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Complex numbers - a geometric view

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Module 6 : Precalculus

601 : Complex Numbers - A Geometric View


In this project we will examine at complex numbers from both an algebraic and geometric point of view. We will look at where the come from, how to define them in Maple, how to perform mathematical operations, and what these operations mean geometrically.


In this project we will use the following command packages. Type and execute this line before begining the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet.

> restart; with(plots):

Warning, the name changecoords has been redefined


A. Defining Complex Numbers


Complex numbers come about naturally as solutions to polynomial equations.

Note that Maple uses a capital I for the imaginery unit, whereas we normally write this as a small letter i in by hand.

> solve( x^2 + 1 = 0);

I, -I

> solve( x^2 + x+ 1 = 0);

-1/2+1/2*I*sqrt(3), -1/2-1/2*I*sqrt(3)

To define a complex numbers directly, enter the number using capital I for the imaginery unit and Maples usual * for multiplication.

Note that we use := to assign a value to a variable. The sqrt command is the square root function.

> z := 4 + I; w := 1 + 3*I;

z := 4+I

w := 1+3*I

> u := 5 - 4*I; v := -5 + sqrt(5)*I;

u := 5-4*I

v := -5+I*sqrt(5)

we can view these complex numbers by using the complexplot command.

> complexplot( {z,w,u,v }, x = -6..6,
y = -6..6, style = point, color=blue);

[Maple Plot]

We can also define complex numbers in a trigonometric format.

> z := cos(Pi/7) + sin(Pi/7) *I;

z := cos(1/7*Pi)+I*sin(1/7*Pi)

> w := 5*cos(2*Pi/3) + 5*sin(2*Pi/3)*I;

w := -5/2+5/2*I*sqrt(3)

> complexplot( {z,w}, x = -6..6, y = 0..6,
style = point, color=blue);

[Maple Plot]


B. Properties of Complex Numbers


Lets explore some properties of complex numbers.

What does the negative of a complex number look like?

> z := 4 + 3*I;

z := 4+3*I

> display( complexplot( {0,z}, x = -6..6,
complexplot( {0,-z}, x = -6..6,
polarplot( abs(z), scaling=
constrained, color = gold));

[Maple Plot]

Every complex number has a modulus and an argument.

The modulus or absolute value of a complex number, |z| given by the maple command abs(z), is the distance from the number to the origin. The argument of a complex number, given by the Maple command argument(z), is the counter-clockwise angle from the x-axis to the number. This angle is given in radians.

> z := 4 + 3*I;

z := 4+3*I

> abs(z);


> evalf(argument(z));


Lets take a look at what is going on.

> z := 4 + 3*I; w := -1 + 2*I;

z := 4+3*I

w := -1+2*I

> display( complexplot( {0,z}, x = -6..6,color=red),
complexplot( {0,w}, x = -6..6,color=blue),
polarplot( {abs(z),abs(w)}, scaling=constrained, color = gold));

[Maple Plot]

The red line shows the position of z, and the length of the red segment is |z|. The blue line indicates the position of w, and the length of the blue segment is |w|.

Lets compare the argument of z and z2

> a1 := evalf( argument(z)) ;

a1 := .6435011088

> a2 := evalf( argument(z^2)) ;

a2 := 1.287002218

> a2 / a1;



C. The Complex Conjugate


Every complex number has a complex conjugate. The conjugate of a+ib is a-ib, and vice-versa.

> z := 5 + 2*I;

z := 5+2*I

> conjugate(z);


The product of a complex number and its conjugate is always a real number. In fact, not just any real number, the square of the modulus.

The evalc command forces Maple to evaluate this expression as a complex number.

> z := 5 + 2*I;

z := 5+2*I

> z*conjugate(z); evalc(%);



> abs(z)^2;


But what does the conjugate look like?

> display( complexplot( {0,z}, x = -6..6, color=blue),
complexplot( {0,conjugate(z)}, x = -6..6, color=green),
polarplot( abs(z),scaling = constrained ));

[Maple Plot]

The blue line indicates the position of z, and the green line indicates the position of w. The conjugate of a complex number is a reflection of the original number through the x-axis.

How are the arguments of the number and its conjugate related?

> z := 5 + 2*I;

z := 5+2*I

> evalf(argument(z));


> evalf(argument(zcon));