COMPLEX NUMBERS

The square-root of -1 is called i. An imaginary number is the product of a real number and i . A common convention for writing imaginary numbers is to write numbers in front of i, and to write variables after i.

A complex number can be written as the linear sum of a real number and an imaginary number, z = x + iy. It can also be expressed as a function of complex-polar coordinates, z = R Exp(iq), where R is called the absolute value of z, and q is called the imaginary argument of z. The complex-polar coordinates are related to the linear (Cartesian) coordinates:

Sometimes it is easier to operate on z in the linear form, and sometimes it is easier in the polar form. For example, it is easier to differentiate z in the linear form than in the polar form:

On the other hand, it is easier to express a power of z in the polar form:

Every complex number z has a complex-conjugate, often written z*. A complex number and its conjugate have the same real coordinate, but their imaginary coordinates have opposite signs:

The product of a complex number and its conjugate is a real number equal to R²:

or in polar coordinates,

This property is useful when resolving fractions with complex denominators.

If a complex number is the root of a polynomial with real coefficients and exponents, then the complex-conjugate of that number is also a root of the polynomial. So if you find one complex root of a real-parametered polynomial, then you automatically know a second root, the complex conjugate.