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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

z 1 + z 2 = ( x 1 + j y 1 ) + ( x 2 + j y 2 ) = ( x 1 + x 2 ) + j ( y 1 + y 2 ) .

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

z 1 z 2 = ( x 1 + j y 1 ) ( x 2 + j y 2 ) = ( x 1 x 2 - y 1 y 2 ) + j ( y 1 x 2 + x 1 y 2 ) .
This cartesian graph has a three line segments extending from the origin. The middle line follow a positive slope to a point labeled z_1+z_2. The upper line follows a more positive slope to a point labeled a_2 and then from that point to point z_1+z_2 there is a dashed line. The bottom line segment has the the least positive but still positive slope and ends at a point labeled z_1. from that point a dashed line proceeds up to point z_1+z_2. Together the lines form a diamond bisected by the middle line that extends from the origin to the point z_1+z_2. This cartesian graph has a three line segments extending from the origin. The middle line follow a positive slope to a point labeled z_1+z_2. The upper line follows a more positive slope to a point labeled a_2 and then from that point to point z_1+z_2 there is a dashed line. The bottom line segment has the the least positive but still positive slope and ends at a point labeled z_1. from that point a dashed line proceeds up to point z_1+z_2. Together the lines form a diamond bisected by the middle line that extends from the origin to the point z_1+z_2.
Adding Complex Numbers

If the polar representations for z 1 and z 2 are used, then the product may be written as cos ( θ 1 + θ 2 ) = cos θ 1 cos θ 2 - sin θ 1 sin θ 2 and sin ( θ 1 + θ 2 ) = sin θ 1 cos θ 2 + cos θ 1 sin θ 2 to derive this result. We have used the trigonometric identities

z 1 z 2 = r 1 e j θ 1 r 2 e j θ 2 = ( r 1 cos θ 1 + j r 1 sin θ 1 ) ( r 2 cos θ 2 + j r 2 sin θ 2 ) = ( r 1 c o s θ 1 r 2 c o s θ 2 - r 1 s i n θ 1 r 2 s i n θ 2 ) + j ( r 1 s i n θ 1 r 2 c o s θ 2 + r 1 c o s θ 1 r 2 s i n θ 2 ) = r 1 r 2 cos ( θ 1 + θ 2 ) + j r 1 r 2 sin ( θ 1 + θ 2 ) = r 1 r 2 e j ( θ 1 + θ 2 ) .

We say that the magnitudes multiply and the angles add. As illustrated in [link] , the product z 1 z 2 lies at the angle ( θ 1 + θ 2 ) .

This cartesian graph contains three line segments extending from the origrin at different positive slopes. The upper line extends at the most positive slop to a point labeled z_1z_2. About three fourths of the way up this line there is the label r_1r_2. The middle line extends from the origin to a point labeled z_2. The bottom line extends from the origin to a point labeled z_1. From the x-axis to the middle of the upper line there is a curved line intesecting all of the previously mentioned line segments. The area of the curved line between the bottom and middle line segment is labeled θ_1+θ_2. This cartesian graph contains three line segments extending from the origrin at different positive slopes. The upper line extends at the most positive slop to a point labeled z_1z_2. About three fourths of the way up this line there is the label r_1r_2. The middle line extends from the origin to a point labeled z_2. The bottom line extends from the origin to a point labeled z_1. From the x-axis to the middle of the upper line there is a curved line intesecting all of the previously mentioned line segments. The area of the curved line between the bottom and middle line segment is labeled θ_1+θ_2.
Multiplying Complex Numbers

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 θ 1 and z 2 is the complex number z 2 = e j θ 2 , then the product of z 1 and z 2 is

z 1 z 2 = z 1 e j θ 2 = r 1 e j ( θ 1 + θ 2 ) .

As illustrated in [link] , z 1 z 2 is just a rotation of z 1 through the angle θ 2 .

The Cartesian graph contains two line segments extending from the origin. The upper line follows a positive slope to point z_1e^{jθ_2}. The bottom line follows a positive slope to a point labeled z_1. There is a curved line in between these lines the labels the angle created by these line segments as θ_2. The Cartesian graph contains two line segments extending from the origin. The upper line follows a positive slope to point z_1e^{jθ_2}. The bottom line follows a positive slope to a point labeled z_1. There is a curved line in between these lines the labels the angle created by these line segments as θ_2.
Rotation of Complex Numbers

Powers. If the complex number z 1 multiplies itself N times, then the result is

( z 1 ) N = r 1 N e j N θ 1 .

This result may be proved with a simple induction argument. Assume z 1 k = r 1 k e j k θ 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 ) θ 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 θ 1 , ... , to N θ 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 + j y = r e j θ is the complex conjugate

z * = x - j y = r e - j θ .

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

The Cartesian graph contains two line segments extending from the origin. The y axis is labeled the imaginary axis and the x-axis is labeled the real axis. The upper line follows a positive slope from the origin to a point labeled z. About half way up on this line is the label r. The bottom line follows a negative slope from the origin to a point labeled z^*. About half way up on this line is the label r. About halfway up on this line is a curve labeling the angle from the line to the x-axis -θ. Another curve extends from the x-axis to the upper line labeling the angle θ. The Cartesian graph contains two line segments extending from the origin. The y axis is labeled the imaginary axis and the x-axis is labeled the real axis. The upper line follows a positive slope from the origin to a point labeled z. About half way up on this line is the label r. The bottom line follows a negative slope from the origin to a point labeled z^*. About half way up on this line is the label r. About halfway up on this line is a curve labeling the angle from the line to the x-axis -θ. Another curve extends from the x-axis to the upper line labeling the angle θ.
A Complex Variable and Its Complex Conjugate

Magnitude Squared. The product of z and its complex conjugate is called the magnitude squared of z and is denoted by | z | 2 :

| z | 2 = z * z = ( x - j y ) ( x + j y ) = x 2 + y 2 = r e - j θ r e j θ = r 2 .

Note that | z | = 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:

z 1 + z 2 = z 2 + z 1 z 1 z 2 = z 2 z 1
( z 1 + z 2 ) + z 3 = z 1 + ( z 2 + z 3 ) z 1 ( z 2 z 3 ) = ( z 1 z 2 ) z 3 z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 .

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

z + 0 = z = 0 + z z 1 = z = 1 z .

In this field, the complex number - z = - x + j ( - y ) is the additive inverse of z , and the complex number z - 1 = x x 2 + y 2 + j ( - y x 2 + y 2 ) is the multiplicative inverse:

z + ( - z ) = 0 z z - 1 = 1 .

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.

This Cartesian graph contains six line segments extending from the origin. Four line segments exist in quadrant I. The uppermost line in this quadrant extends from the origin to a point labeled prod. The line below this on is shorter and extends to a point labeled z2. The line below this one is the longest line and extends to a point labeled sum. The bottom line is the shortest and extends to a point labeled z1. In quadrant II there is only one line segment. The line segment follows a negative slope from the origin to perp(z2). There are no lines in quadrand III. Quandrants four also contain a single line segment. It follows a negative slope and extends from the origin to a point labeled conj(z1). This Cartesian graph contains six line segments extending from the origin. Four line segments exist in quadrant I. The uppermost line in this quadrant extends from the origin to a point labeled prod. The line below this on is shorter and extends to a point labeled z2. The line below this one is the longest line and extends to a point labeled sum. The bottom line is the shortest and extends to a point labeled z1. In quadrant II there is only one line segment. The line segment follows a negative slope from the origin to perp(z2). There are no lines in quadrand III. Quandrants four also contain a single line segment. It follows a negative slope and extends from the origin to a point labeled conj(z1).
Complex Numbers (Demo 1.2)

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] .

A graph of two concentric swirl designs. A graph of two concentric swirl designs.
Powers of z

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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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