The **Real Numbers** R are defined by **Completing** the rational
numbers.

This means we add limits of sequences of rational numbers to the field . We

should then check that all the field axioms hold and that the ordering
properties

persist. The Real Numbers are characterized by the properties of

Complete Ordered Fields. In these notes we give definitions of these terms.

**Definition 0.1** A sequence of real numbers is an assignment of the set of

counting numbers of a set ,
of real numbers,
.

**Definition 0.2** A sequence
of real numbers has a limit a if, for every

positive number ε > 0, there is an integer N = N(ε) such that

for all with n > N.

Example 1: The sequence = 1/n has limit 0 since we can take N =

[1/ε] + 1. This says

1/n < 1/(1/[epsilon] + 1) < ε, for all n > N

Example 2: The sequence has limit 0 since we can take N >

. This says

1/2^{n} < ε, for all n > N

Here is a list of equivalent statements . We can choose any one of them

as an axiom for completeness. Choosing one, we can prove that all the other

properties hold:

1. A field is complete if every infinite continued fraction has a limit .

(Nested Intervals) If is an infinite sequence of closed intervals

with in the field such that
In then the field is

complete if the infinite intersection of the intervals is non-empty, that

is, .

2. (Dedekind Cuts) A subset of a field is called a cut if: 1) It is non-empty,

but is not the whole set of rationals 20 every rational number of the

set is smaller than every rational number not in the set, 3) it does no

contain a number that is greater than any other number of the set. A

field is complete if it contains cuts.

3. (Greatest lower bound or Least upper bound) A lower
bound for a set

is a number less than every number in the set, that is, if B ≤ x for all

x in the set, B is a lower bound. G is a greatest lower bound for the

set if it is a lower bound and every lower bound B satisfies B ≤ G.

Least upper bounds are defined similarly. For a complete field every

set that has a lower bound (upper bound) has a greatest lower bound

(least upper bound). The bound may or may not be in the set.

4. A field is complete if every bounded monotonic sequence has a limit.

5. If is a sequence with the following property: given ε > 0 there exists

an N = N(ε) such that for all m, n > N we have
then

the sequence is called a Cauchy Sequence. A field is complete if every

Cauchy sequence has a limit.

Example 3: Let be the finite decimal whose n entries are the first n digits

of the infinite decimal for . Then
is
monotonic increasing, bounded

by 2, and has limit (the least upper bound).
Example 4: Consider the

sequence of continued fractions. Let
. Then

it is not hard to show that the intervals are nested. The diameters

of the intervals goes to zero so the infinite continued fraction is the unique

point in the intersection. One way to prove this uses the cutting sequence

technique. and
have cutting sequences that agree until the

last set of L's (or R's). If the sequence ends in L's it is to the left of the

number and if it ends in R's it is to the right. Note that
and

must be neighbors.

Example 5: Consider the sequence
= [1, 1, ... , 1] where there are n 1's

in the continued fraction. Let
. It is easy to check that

and where is the n^{th}
Fibonacci number. Then the sequence

is a Cauchy sequence. The limit is .

For real numbers we can talk about continuous functions and define

derivatives. Consider the polynomials x^{2} - r = 0 for r > 0. We can graph

them. Because of the completeness of the reals, the graph is continuous and

must cross the x axis in two places. These are the roots of the equation . For

all odd degree polynomials we can see that they must cross the x axis in at

least one place so they have at least one real root. Note though that not all

polynomials have real roots. For example x^{2} + 1 = 0.

Complex Numbers

If we want to solve for roots of all algebraic equations , f(x) = 0, we need

to introduce complex numbers. We look at pairs of real numbers (x, y) and

define the following operations for them :

Taking the additive identity as (0, 0) and the multiplicative identity as (1, 0)

it is possible to check that the pairs of numbers with this operation define

a field. Note that it is NOT an ordered field! We set z = (x, y). We can

plot the complex numbers as points in a plane using the first and second

coordinate as the horizontal and vertical coordinates. Using the standard

distance in the plane we can define the distance between two points

Using this distance we can define Cauchy sequences. One can prove that

all Cauchy sequences have limits using the fact that the coordinates are real

numbers. It follows that the complex numbers are a COMPLETE field.

The distance from a point z to the origin (additive identity), (0, 0) is

lzl = sqrtx^{2} + y^{2}. We can specify a point z by its distance from the origin

and the angle a line joining the point to the origin makes with the horizontal

axis. Call this angle θ.

Exercise: Show that if r = lzl and θ is the angle defined above,

z = (x, y) = (r cos θ, r sin θ)

Addition of complex numbers can be thought of as addition of vectors. To

find the sum , draw the line from the origin
to , then translate the

line from the origin to so that it starts at the end of the first line. The
final

endpoint is the sum. Multiplication of complex numbers also has a geometric

interpretation: if and the corresponding
angles are

then and the angle that
makes with the horizontal is

This leads us to define

e^{iθ} = (cos θ, sin θ)

to describe points on the unit circle . It is clear that
le^{iθ}l = 1 since cos^{2} θ +

sin^{2} θ = 1. We see from the multiplication law that we have DeMoivre’s

theorem:

e^{inθ} = (cos nθ, sin nθ)

Note that there is another notation that we often use:

(x, y) = x + iy

We think of (0, 1) = i as the imaginary number that solves x^{2} +1 = 0. That

is, i^{2} = -1. We have the following interesting formula:
.