2.3 Complex numbers

A complex number may be written in the form:

where and are real numbers and i is a special quantity with the property that .

Each complex number has a real part, Re, and an imaginary part, Im.

Complex numbers can be added,

and multiplied,

using the usual rules of algebra along with .

The complex conjugate of is (pronounced "z star").

This results in showing that is a positive real number, (unless ).

The modulus of the complex number is defined as

which is a real, non-negative quantity.

Complex numbers can also be written in polar form,

where and are real numbers. The relationship between and , and is shown in Figure 3. Here is the modulus of as defined in Equation (4); is known as the phase, and is a phase factor.

Figure 3 A diagram showing the relationship between Cartesian coordinates and , and polar coordinates and

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The fighure shows a graph with two axes, labelled x-axis and y-axis, which intersect near the middle of the diagram, marked 0. An arrow is drawn from the origin at 0 into the upper right quadrant of the graph. The length of the arrow is labelled r and the angle it makes with the horizontal x-axis is labelled theta. A horizontal dashed line from the end of this arrow intersects the y-axis at a point marked y. A vertical dashed line from the end of this arrow intersects the x-axis at a point marked x.

Figure 3 A diagram showing the relationship between Cartesian coordinates and , and polar coordinates and