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AppendixAComplex Numbers

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

A = D 0 − 110 E

has characteristic polynomial f ( λ )= λ 2 + 1. A zero of this function is a square root of − 1. 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 − 1.

Definition

  1. The imaginary number i is defined to satisfy the equation i 2 = − 1.
  2. A complex number is a number of the form a + bi , where a , b are real numbers.

The set of all complex numbers is denoted C .

The real numbers are just the complex numbers of the form a + 0 i , so that R is contained in C .

We can identify C with R 2 by a + bi ←→ A ab B . So when we draw a picture of C , we draw the plane:

realaxis imaginaryaxis 1 i 1 − i
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.

  • Addition is performed componentwise:
    ( a + bi )+( c + di )=( a + c )+( b + d ) i .
  • Multiplication is performed using distributivity and i 2 = − 1:
    ( a + bi )( c + di )= ac + adi + bci + bdi 2 =( ac − bd )+( ad + bc ) i .
  • Complex conjugation replaces i with − i , and is denoted with a bar:
    a + bi = a − bi .
    The number a + bi is called the complex conjugate of a + bi . One checks that for any two complex numbers z , w , we have
    z + w = z + w and zw = z · w .
    Also, ( a + bi )( a − bi )= a 2 + b 2 , so z z is a nonnegative real number for any complex number z .
  • The absolute value of a complex number z is the real number | z | = A z z :
    | a + bi | = C a 2 + b 2 .
    One checks that | zw | = | z |·| w | . Beware that | z + w | is usually less than | z | + | w | , in contrast to the situation for positive real numbers.
  • Division by a nonzero real number proceeds componentwise:
    a + bi c = a c + b ci .
  • Division by a nonzero complex number requires multiplying the numerator and denominator by the complex conjugate of the denominator:
    z w = z w w w = z w | w | 2 .
    For example,
    2 + i 1 − i =( 2 + i ) ( 1 − i )( 1 + i )( 1 + i )=( 2 + i )( 1 + i ) 1 2 +( − 1 ) 2 = 1 + 3 i 2 = 1 2 + 3 2 i .
  • The real and imaginary parts of a complex number are
    Re ( a + bi )= a Im ( a + bi )= b .

The point of introducing complex numbers is to find roots of polynomials. It turns out that introducing i 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:

f ( x )= x 2 + bx + c = ⇒ x = − b ±A b 2 − 4 c 2.

For example, if f ( x )= x 2 −A 2 x + 1, then

x = A 2 ±A − 2 2 = A 2 2 ( 1 ± i )= 1 ± i A 2.

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

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

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

0 = f ( λ )= λ n + a n − 1 λ n − 1 + ··· + a 1 λ + a 0 = λ n + a n − 1 λ n − 1 + ··· + a 1 λ + a 0 = f A λ B .

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, f ( x )= x 3 − x = x ( x − 1 )( x + 1 ) has three real roots; its graph looks like this:

On the other hand, the polynomial

g ( x )= x 3 − 5 x 2 + x − 5 =( x − 5 )( x 2 + 1 )=( x − 5 )( x + i )( x − i )

has one real root at 5 and a conjugate pair of complex roots ± i . Its graph looks like this: