Definition
- The imaginary number is defined to satisfy the equation
- A complex number is a number of the form where are real numbers.
The set of all complex numbers is denoted
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
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:
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.
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.
Every polynomial of degree has exactly (real and) complex roots, counted with multiplicity.
Equivalently, if is a polynomial of degree then factors as
for (not necessarily distinct) complex numbers
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.
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: