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Complex Numbers: Geometry of Complex Numbers

Module by: Louis Scharf. E-mail the author

Note:

This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

The most fundamental new idea in the study of complex numbers is the “imaginary number” jj. This imaginary number is defined to be the square root of - 1 -1:

j = - 1 j = - 1
(1)
j 2 = - 1 . j 2 = - 1 .
(2)

The imaginary number jj is used to build complex numbers xx and yy in the following way:

z=x+jy .z=x+jy.
(3)

We say that the complex number zz has “real part” xx and “imaginary part” yy:

z = Re [ z ] + j Im [ z ] z = Re [ z ] + j Im [ z ]
(4)
Re [ z ] = x ; Im [ z ] = y . Re [ z ] = x ; Im [ z ] = y .
(5)

In MATLAB, the variable xx is denoted by real(z), and the variable yy is denoted by imag(z). In communication theory, xx is called the “in-phase” component of zz, and yy is called the “quadrature” component. We call z=z=x+jyx+jy the Cartesian representation of zz, with real component xx and imaginary component yy. We say that the Cartesian pair (x,y)(x,y)codes the complex number zz.

We may plot the complex number zz on the plane as in Figure 1. We call the horizontal axis the “real axis” and the vertical axis the “imaginary axis.” The plane is called the “complex plane.” The radius and angle of the line to the point z=x+jyz=x+jy are

r = x 2 + y 2 r = x 2 + y 2
(6)
θ = tan - 1 ( y x ) . θ = tan - 1 ( y x ) .
(7)

See Figure 1. In MATLAB, rr is denoted by abs(z), and θθ is denoted by angle(z).

Figure 1: Cartesian and Polar Representations of the Complex Number z
This Cartesian graph contains a line segment extending from the origin to a point labeled z=x-jy=re^{jθ}. The line segment has a positive slope and is labeled r. The angle between this line and the x axis is marked by a curved line and it labeled θ. There is a point in the middle of the positive side of the x axis and it is labeled x=rcosθ. There is a similar point on the positive side of the y axis and it is labeled rsinθ=y. The y axis is labeled imaginary axis (j) and the x axis is labeled real axis.

The original Cartesian representation is obtained from the radius rr and angle θθ as follows:

x=rcos θx=rcosθ
(8)
y = r s i n θ . y = r s i n θ .
(9)

The complex number zz may therefore be written as

z = x + j y = r cos θ+jrsin θ = r ( cos θ + j s i n θ ) . z = x + j y = r cos θ+jrsin θ = r ( cos θ + j s i n θ ) .
(10)

The complex number cosθ+jsinθcosθ+jsinθ is, itself, a number that may be represented on the complex plane and coded with the Cartesian pair (cosθ,sin θ) (cosθ,sinθ). This is illustrated in Figure 2. The radius and angle to the point z=cosθ+jsin θz=cosθ+jsinθ are 1 and θθ. Can you see why?

Figure 2: The Complex Number cosθ+jsinθcosθ+jsinθ
 This Cartesian graph contains a line segment extending from the origin to a point labeled cosθ+jsinθ=e^{jθ}. The line segment has a positive slope and is labeled 1. The angle between this line and the x axis is marked by a curved line and it labeled θ. There is a point in the middle of the positive side of the x axis and it is labeled cosθ. There is a similar point on the positive side of the y axis and it is labeled sinθ. The y axis is labeled imaginary axis and the x axis is labeled real axis.

