Appendix B - Complex Numbers

Complex numbers arise naturally from an attempt to solve the equation

a^{2}+1=0.\,\!

It's easy enough to write such an equation down, but how would you solve it? The answer is

a=\pm {\sqrt {-1}}=\pm i.\,\!

We let the symbol $i\,\!$ represent ${\sqrt {-1}}\,\!$ , so that $i^{2}=-1\,\!$ . Then any number of the form


	$z=x+iy,\,\!$	(B.1)

where $x\,\!$ and $y\,\!$ are real is called a complex number. Complex numbers arise in a variety of applications, including quantum mechanics. Suppose we have two complex numbers to be $z_{1}=x_{1}+iy_{1}\,\!$ and $z_{2}=x_{2}+iy_{2}\,\!$ . Then the two complex numbers are equal,

z_{1}=z_{2}.\,\!

This means that

x_{1}+iy_{1}=x_{2}+iy_{2},\,\!

is true if and only if

x_{1}=x_{2},\;\;{\mbox{ and }}\;\;y_{1}=y_{2}.\,\!

We refer to $x\,\!$ as the real part of the complex number $z\,\!$ and $y\,\!$ as the imaginary part. Sometimes these are written as Re( $z\,\!$ ) and Im( $z\,\!$ ) respectively. That is, $x=\,\!$ Re( $z\,\!$ ) and $y=\,\!$ Im( $z\,\!$ ).

We may restate the equivalence condition as $z_{1}=z_{2}\,\!$ if and only if the real part of $z_{1}\,\!$ is equal to the real part of $z_{2}\,\!$ and the imaginary part of $z_{1}\,\!$ is equal to the imaginary part of $z_{2}\,\!$ .

To add two complex numbers, add the real parts and the imaginary parts separately:

z_{1}+z_{2}=(x_{1}+iy_{1})+(x_{2}+iy_{2})=(x_{1}+x_{2})+i(y_{1}+y_{2}).\,\!

Complex numbers are multiplied like any other binomial expression (recall FOIL: First two, Outside two, Onside two, Last two):

{\begin{aligned}z_{1}z_{2}&=(x_{1}+iy_{1})(x_{2}+iy_{2})\\&=x_{1}x_{2}+iy_{1}x_{2}+ix_{1}y_{2}-y_{1}y_{2}\\&=x_{1}x_{2}-y_{1}y_{2}+i(y_{1}x_{2}+x_{1}y_{2}),\end{aligned}}\,\!

where we have used $i^{2}=-1\,\!$ .

The complex conjugate of the complex number $z\,\!$ is denoted $z^{*}\,\!$ and is given by


	$z^{*}=x-iy.\,\!$	(B.2)

Taking the complex conjugate of the number means just changing the sign in front of the $i\,\!$ . One reason for defining this is that a number times its own complex conjugate is real,

zz^{*}=(x+iy)(x-iy)=x^{2}+y^{2}+i(xy-yx)=x^{2}+y^{2}.\,\!

The complex conjugate of the complex conjugate is the original complex number, and

z^{*}z=(x-iy)(x+iy)=x^{2}+y^{2}+i(yx-xy)=x^{2}+y^{2}=zz^{*}.\,\!

We also call this the modulus squared . The modulus is


	$\|z\|={\sqrt {(z^{*}z)}}={\sqrt {x^{2}+y^{2}}}.\,\!$	(B.3)

Note that the complex conjugate of a product is the product of the complex conjugates:

(z_{1}z_{2})^{*}=z_{1}^{*}z_{2}^{*}.\,\!

It is often useful to look at a graph for a complex number, consisting of an x-axis for the real part and a y-axis for the complex part. This is shown in Fig. B.1. In this way, we can think of $z\,\!$ as a two-dimensional vector with the magnitude (length) being equivalent to the modulus of the complex number, $|z|={\sqrt {x^{2}+y^{2}}}\,\!$ . In some sense (like looking at the complex number as a vector), the modulus of the complex number is the length, or magnitude, of the complex number.

Figure B.1: A complex number in Cartesian coordinates.

Another useful way to represent this is with polar coordinates. We can do this by writing

z=|z|(\cos \theta +i\sin \theta ).\,\!

Clearly if we wanted to rewrite this in terms of $x\,\!$ and $y\,\!$ , we find that $x=|z|\cos \theta \,\!$ and $y=|z|\sin \theta \,\!$ .

A very famous identity exists between the exponential function and sine and cosine,


	$e^{i\theta }=\cos \theta +i\sin \theta ,\,\!$	(B.4)

where $e\,\!$ is the exponential function, which is a number that is approximately 2.71828. Using this equation, we could also write

z=|z|e^{i\theta }.\,\!

This is really just a short-hand notation for us and most of the time, when we calculate something, we will use the $\sin \theta ,\cos \theta \,\!$ functions. But is often the case that people will write this as


	$z=re^{i\theta },\,\!$	(B.5)

where $r={\sqrt {x^{2}+y^{2}}}\,\!$ as is usual for polar coordinates. So everything is just like polar coordinates with the exception of the inclusion of the factor $i\,\!$ . (See Fig. B.2.)

Figure B.2: A polar coordinate representation of a complex number.

Exercises

Let $z_{1}=5+2i\,\!$ and $z_{2}=3+i\,\!$ . For 2-6, write the answer in terms of the real and complex components, i.e., in the form $x+iy\,\!$ , where $x\,\!$ and $y\,\!$ are real numbers.