; complex numbers spread out over a plane. A complex number z = a + bi has real part a and imaginary part b, and is shown as the point (a, b) on the Argand plane. When two complex numbers are added, the parallelogram rule for vectors applies. Multiplication follows a more surprising rule: the sizes multiply and the angles add. That is why multiplying by i is a 90° anticlockwise rotation (i·i = −1 = a 180° turn). Complex numbers appeared in the 16th century in the solution of cubic equations, but for a long time mathematicians regarded them as "imaginary" numbers. Today they are indispensable in modern physics (quantum mechanics, electromagnetic waves), electrical engineering (AC circuit analysis, signal processing) and computer graphics (generating fractals).
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