2.2 Complex numbers: algebra of complex numbers

The complex numbers form a mathematical “field” on which the usual operations of addition and multiplication are defined. Each of these operationshas a simple geometric interpretation.

Addition and multiplication.

The complex numbers $z_1$ and $z_2$ are

added according to the rule

We say that the real parts add and the imaginary parts add. As illustrated in [link] , the complex number $z_1+z_2$ is computed from a “parallelogram rule,” wherein $z_1+z_2$ lies on the node of a parallelogram formed from $z_1$ and $z_2$ .

The product of z ₁ and z ₂ is

If the polar representations for $z_1$ and $z_2$ are used, then the product may be written as $cos({\theta }_1+{\theta }_2)=cos{\theta }_1$ cos ${\theta }_2-$ sin θ ₁ sin θ ₂ and $sin({\theta }_1+{\theta }_2)=sin{\theta }_1$ cos ${\theta }_2+cos{\theta }_1$ sin θ ₂ to derive this result. We have used the trigonometric identities

We say that the magnitudes multiply and the angles add. As illustrated in [link] , the product $z_1{z}_2$ lies at the angle $({\theta }_1+{\theta }_2)$ .

Rotation. There is a special case of complex multiplication that will become very important in our study of phasors in the chapter on Phasors . When $z_1$ is the complex number $z_1=r_1{e}^{j{\theta }_1}$ and $z_2$ is the complex number $z_2=e^{j{\theta }_2}$ , then the product of $z_1$ and $z_2$ is

As illustrated in [link] , $z_1{z}_2$ is just a rotation of $z_1$ through the angle ${\theta }_2$ .

Powers. If the complex number $z_1$ multiplies itself $N$ times, then the result is

This result may be proved with a simple induction argument. Assume $z_1^k=r_1^k{e}^{jk{\theta }_1}$ . (The assumption is true for $k=1.$ ) Then use the recursion $z_1^{k+1}=z_1^k{z}_1=r_1^{k+1}{e}^{j(k+1){\theta }_1}$ . Iterate this recursion (or induction) until $k+1=N$ . Can you see that, as n ranges from $n=1,...,N$ , the angle of z§ranges from θ ₁ to $2{\theta }_1,...$ , to $N{\theta }_1$ and the radius ranges from r ₁ to $r_1^2,...$ , to $r_1^N$ ? This result is explored more fully in Problem 1.19.

Complex Conjugate. Corresponding to every complex number $z=$ $x+jy=r{e}^{j\theta }$ is the complex conjugate

The complex number $z$ and its complex conjugate are illustrated in [link] . The recipe for finding complex conjugates is to “change $jto-j$ . This changes the sign of the imaginary part of the complex number.

Magnitude Squared. The product of z and its complex conjugate is called the magnitude squared of z and is denoted by ${\left|z\right|}^2$ :

Note that $\left|z\right|=r$ is the radius, or magnitude, that we defined in "Geometry of Complex Numbers" .

Commutativity, Associativity, and Distributivity. The complex numbers commute, associate, and distribute under addition and multiplication as follows:

Identities and Inverses. In the field of complex numbers, the complex number $0+j0$ (denoted by 0) plays the role of an additive identity, and the complex number $1+j0$ (denoted by 1) plays the role of a multiplicative identity:

In this field, the complex number $-z=-x+j(-y)$ is the additive inverse of z , and the complex number $z^{-1}=\frac{x}{x^2+y^2}+j\left(\frac{-y}{x^2+y^2}\right)$ is the multiplicative inverse:

Demo 1.2 (MATLAB). Create and run the following script file (name it Complex Numbers) If you are using PC-MATLAB, you will need to name your file cmplxnos.m.

With the help of Appendix 1, you should be able to annotate each line of this program. View your graphics display to verify the rules for add, multiply,conjugate, and perp. See [link] .