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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=D0−110E

has characteristic polynomial f(λ)=λ2+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 i2=−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+0i, so that R is contained in C.

We can identify C with R2 by a+bi←→AabB. So when we draw a picture of C, we draw the plane:

realaxisimaginaryaxis1i1−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 i2=−1:
    (a+bi)(c+di)=ac+adi+bci+bdi2=(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+wandzw=z·w.
    Also, (a+bi)(a−bi)=a2+b2, so zz is a nonnegative real number for any complex number z.
  • The absolute value of a complex number z is the real number |z|=Azz:
    |a+bi|=Ca2+b2.
    One chacks that |zw|=|z|·|w|.
  • Division by a nonzero real number proceeds componentwise:
    a+bic=ac+bci.
  • Division by a nonzero complex number requires multiplying the numerator and denominator by the complex conjugate of the denominator:
    zw=zwww=zw|w|2.
    For example,
    1+i1−i=(1+i)212+(−1)2=1+2i+i22=i.
  • The real and imaginary parts of a complex number are
    Re(a+bi)=aIm(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)=x2+bx+c=⇒x=−b±Ab2−4c2.

For example, if f(x)=x2−A2x+1, then

x=A2±A−22=A22(1±i)=1±iA2.

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)=λn+an−1λn−1+···+a1λ+a0=λn+an−1λn−1+···+a1λ+a0=fAλB.

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)=x3−x=x(x−1)(x+1) has three real roots; its graph looks like this:

On the other hand, the polynomial

g(x)=x3−5x2+x−5=(x−5)(x2+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: