Complex Arithmetic

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In the introductory talk we introduced complex numbers, and mentioned that they are usually represented as points in a plane. This exercise is designed to help you visualize the effect of various algebraic operations on complex numbers.

The blue command line:

is the place where you may define complex number variables, and manipulate them. We will highlight commands that you type with the color blue.  Responses will be shown in red. You should first execute the following commands on the command line.  After each command, press Enter to execute it.  Notice, as indicated below, we always use an asterisk  to denote multiplication in commands.

readfloat;
precision 7;

There is a special function called Complex that you may use to define complex variables in the following way. If you type, say:

make v complex(1/2,1);

then the variable v is assigned to the complex number .  In fact, the system prints:

Defined: v =  in the Response field.

This is the decimal form of the variable.

If you want the exact rational form, type instead:

make v exact complex(1/2,1);

and it prints in the response field:

Defined: v =  in the Response field.

The word "exact" forces the rational form to be kept. As you make these variable definitions in decimal form, i.e.

make v complex(1/2,1);

the system will also draw the point in the plane on the right (It does not draw the point in the case that you use the exact form). For that, and for all drawing, it uses the present color in the color palette.

So, if you want to draw in a new color, select the color first, by clicking the Choose Color button.

Now the idea is to create a few variables, and then to plot various combinations and functions of them. This is how to do that. Suppose you created variables r and s by typing:

make r complex(-1/2, sqrt(3)/2);
make s complex(-1/2, -sqrt(3)/2);

If the palette was set to red when you created them, and if you then set the palette to light blue and typed:

show(4*r);
show (4*s);

then set it to light green and typed:

show(4*r*s);

you would see the following.

-2+3.464102*i
-2-3.464102*i
4+0*i

In this way, you get both algebraic (symbolic) and pictorial representations of these operations. Suppose next you clear the screen by pressing the Clear button, and you type:

show(2*r);
show(-2*s);
show(2*(r-s));

you would see the following.

-1+1.732051*i
1+1.732051*i
0+3.464102*i

This illustrates the law of addition of vectors. What can you do with these complex variables? You may multiply them by (real) constants, as in . You may add together any "linear" combination, such as , etc. You may multiply any number and combination, as in  and so on. You may raise them to positive integer powers. For example:

show( (2*r)^2 );

gives:

and prints:  

To raise complex numbers to negative powers, you must use the inv() function. If you type:

show(r);
show( inv(r) );

the system prints:

and draws:

showing, by the way, that s is the inverse of r.

In addition to the inv() function, there is the built-in conjugate() function. It produces the conjugate of its argument. The important property of conjugation as we pointed out, is this:

    , or in standard notation:

Thus,

show(s + conjugate(s));

gives -1 as would be expected.

These complex numbers are actually represented internally as matrices. You may define your own functions of one or several variables to manipulate these complex variables. Just use the following form for example to define the function of complex numbers:

 

make function f (ma x) of type matrix x^2-3*x;

Then you may for example, calculate expressions like:

show( f(r+s) );

Finally, we may calculate the norm (for example,  ) of any complex expression. For example, if you type:

calc norm(RS);

the system will print: 1.732051 Now with all of this said, it is time to use this exercise.

1) The equation:  has exactly n solutions in the complex plane. How are they arranged geometrically? Hint: Notice that if , then

 

Now use Euler's Formula (previous page) to arrive at the answer.

2) Show that both r and s are solutions to the cubic equation: .  These are two of the cube roots of unity. What is the third? We call the numbers r and s primitive cube roots of unity because they each generate, through their powers, all other cube roots of unity. What, explicitly are the solutions to:

   (the fourth roots of unity)

   (the fifth roots of unity)

3) Show the sum of the nth roots of unity must be 0 for any n > 1, but try it out here first with some example n.

4) Show that the conjugate of an nth root of unity is an nth root of unity, and is in fact the inverse of the original one!

5) Check the identity:

These exercises, which may all be checked on this exploration page should prepare you for what is coming next. We are nearly ready to discuss the general strategy for solving cubic and quartic equations by introducing a new type of number in the Algebra of Cubic Numbers.