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The history of the complex numbers is very interesting. By the 16th Century, although no-one understood exactly what a complex number was, it was found that complex numbers were a useful tool for solving problems. Later, mathematicians tried to understand the complex numbers. This led in turn to investigations of the real numbers, the rational numbers, the integers and finally the natural numbers. So historically there was a reverse development: the more complicated system was found to be useful early on, and the study of the simplest systems were left till later.
Complex numbers are defined as ordered pairs z = (x, y) of real numbers x and y. We shall define addition and multiplication of these numbers shortly. We identify the pairs (x, 0) with the real numbers x. This means that the real numbers can be thought of as a subset of the complex numbers. Complex numbers of the form (0, y) are said to be pure imaginary numbers. The numbers x and y are now called the real and imaginary parts of z respectively, and we write x = Re z, y = Im z. Equality Two complex numbers are equal if they have the same real and imaginary parts. Thus (x1, y1) = (x2, y2) if and only if x1 = x2, y1 = y2. Sum and Product The sum z1 + z2 and product z1.z2 of z1 = x1 + y1 and z2 = x2 + y2 are defined by: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2), (x1, y1).(x2, y2) = (x1x2 – y1y2, x1y2 + x2y1). This last definition looks pretty weird, but we shall shortly see the reason for it. With these definitions of addition and multiplication, we have (x, y) = (x, 0) + (0, 1)(y, 0). The set of complex numbers of the form (x, 0) act just like the real numbers R. Further, setting i = (0, 1) gives z = (x, y) = x + iy, and we have i 2 = (0, 1).(0, 1) = (–1, 0), that is, i 2 = –1. Addition and multiplication can now be rewritten as the more usual: (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2), 1. Why did we not just start here? Notice that our derivation of i is firmly based on the real numbers. There is no ‘magic’ about 2. We can now perform complex operations by treating the terms as real, and substituting i 2 = –1.
The complex numbers behave in much the same way as the real numbers. The Commutative Laws: z1 + z2 = z2 + z1; z1z2 = z2z1; The Associative Laws: (z1 + z2) + z3 = z1 + (z2 + z3); (z1z2)z3 = z1(z2z3); The Distributive Law: z1(z2 + z3) = z1z2 + z1z3. These are easily proved by writing the z s in the form x + iy and using the corresponding properties of the real numbers.
We now use the idea of inverse to define division of complex numbers in the following way: z1/z2 = z1 . z2–1 (z2 Try writing this out in terms of the real and imaginary parts! Later we shall give an easier way of dividing complex numbers. Clearly division by zero is not allowed, as z2–1 is undefined when z2 = 0. The expressions below follow from this definition:
As mentioned earlier, complex numbers (like the reals) form a field, C. However, it is not possible to order C. Thus expressions like z > 0, z1 < z2 are meaningless unless the complex numbers are real. We set up the natural correspondence z = x + iy The plane of such representative points is called the Argand diagram, or the complex plane or the z-plane. The x-axis is called the real axis and the y-axis is called the imaginary axis.
Notice the direction of the arrow representing z1 – z2. The modulus or absolute value of a complex number
The (real) number |z| denotes the length of the vector representing z, or the distance of the point (x, y) from the origin O. Even though z1 < z2 is meaningless, we can note that |z1| < |z2| is a valid statement, meaning that z1 is closer to the origin than z2. We can extend this idea to see that | z1 – z2 | = is the distance between the points representing z1 and z2.
from which it follows that | z | The complex conjugate or conjugate of z = x + iy is the number z = x – iy. The number We observe that for all z,
The following properties are easy to establish from the definitions:
We also have Re z = Finally, z. so for z2 This gives us a nice way to find the quotient of two complex numbers.
