Complex numbers

Definitions. A point [x,y] in the plane is called a complex number Addition and multiplication of complex numbers are defined by and . The complex numbers [1,0] and [0,1] are written 1 and I respectively, and so each complex number [a,b] = . The real part of is a; the imaginary part is b. The conjugate of is defined as . The absolute value of z is defined as . The argument of z the angle , < that the ray from 0 to z makes with the positive x-axis. So each nonzero complex number z can be written in the form , where and . The set of C complex numbers is called the complex plane . The x-axis is called the real axis and the y-axis is called the imaginary axis.

Theorem. The complex numbers form a field under addition and multiplication. The x-axis (i.e., the complex numbers with imaginary part = 0) is a subfield isomorphic with the real numbers.

vocabulary

> z := 2+3*I;

> z*(3+9*I);

> z+(4-13*I);

> 1/z;

> seq(z^i,i=1..5);

> Re(z);

> Im(z);

> abs(z) ;

> argument(z);

Here is a word to plot the location of a given complex number or a list or set of complex numbers.

> drawpts := proc(z,clr)
if type(z,complex) then plot({[Re(z),Im(z)]},style=point,symbol=circle,color=clr)
else plot({seq([Re(z[i]),Im(z[i])],i=1..nops(z))},style=point,symbol=circle,color=clr) fi end;

> drawpts({seq((cos(Pi/13)+I*sin(Pi/13))^i,i=1..26)},red);

Problems.

1. Describe geometrically the following sets of complex numbers.

a) all z with positive real part.

b) all z such that .

c) all z such that .

d) all z such that < 3.

e) all z such that .

2. Show that the conjugate of the sum or product of two complex numbers is the sum or product of their conjugates.

3. Show that each nonzero complex number z has a reciprocal 1/z.

4. Let S be the set of complex numbers of absolute value 1. Show that if z S then 1/z S. Show that S is a semigroup under multiplication.

5. Take a complex number z and form the sequence of its positive integeral powers , etc. Describe this sequence in qualitative geometric terms as a function of z. Use the word drawpts to experiment. Hint: check out , , and

> frame := (z,n) -> drawpts({seq(z^i,i=1..n)},red);

> movie := (z,len)-> plots[display]([seq(frame(z,n),n=1..len)],insequence=true);

> movie(cos(Pi/20)+I*sin(Pi/20),40);

Polynomial functions from C to C.

Let p be a polynomial in one indeterminant x with complex coefficients. Then p can be used to define a function p: C -> C by p(z) is the complex number obtained by substituting x=z into p and computing the resulting complex number p(z). We want to investigate these 'polynomial functions' from C to C by seeing where they take certain curves in C. A lot can be learned by looking at the image of circles centered at 0 under the function p.

> image := proc(p,rs,trng,clr)
local x,y,z,pz,r;
z:= unapply(r*cos(theta)+r*sin(theta)*I,r);
pz := unapply(evalc(p(z(r))),r);
x := unapply(Re(pz(r)),r); y := unapply(Im(pz(r)),r);
plot({seq([x(rs[i]),y(rs[i]),theta=trng],i=1..nops(rs))},color=clr,scaling=constrained); end;

>

> image(z->z^2+1,[1,1.5,2],0..Pi/2,red);

Problem. Investigate several polynomials using image. Make some conjectures. Prove one.