7.1: A - Complex Numbers

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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

\[ A =\left(\begin{array}{cc}0&-1\\1&0\end{array}\right) \nonumber \]

has characteristic polynomial \(f(\lambda) = \lambda^2 + 1\). 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 \(\PageIndex{1}\): Imaginary Number and Complex Number

1. The imaginary number \(i\) is defined to satisfy the equation \(i^2 = -1\).
2. A complex number is a number of the form \(a+bi\text{,}\) where \(a,b\) are real numbers.

The set of all complex numbers is denoted \(\mathbb{C}\).

The real numbers are just the complex numbers of the form \(a + 0i\text{,}\) so that \(\mathbb{R}\) is contained in \(\mathbb{C}\).

We can identify \(\mathbb{C}\) with \(\mathbb{R}^2 \) by \(a+bi \longleftrightarrow {a\choose b}\). So when we draw a picture of \(\mathbb{C}\text{,}\) we draw the plane:

Figure \(\PageIndex{1}\)

Note \(\PageIndex{1}\): 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 component-wise:

  \[ (a + bi) + (c + di) = (a + c) + (b + d)i. \nonumber \]

• Multiplication is performed using distributivity and \(i^2=-1\text{:}\)

  \[ (a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac-bd) + (ad+bc)i. \nonumber \]

• Complex conjugation replaces \(i\) with \(-i\text{,}\) and is denoted with a bar:    

  \[ \overline{a+bi} = a - bi. \nonumber \] The number \(\overline{a+bi}\) is called the complex conjugate of \(a+bi\). One checks that for any two complex numbers \(z,w\text{,}\) we have

  \[ \overline{z+w} = \overline{ z} + \overline{ w} \quad\text{and}\quad \overline{zw} = \overline{z}\cdot\overline{w}. \nonumber \] Also, \((a+bi)(a-bi) = a^2 + b^2\text{,}\) so \(z\bar z\) is a nonnegative real number for any complex number \(z\).

• The absolute value of a complex number \(z\) is the real number \(|z| = \sqrt{z\overline{ z}}\text{:}\)

  \[ |a+bi| = \sqrt{a^2 + b^2}. \nonumber \] One checks that \(|zw| = |z|\cdot|w|.\)

• Division by a nonzero real number proceeds component-wise:

  \[ \frac{a+bi}c = \frac ac + \frac bci. \nonumber \]

• Division by a nonzero complex number requires multiplying the numerator and denominator by the complex conjugate of the denominator:

  \[ \frac zw = \frac{z\overline{ w}}{w\overline{ w}} = \frac{z\overline{ w}}{|w|^2}. \nonumber \] For example,

  \[ \frac{1+i}{1-i} = \frac{(1+i)^2}{1^2+(-1)^2} = \frac{1+2i+i^2}2 = i. \nonumber \]

• The real and imaginary parts of a complex number are   

  \[ \Re(a+bi) = a \qquad \Im(a+bi) = b. \nonumber \]

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.

Theorem \(\PageIndex{1}\): Fundamental Theorem of Algebra

Every polynomial of degree \(n\) has exactly \(n\) (real and) complex roots, counted with multiplicity.

Equivalently, if \(f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) is a polynomial of degree \(n\text{,}\) then \(f\) factors as

\[ f(x) = (x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_n) \nonumber \]

for (not necessarily distinct) complex numbers \(\lambda_1,\lambda_2,\ldots,\lambda_n\).

Note \(\PageIndex{2}\): Degree-2 Polynomials

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

\[ f(x) = x^2 + bx + c \implies x = \frac{-b \pm \sqrt{b^2 - 4c}}2. \nonumber \]

For example, if \(f(x) = x^2 - \sqrt 2x + 1\text{,}\) then

\[ x = \frac{\sqrt 2\pm\sqrt{-2}}2 = \frac{\sqrt 2}2(1\pm i) = \frac{1\pm i}{\sqrt 2}. \nonumber \]

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 \(z = \bar z\). This leads to the following observation.

Note \(\PageIndex{3}\)

If \(f\) is a polynomial with real coefficients, and if \(\lambda\) is a complex root of \(f\text{,}\) then so is \(\overline{\lambda}\text{:}\)

\[\begin{aligned}0=\overline{f(\lambda )}&=\overline{\lambda^n+a_{n-1}\lambda^{n-1}+\cdots +a_1\lambda +a_0} \\ &=\overline{\lambda}^n+a_{n-1}\overline{\lambda}^{n-1}+\cdots +a_1\overline{\lambda}+a_0=f(\overline{\lambda}).\end{aligned}\]

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

Note \(\PageIndex{4}\): 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:

Figure \(\PageIndex{2}\)

On the other hand, the polynomial

\[ g(x) = x^3-5x^2+x-5 = (x-5)(x^2+1) = (x-5)(x+i)(x-i) \nonumber \]

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

Figure \(\PageIndex{3}\)