The complex number cosθ+jsinθcosθ+jsinθ is of such fundamental importance to our study of complex numbers that we give it the special symbol ejθejθ :

ejθ=cosθ+jsinθ.ejθ=cosθ+jsinθ.
(11)

As illustrated in Figure 2, the complex number ejθejθ has radius 1 and angle θθ. With the symbol ejθejθ, we may write the complex number zz as

z=rejθ.z=rejθ.
(12)

We call z=rejθz=rejθ a polar representation for the complex number zz. We say that the polar pair rθrθcodes the complex number zz. In this polar representation, we define |z|=r|z|=r to be the magnitude of zz and arg(z)=θarg(z)=θ to be the angle, or phase, of zz:

| z | = r | z | = r
(13)
a r g ( z ) = θ . a r g ( z ) = θ .
(14)

With these definitions of magnitude and phase, we can write the complex number zz as

z=|z|ejarg(z).z=|z|ejarg(z).
(15)

Let's summarize our ways of writing the complex number z and record the corresponding geometric codes:

z = x+jy = r e j θ = | z | e j arg ( z ) . ( x , y ) r θ z = x+jy = r e j θ = | z | e j arg ( z ) . ( x , y ) r θ
(16)

In "Roots of Quadratic Equations" we show that the definition ejθ=cosθ+jsinθejθ=cosθ+jsinθ is more than symbolic. We show, in fact, that ejθejθ is just the familiar function e x e x evaluated at the imaginary argument x=jθx=jθ. We call ejθejθ a “complex exponential,” meaning that it is an exponential with an imaginary argument.

Exercise 1

Prove (j)2n=(-1)n(j)2n=(-1)n and (j)2n+1=(-1)nj(j)2n+1=(-1)nj. Evaluate j3,j4,j5j3,j4,j5.

Exercise 2

Prove ej[(π/2)+m2π]=j,ej[(3π/2)+m2π]=-j,ej(0+m2π)=1ej[(π/2)+m2π]=j,ej[(3π/2)+m2π]=-j,ej(0+m2π)=1, and ej(π+m2π)=-1ej(π+m2π)=-1. Plot these identities on the complex plane. (Assume mm is an integer.)

Exercise 3

Find the polar representation z=rejθz=rejθ for each of the following complex numbers:

  1. z=1+j0z=1+j0;
  2. z=0+j1z=0+j1;
  3. z=1+j1z=1+j1;
  4. z=-1-j1z=-1-j1.
Plot the points on the complex plane.

Exercise 4

Find the Cartesian representation z=x+jyz=x+jy for each of the following complex numbers:

  1. z=2ejπ/2z=2ejπ/2 ;
  2. z=2ejπ/4z=2ejπ/4;
  3. z=ej3π/4z=ej3π/4 ;
  4. z=2ej3π/2z=2ej3π/2.
Plot the points on the complex plane.

Exercise 5

The following geometric codes represent complex numbers. Decode each by writing down the corresponding complex number z:

  1. (0.7,-0.1)z=(0.7,-0.1)z= ?
  2. (-1.0,0.5)z=(-1.0,0.5)z= ?
  3. 1.6π/8z=1.6π/8z=?
  4. 0.47π/8z=0.47π/8z=?

Exercise 6

Show that Im[jz]= Re [z]Im[jz]= Re [z] and Re [-jz]= Im [z] Re [-jz]= Im [z]. Demo 1.1 (MATLAB). Run the following MATLAB program in order to compute and plot the complex number ejθejθ for θ=i2π/360,i=1,2,...,360θ=i2π/360,i=1,2,...,360:


          j=sqrt(-1)
          n=360
          for i=1:n,circle(i)=exp(j*2*pi*i/n);end;
          axis('square')
          plot(circle)
Replace the explicit for loop of line 3 by the implicit loop

          circle=exp(j*2*pi*[1:n]/n);
to speed up the calculation. You can see from Figure 3 that the complex number ejθejθ, evaluated at angles θ=2π/360,2(2π/360),...θ=2π/360,2(2π/360),..., turns out complex numbers that lie at angle θθ and radius 1. We say that ejθejθ is a complex number that lies on the unit circle. We will have much more to say about the unit circle in Chapter 2.

Figure 3: The Complex Numbers ejθejθ for 0θ2π0θ2π (Demo 1.1)
This Cartesian graph contains a circle that is equally present in all four quadrants. The width of the circle is from -1 to 1 and the height is from -1 to 1 as well.

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