The relation z | z1.z2 | = | z1 |.| z2 |, | z1/z2 | = | z1 | / | z2 |. It is also not too hard to derive the Triangle Inequality: | z1 + z2 |
It is easy to extend this to larger finite sums: |z1 + z2 + z3|
Another useful result is the ModDiff Inequality: | |z1| – |z2| | We can think of this as providing a lower bound for |z1 + z2|. Proof: |z1| = |(z1 + z2) + (–z2)| so |z1| – |z2| Similarly, |z2| – |z1| We say that z = (x, y) has polar coordinates (r, z = r cos
Note that r = |z|. We write The angle \theta is not unique: thus We use Arg z to denote the value of arg z satisfying – If z = 0, arg z is undefined. We adopt the convention that if z is expressed in polar form, then z We begin with a natural extension: z – z0 = cis The adjacent figure illustrates the situation. The polar form is important because it gives a very simple method of multiplying and dividing complex numbers. This is because arg (z1.z2) = arg z1 + arg z2 (*) We shall prove this shortly, but notice that this relation may fail for the principal value ‘Arg’.
We now prove the identity (*): arg(z1z2) = arg z1 + arg z2. Set z1 = r1 cis z1z2 = r1r2 (cos = r1r2 [cos = r1r2 [cos ( = r1r2 cis ( So any argument of z1 plus any argument of z2 is an argument of z1z2. On the other hand, if arg(z1z2) = Euler’s Formula is:
We will justify this later. But note that
corresponding to cis In particular,
But as an example, consider the mapping z Here we have |i| = 1, arg i = z = r cis That is, multiplication of z by i effects a rotation through This is the Engineers’ j - operator. By induction, it is easy to obtain
where there are n factors on the left. Equivalently, we have the well-known de Moivre's Theorem: (cis It follows that
(Feel free to work with either form.)
We can easily extend the above method to finding n th roots of any w = For if z = r cis Thus r =
We have already mentioned the complex plane. We look at some terms which describe certain sets in the plane. A neighbourhood or more specifically, an A point z is said to be interior to set S if there is some neighbourhood of z which just contains points of S. A point z is said to be exterior to set S if there is some neighbourhood of z which contains no points of S. A point z is said to be a boundary point of set S if every neighbourhood of z contains points in S and points not in S. A set is open if it contains none of its boundary points. In fact, this is the same as saying that every point of S is an interior point. A set is closed if it contains all of its boundary points. Of course a set need not be open or closed. An example would be the ring 1 < |z| Given any set S, we can take all the points of S together with all the boundary points of S to obtain a new closed set called the closure of S, and denoted An open set is connected if each pair of points of S can be joined by a polygonal path – a finite set of line segments joined end to end, which lies entirely within S. Note You may wonder why we insisted on a polygonal path in the above definition. Why not any continuous path? The answer is that a ‘polygonal path’ is easily defined and understood. A ‘continuous path’ is a much more difficult concept. An open, connected set is called a domain. A connected set is called a region.
A set S is bounded if every point of S lies inside some circle |z| = R; otherwise it is unbounded.
We say that point z0 is an accumulation point of a set S, if each neighbourhood of z0 contains at least one point of S distinct from z0. This is a more difficult concept, and is obviously associated with a limiting process. Thus we would expect to be able to find a sequence of points of S lying closer and closer to z0. Lemma S is closed Proof. ( ( |
(–1), contrary to the use of the historical word ‘imaginary’.
0).
, 
, 
.
(x, y) between the complex number z and the point (x, y) in the cartesian plane. Each complex number corresponds to exactly one point in the plane, and conversely. For example, 
We now have an immediate geometric interpretation of the sum and difference of two complex numbers:
From the definition of | z |, we have
|Re z| 
is represented in the complex plane by the point (x, –y) – the reflection in the 
= z and | 
= 
, 
= 
= 
(z + 
(z – 
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| z1 | + | z2 |.
)
| z1 |2 + 2| z1 |.| z2 | + | z2 |2
The triangle inequality becomes clear when seen geometrically, as in the adjacent figure. It just says that the length of any side of a triangle is not more than the sum of the lengths of the remaining two sides.
} when x = r cos 
/3 + i sin 
. Here, r = |z – z0|, and 
We can use complex numbers to represent geometric transformations in the plane, and we shall develop this idea later. 
f(z) = iz.
n
cis 
(the positive nth root),and 
.
- neighbourhood of z0, is the set of points satisfying |z – z0| < 
. (Not to be confused with conjugates!)
S contains all its points of accumulation.
) Suppose now that S contains all its points of accumulation. Does S contain all its boundary points? Let z0 be a boundary point of S not in S. By definition of boundary point, each neighbourhood of z0 contains a point of S, so z0 is a point of accumulation of