Review of Complex Numbers

Subsections

# Review of Complex Numbers

## Cartesian Form and the Complex Plane

• Complex numbers and functions contain the number .

• Any complex number or function can be written in Cartesian form,
 (1)

where is the real part of and is the imaginary part of , often denoted and , respectively. Note that and are both real numbers.

• The form of Eq. 1 is called Cartesian, because if we think of as a two dimensional vector and and as its components, we can represent as a point on the complex plane.

## Polar Form

• As with a two dimensional vector, a complex number can be written in a second form, as a magnitude and angle ,
 (2) (3)

where is called the complex phase of .

## Exponential Form

• Consider the relation
 (4)

This can be shown by comparing the Taylor series expansions of , , and . It follows that can be written in a third form,
 (5)

• Eq. 5 provides a useful way of looking at multiplication of complex numbers. The product is obtained by multiplying magnitudes and adding complex phases,
 (6)

• Raising complex numbers to powers is also simplified by Eq. 5,
 (7)

For example, we can evaluate , noting that

and using Eq. 7, we find

## Complex Conjugation and the Complex Square

• The complex conjugate of is

It is obtained by changing the sign of wherever it appears in .
• To calculate the magnitude directly from written in any form, we use the complex square,

The complex square in terms of and is

and in terms of and

Hence,
 (8)

• We can also use complex conjugation to separate the real and imaginary parts of .

so
 (9)

similarly
 (10)

For example, it follows from Eq.'s 9 and 10 together with Eq. 4 that
 (11)

## Finding Roots

• has unique values for integer . For example, . In general, some or all of the roots are complex numbers.

• The cyclical nature of angles means that

all represent the same number.

• However, if we take the nth root of these representations of , we find that there are unique results with complex phase angles less than .

• Example 1
• The first 6 representations of are

Taking the 6th root, we obtain

The rest of the roots have complex phase and all of them are alternate representations of the six roots above.

• Graphically,

• In general, to find the roots of a number , start with . The remaining roots lie, along with the first, on a circle of radius in the complex plane at an equal spacing of in phase angle.

 Copyright © 2002-2004, Lewis A. Riley Updated Mon Jan 19 13:29:10 2004