     COMPLEX NUMBERS AND FUNCTIONS Sessions  1-    History of numbers

A long time ago people used their fingers for counting (some still do!). We begin here with the counting numbers (natural numbers, positive integers) . Clearly this set is closed under the operations of addition and multiplication but not under subtraction. Thus it is natural to ask the following question:

Question 1 Can the above set be extended so that the extended set is closed under the three previously mentioned operations?

The answer is clearly yes. Take the extended set to be the set of integers . The set of integers is not closed under division though. So we ask a natural question:

Question 2 Can the set of integers be extended so that the extended set is closed under the four basic operations?

The answer is yes. We denote the set of rational numbers by .

However, is not closed under the operation of taking square roots. So here we go again and we ask:

Question 3 Can the set of integers be extended so that the extended set is closed under the four basic operations as well as the additional operation of taking square roots?

The answer as you might expect is yes. Take the set of real numbers which is by definition the union of the sets of rational an irrational numbers (the latter set is nonempty since is not rational as one can see by a standard contradiction argument; On the  other hand, the rational number system has certain gaps that were filled by the mathematician Dedekind who gave use the real number system.
Since the equation has no solutions in the set of real numbers, we can ask:

Question 4 Can the set of real numbers be extended so that the extended set is closed under the four basic operations, taking square roots, and, in addition, so that equations like would have a solution in the extended set?

A little comparison with a method of solving suggests that the answer is yes after introducing the symbol , and here we witness the birth of the set of

Complex numbers C defined as

C={x+iy: R}.
Notice that
C clearly extends R and has the required properties.

Addition and subtraction of complex numbers are defined in a natural way. Thus if and , then and . We now define multiplication of two complex numbers in such a way that the existing laws (like the foil method) still hold which explains why multiplication is defined as follows: .

Division is defined via conjugation (the conjugate of is defined as ). Thus if is a nonzero complex number, 2- Triangle Inequality

We now prove an important inequality called the triangle inequality

For C and C, we have To see this, first note that or But So and therefore and consequently It is worth noting now that the generalized triangle inequality 3-   Limits and Continuity

Definitions As in real analysis, means for there is such that A function is said to have a limit at denoted by , if given there exists a large positive number such that when In addition, means given an arbitrary there exists a such that for Definitions A function is said to be continuous    at if 4-    Differentiable  and holomorphic  functions

Definitions A function is differentiable  at if the partial derivatives are continuous and satisfy the Cauchy-Riemann equations and . The function is differentiable in a region if it is differentiable at each point in the region.

A function is analytic (or holomorphic)  at if is differentiable in some disk . A function is analytic in a region if it is analytic at each point in the region. A function is called an entire  function if it is analytic in the whole plane C. 