Complex numbers — concept

A number with two coordinates

Real numbers fill a line. A complex number has two coordinates, a horizontal position and a vertical position, and is therefore a point in a plane. The point below is draggable, and the dashed lines mark its two coordinates on the axes.

z = a + b\,i

$a$ is the horizontal coordinate, called the real part; $b$ is the vertical coordinate, called the imaginary part. The names are historical labels for the two axes, and both coordinates are ordinary real numbers. The symbol $i$ marks the vertical unit: $b\,i$ means $b$ units in the vertical direction, and the point at $(0, 1)$ is $i$ itself.

Multiplying by $i$ is a quarter turn

Multiplication by $i$ is defined as a quarter turn counterclockwise about the origin. The widget applies it three times to the draggable point: $i\,z$ is $z$ turned $90^\circ$ , $i^2 z$ is $z$ turned $180^\circ$ , and $i^3 z$ is $z$ turned $270^\circ$ . At every position of $z$ , the readouts show the same relation: $i^2 z$ is exactly $-z$ , the point opposite $z$ .

i^2 = -1

Two quarter turns are a half turn, and a half turn sends every point to its negative. The $i\,z$ readout shows the coordinate rule for one quarter turn: $i\,(a + bi) = -b + a\,i$ . The coordinates swap, and the new real part changes sign.

Addition is tip-to-tail

Adding complex numbers adds their coordinates. Geometrically, one arrow's tail is placed at the other's tip, and the sum runs from the origin to the combined endpoint; the dashed copies below complete the parallelogram. When either arrow is dragged, the coordinates add independently: real parts with real parts, and imaginary parts with imaginary parts.

(a + b i) + (c + d i) = (a + c) + (b + d)\,i

Real parts add and imaginary parts add, with no interaction between the axes. The rule is the same as for adding any pair of arrows coordinate by coordinate.

Multiplication rotates and scales

Multiplication by a general complex number extends the quarter turn of section 2. Each number is described by the length of its arrow, $|z| = \sqrt{a^2 + b^2}$ , and by its angle from the positive real axis. The product follows two rules, and the widget's readouts verify both as the arrows move:

|z_1 z_2| = |z_1|\,|z_2|, \qquad \angle(z_1 z_2) = \angle z_1 + \angle z_2

Lengths multiply and angles add. Multiplying by a length-1 number is therefore a pure rotation; section 2 was the special case $i$ , with length 1 and angle $90^\circ$ . In coordinates the same operation is $(a+bi)(c+di) = (ac - bd) + (ad + bc)\,i$ , which follows from distributing and applying $i^2 = -1$ .

The green arrow is the product, computed from the coordinate formula. Setting both lengths to 1 keeps the product on the unit circle: the multiplication is then a rotation without scaling.

Check your understanding

In the web

Builds on the assumed floor only — a foundation.