Defreal Slides

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The Set of Real NumbersR

Philippe B. Laval

KSU

Fall 2012

Philippe B. Laval (KSU) R Fall 2012 1 / 20
Introduction

Though it is possible to construct the set of real numbers from scratch (asdid Cantor and Dedekind) and derive its properties from the fundamentalaxioms of set theory, this process is too big a task and beyond the scope ofan introductory course in real analysis. Instead, we will make certainassumptions and derive all the important properties of R from theseassumptions. Most of the properties we will see are already known to you.The goal here is to see how they can be proven using only our assumptionsand what has already been proven.The properties of R we will discuss fall in three categories:

1 Algebraic properties

2 Order properties3 Metric properties

Introduction


1 Algebraic properties2 Order properties

3 Metric properties

Introduction


1 Algebraic properties2 Order properties3 Metric properties

Introduction

We have the following number systems:

1 The set of natural numbers: N = f1; 2; 3; :::g. These are thenumbers used to count.

2 Trying to solve x + 5 = 2 gives x = 3 which is not in N. Hence, wedene a new set, the integers, denoted Z = f0;1;2; :::g.

3 Division does not work well in N or Z as the result is usually not anelement of these sets. So, we are forced to deal with fractions orrational numbers. The set of these numbers isQ =

nmn: m 2 Z, n 2 Z and n 6= 0

o.

4 Pythagora realized that if we build a right triangle whose sides havelength 1, then the hypotenuse has length

p2, and he proved that

p2

was not rational. There are many other numbers which are notrational. They include , e,

pp where p is prime , :::. Numbers

which are not rational are called irrational. The set of real numbers,R is the set of rational together with the irrational numbers.

Introduction





nmn: m 2 Z, n 2 Z and n 6= 0

o.



p2




Introduction





nmn: m 2 Z, n 2 Z and n 6= 0

o.



p2




Introduction





nmn: m 2 Z, n 2 Z and n 6= 0

o.



p2




Introduction

Now, lets focus on R. We assume we are given the following:1 The set of real numbers R.

2 Two binary operations (addition, denoted + and multiplication,denoted :)

3 An order relation 0 then x < 0.

TheoremLet a 6= 0 2 R.

1 a2 > 02 1 > 0

Additional Order Properties

TheoremLet a, b, c be elements of R.

1 a < b () a b < 02 a > b and c < 0) ac < bc3 a > 0) 1=a > 04 a < 0) 1=a < 0

Theorem

If a and b are any real numbers, then a > b =) a > 12(a+ b) > b

See the class notes for additional important properties.

Metric Properties - Absolute ValueDenitions

The trichotomy property assures that if a 6= 0 then either a > 0 or a < 0.Thus, we dene:

DenitionThe absolute value of a real number a, denoted jaj is dened by:

jaj =

a if a 0a if a < 0

Metric Properties - Absolute ValueRemarks

The domain of this function is R, its range is the set of non-negativereal numbers, and it maps the elements a and a into the sameelement.

Geometrically, the absolute value of a number x can be interpreted ashow far from 0 x is.

Similarly, jx aj can be interpreted as the distance between x and aand therefore, one can think of the absolute value as a tool tomeasure distances between numbers.

In particular, for a xed number a, the set dened byfx 2 R such that jx aj < g denotes the set of numbers within adistance of a.

The inequality jx aj < is equivalent to < x a < ora < x < a+ that is x 2 (a ; a+ ). This will be used a lotthroughout this class.

Metric Properties - Absolute ValueProperties

TheoremThe following properties hold:

1 jaj = 0, a = 02 jaj = jaj for all a in R3 jabj = jaj jbj for all a; b in R4

1a = 1jaj for all a 6= 0 in R

5

ab

= jajjbj for all a; b 6= 0 in R6 If c 0, then jaj c , c a c7 jaj a jaj8pa2 = jaj

Metric Properties - Absolute ValueMore Theorems

The next theorem is known as the triangle inequality.

TheoremIf a and b are any real numbers, then

jjaj jbjj ja+ bj jaj+ jbjThe next theorem is known as the reverse triangle inequality

TheoremIf a and b are any real numbers, then

jjaj jbjj ja bjThis last theorem is very important and will be used a lot.

TheoremLet x be a real number. If jx j < for each > 0 then x = 0.

Metric Properties - Absolute ValueEquations and Inequalities Involving the Absolute Value

We remind students of the basic principles used when solving equationsand inequalities involving absolute values. For what follows, let a and bdenote any expression.

The equation jaj = b is equivalent to the two equations a = b anda = b. However, solutions must be checked, some may not be valid.The equation jaj = jbj is equivalent to the two equations a = b anda = b. However, solutions must be checked, some may not be valid.The inequality jaj < b is equivalent to b < a < bThe inequality jaj b is equivalent to b a bThe inequality jaj > b is equivalent to a > b or a < bThe inequality jaj b is equivalent to a b or a b

Metric Properties - Absolute ValueConclusion

As we noted earlier, the absolute value can be used to measure distancesbetween numbers. jx y j denotes the distance between x and y on thereal line. For the rest of these notes, we will use the absolute value everytime we need to measure distances between two real numbers. However, ifthe set in which our elements come from were not R, the absolute valuemay not be appropriate to measure distances. For example, you alreadyknow that in R2, the distance between two points of coordinates (x1; y1)

and (x2; y2) is given byq(x2 x1)2 + (y2 y1)2. In more general

settings, we replace the absolute value by a function we call a distancefunction or a metric. We give a formal denition for a metric.


DenitionLet X be a set. A metric or a distance on X is a functiond : X X ! R which satises the properties below. In the propertiesbelow, we let x ; y and z be elements of X .

1 d (x ; y) 0 (non-negativity)2 d (x ; y) = 0 () x = y3 d (x ; y) = d (y ; x) (symmetry)4 d (x ; z) d (x ; y) + d (y ; z) (triangle inequality)

DenitionA space X together with a metric d is called a metric space. We say that(X ; d) is a metric space.


ExampleIt is easy to see that the absolute value function is a metric for R, hencewe say that (R; j:j) is a metric space.

Example

On R2, the function d : R2 R2 ! R dened byd (P;Q) =

q(x2 x1)2 + (y2 y1)2 where P and Q are two points of

respective coordinates (x1; y1) and (x2; y2).

Remark Real analysis can be taught in a more abstract setting, thesetting of a metric space. Instead of using the absolute value,we would use any distance function. Students are encouragedto check their local library and nd books on metric spaces.

Exercises

See the problems at the end of the notes on denition and basic propertiesof R. These notes have several sections, there is a problem set at the endof each section.

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