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 $\lambda_1,\lambda_2,\ldots,\lambda_n$.

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.

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 $\PageIndex{3}$