In this Appendix we give a brief review of the arithmetic and basic properties of the complex numbers.

As motivation, notice that the rotation matrix

has characteristic polynomial A zero of this function is a square root of If we want this polynomial to have a root, then we have to use a larger number system: we need to declare by fiat that there exists a square root of

##### Definition

1. The imaginary number is defined to satisfy the equation
2. A complex number is a number of the form where are real numbers.

The set of all complex numbers is denoted

The real numbers are just the complex numbers of the form so that is contained in

We can identify with by So when we draw a picture of we draw the plane:

##### Arithmetic of Complex Numbers

We can perform all of the usual arithmetic operations on complex numbers: add, subtract, multiply, divide, absolute value. There is also an important new operation called complex conjugation.

• Multiplication is performed using distributivity and
• Complex conjugation replaces with and is denoted with a bar:
The number is called the complex conjugate of One checks that for any two complex numbers we have
Also, so is a nonnegative real number for any complex number
• The absolute value of a complex number is the real number
One chacks that
• Division by a nonzero real number proceeds componentwise:
• Division by a nonzero complex number requires multiplying the numerator and denominator by the complex conjugate of the denominator:
For example,
• The real and imaginary parts of a complex number are

The point of introducing complex numbers is to find roots of polynomials. It turns out that introducing is sufficent to find the roots of any polynomial.

##### Degree-2 Polynomials

The quadratic formula gives the roots of a degree-2 polynomial, real or complex:

For example, if then

Note that if are real numbers, then the two roots are complex conjugates.

A complex number is real if and only if This leads to the following observation.

If is a polynomial with real coefficients, and if is a complex root of then so is

Therefore, complex roots of real polynomials come in conjugate pairs.

##### Degree-3 Polynomials

A real cubic polynomial has either three real roots, or one real root and a conjugate pair of complex roots.

For example, has three real roots; its graph looks like this:

On the other hand, the polynomial

has one real root at and a conjugate pair of complex roots Its graph looks like this: