Real numbers

We use to denote the collection of all the real numbers, so that ? ? simply means that ? is a real number. We use curved brackets to denote the open end of an interval, and square brackets to denote the closed end of an interval, so for example

and

One end of the interval is allowed to be ?, for example

Complex numbers

A complex number is a number z of the form z = x + i y, where . Any complex number can be split into its real and imaginary parts, Re[ z] and Im[ z], where

The rules for addition and multiplication of complex numbers follow from applying standard algebra and the fact that i ² = ?1; we have

and

The complex conjugate of a complex number z = x + i y is written z* and is given by z* = x ? i y. Adding z and its complex conjugate yields twice the real part of z,

while their difference gives i multiplied by twice the imaginary part of z,

Also useful is the fact that the complex conjugate of a product is the product of the complex conjugates, ( wz)* = w* z