Exercises
1. Find the fol lowing complex numbers in the form x + iy:
2. Find all complex z = (x, y) such that
3. Prove that if wz = 0, then w = 0 or z = 0.
1.2. Geometry. We now have this collection of all ordered pairs of real numbers ,
and so
there is an uncontrollable urge to plot them on the usual coordinate axes . We
see at once
then there is a one-to- one correspondence between the complex numbers and the
points in
the plane. In the usual way, we can think of the sum of two complex numbers, the
point in
the plane corresponding to z + w is the diagonal of the paralle logram having z
and w as
sides:
We shall postpone until the next section the geometric inter pretation of the
product of two
complex numbers.
The modulus of a complex number z + x + iy is defined to be the non negative real
number
which is, of course, the length of the vector interpretation of z.
This modulus is
traditionally denoted |z|, and is sometimes called the length of z. Note that
and so |•| is an excellent choice of notation for the
modulus.
The conjugate 
of a complex number 
is defined by
Thus 
Geometrically, the conjugate of z is simply the reflection of z in the
horizontal axis:
Observe that if z = x + iy and w = u + iv, then
In other words, the conjugate of the sum is the sum of the conjugates. It is
also true that
If z = x + iy, then x is called the
real part of z, and y is called
the imaginary
part of z. These are usually denoted Rez and Imz, respectively. Observe then
that
Rez and 
Imz.
Now, for any two complex numbers z and w consider
In other words,
the so-called triangle inequality. (This inequality is an obvious geometric
fact–can you
guess why it is called the triangle inequality?)
Exercises
4. a)Prove that for any two complex numbers,
b)Prove that 
c)Prove that 
5. Prove that 
and that
6. Sketch the set of points satisfying
1.3. Polar coordinates. Now let’s look at polar coordinates
of complex
numbers.
Then we may write 
In complex analysis, we do not allow r
to be
negative; thus r is simply the modulus of z. The number θ is called an argument
of z, and
there are, of course, many different possibilities for θ. Thus a complex numbers
has an
infinite number of arguments, any two of which differ by an integral multiple of
2π. We
usually write θ = arg z. The principal argument of z is the unique argument that
lies on
the interval 
, .
Example. For 1 - i, we have
etc., etc., etc. Each of the numbers
and 
is an argument of 1 - i, but the
principal argument is 
We have the nice result that the product of two complex numbers is the complex
number
whose modulus is the product of the moduli of the two factors and an argument is
the sum
of arguments of the factors. A picture:
We now define 
by
We shall see later as the drama of the term unfolds that this very suggestive
notation is an
excellent choice. Now, we have in polar form
where r = |z| and θ is any argument of z. Observe we have just shown that
.
It follows from this that
. Thus
It is easy to see that
Exercises
7. Write in polar form 
8. Write in rectangular form—no decimal approximations , no trig functions :
9. a) Find a polar form of
.
b) Use the result of a) to find
and
10. Find the rectangular form of
11. Find all z such that z3 = 1. (Again, rectangular form, no trig functions.)
12. Find all z such that z4 = 16i. (Rectangular form, etc.)
