Complex Conjugate - Use As A Variable

Use As A Variable

Once a complex number or is given, its conjugate is sufficient to reproduce the parts of the z-variable:

• Real part:
• Imaginary part:
• Modulus/Absolute Value:
• Argument:, so

Thus the pair of variables and also serve up the plane as do x,y and and . Furthermore, the variable is useful in specifying lines in the plane:

is a line through the origin and perpendicular to since the real part of is zero only when the cosine of the angle between and is zero. Similarly, for a fixed complex unit u = exp(b i), the equation:

determines the line through in the direction of u.

These uses of the conjugate of z as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.