Elementary Linear Algebra, 11th Edition

APPENDIX BCOMPLEX NUMBERS

Complex numbers arise naturally in the course of solving polynomial equations. For example, the solutions of the quadratic equation $a{x}^2+bx+c=0$which are given by the quadratic formula

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

are complex numbers if the expression inside the radical is negative. In this appendix we will review some of the basic ideas about complex numbers that are used in this text.

Complex Numbers

To deal with the problem that the equation $x^2=-1$

$$i=\sqrt{-1}$$

which is assumed to have the property

$$i^2=(\sqrt{-1}{)}^2=-1$$

but which otherwise has the algebraic properties of a real number. An expression of the form

$$\begin{array}{lll}a+bi\hfill & \text{or}\hfill & a+ib\hfill \end{array}$$

in which $a$$b$real numbers is called a complex number. Sometimes it will be convenient to use a single letter, typically $z$

$$\begin{array}{lll}z=a+bi\hfill & \text{or}\hfill & z=a+ib\hfill \end{array}$$

The number $a$real part of $z$$\mathrm{Re}\left(z\right)$$b$imaginary part of $z$$\mathrm{Im}\left(z\right)$

$$\begin{array}{ll}\text{Re}\left(3+2i\right)=3\text{,}\hfill & \text{Im}\left(3+2i\right)=2\hfill \\ \text{Re}\left(1-5i\right)=1\text{,}\hfill & \text{Im}\left(1-5i\right)=\text{Im}(1+(-5\left)i\right)=-5\hfill \\ \text{Re}\left(7i\right)=\text{Re}(0+7i)=0\text{,}\hfill & \text{Im}\left(7i\right)=\text{Im}(0+7i)=7\hfill \\ \text{Re}\left(4\right)=\text{Re}(4+0i)=4\text{,}\hfill & \text{Im}\left(4\right)=\text{Im}(4+0i)=0\hfill \end{array}$$

Two complex numbers are considered equal if and only if their real parts are equal and their imaginary parts are equal; that is,

$$\begin{array}{lllll}a+bi=c+di\hfill & \text{if and only if}\hfill & a=c\hfill & \text{and}\hfill & b=d\hfill \end{array}$$

A complex number $z=bi$pure imaginary. A complex number $z=a$

Complex numbers are added, subtracted, and multiplied in accordance with the standard rules of algebra but with $i^2=-1$

(1)$$\boxed{\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i}$$

(2)$$\boxed{\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i}$$

(3)$$\boxed{\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i}$$

Multiplication formula (3) is obtained by expanding the left side and using the fact that $i^2=-1$$b=0$

(4)$$\boxed{a\left(c+di\right)=ac+adi}$$

The set of complex numbers with these operations is commonly ...