We have been working with real numbers since

elementary school, however they are not an easy subject to define. In a similar

way, when a biologist finds a new bacteria, it will try to classify it

according to its features (morphology, intra/extra cellular structures,

metabolism, growth, reproduction, etc.. ). With this analogy, we will define

the real numbers, , as the set of points (numbers) that satisfy the following five

properties (also called field properties) plus what is called axiom of

order and least upper bound property:

1. Commutativity: For any , we have that

2. Associativity: For every , we have that

3. Distributivity: For every we have

4. Existence of neutral elements: There are

distinct elements 0 and 1 of such that for all we have and .

5. Existence of multiplicative and additive

inverses: For any , there is an element in denoted by such that , and for any nonzero there is an element of denoted by , such that .

6. Axiom of order: There is a subset of such that

(a) If , then

(b) For

any , one and only one of the following options is true

7. The least upper bound property: A non empty

set of real numbers that is bounded from above as a least upper bound.

There are three objects involved in this definition,

the set of points (denoted by ), and the two operations denoted by . Properties 1-5 may sound familiar (trivial), but remember that in

principle can be any set, and may not represent the usual sum and

multiplication. The last two properties of the real numbers differentiate them

from other sets (fields) in the following way:

1. The axiom of order is what makes us think

that real numbers are aligned up in a row, one number following the other. In

other words, given two numbers you can always decide if they are equal or in

other case, which one is bigger.

2. The least upper bound property basically says

that there are no holes or gaps in our model of real numbers as a straight

line.

Subsets of the

real numbers are denoted with square brackets and parenthesis as follows: Given

two real numbers,

1. (these are called open) is

the set of values of x with

2. (thess are called closed) is

the set of values of x with

3. is the set of values of x with

4. is the set of values of x with

5. is the set of values of x with

6. is the set of values of x with

7. is the set of values of x with

It is possible

to use the same algebraic properties of numbers when solving inequalities of

real numbers. One of the most useful inequlities is called

Theorem 1.1 For any real numbers