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Posted 11 July 2011

REAL NUMBERS

Introduction

I will not give a mathematical definition of “real number”.  There are several equivalent definitions of real number all of which are quite complicated.   Mathematicians rarely think about real numbers in terms of these definitions; what they have in mind when they work with them are their familiar algebraic and topological properties.

The following two facts about real numbers should be familiar to you and can serve as a starting place for understanding them. 

¨  A real number is any number, positive, negative or zero, that can measure the length of a directed line segment.  “Directed” means you have chosen one end to be “bottom” or “left” and the other end to be “top” or “right”.  Then if you measure it top to bottom or right to left you get a negative number.  For example the length of the diagonal of a square whose sides have length 2 is .  If you regard “left to right” as the positive direction, then if you measure it from right to left you get .

¨  A real number has a decimal representation.    For example, 2 is 2.000… (infinite number of zeroes) and the decimal representation of  begins with 2.828427…

 Integers and rational numbers are real numbers, but there are real numbers that are not integers or rationals.   One such number is .

Usage: “Real number” is a technical term.  Real numbers are not any more “genuine” that any other numbers.

Images and metaphors for real numbers

Real numbers are quantities

Real numbers are used to measure continuous quantities, such as

¨  The temperature at a given place and a given time.

¨  The speed of a moving car.

¨  The amount of water in a particular jar.

The name “continuous” for these quantities indicates that the quantity can change from one value to another without “jumping”. 

Example.  If you have 1.334 cm3 of water in a jar you can add any additional small amount into it or you can withdraw any small amount from it.  The volume does not suddenly jump from 1.334 to 1.335  as you put in the water it goes up gradually from 1.334 to 1.335.

Caveat.  This whole explanation of “continuous quantity” is done in terms of how we think about continuous quantities, not in terms of a mathematical definition.  In fact since you can’t measure an amount smaller than one molecule of water, the volume does jump up in tiny discrete amounts.   Because of quantum phenomena, temperature and speed change in tiny jumps, too (much tinier than molecules). 

Text Box: Some authors refer to the custom of treating physical quantities as continuous as the “continuum hypothesis”.  This is not what “continuum hypothesis” means to most mathematicians.Quantum jumps and individual molecules are ignored in large-scale physical applications because the scale at which they occur is so tiny it doesn’t matter.  For such applications, physicists and chemists (and cooks and traffic policemen!) think of the quantities they are measuring as continuous, even though at tiny scales they are not.

The real line

Text Box: . . .It is customary to visualize the set of real numbers as the real line.  Each real number represents a location on the real line.  Some locations are shown here:

 


Text Box: . . .The locations are commonly called points on the real line.  This can lead to a seriously mistaken mental image of the reals as a row of points, like beads.  Just as in the case of the rationals, there is no real number “just to the right” of a given real number.  See density of the reals.

Properties of real numbers

Ordering

 

The real numbers are ordered by the familiar relationship “<” (less than). 

¨  r < s means there is a positive real number t such that . 

¨  r > s means that s < r.

¨   means that either r < s or r = s.  Fine point:  This means that both the statements “” and “” are correct!  (See inclusive or).

¨   means that .

A real number r is either greater than 0 (then it is positive), equal to 0, or less than 0 (then it is negative).  As in the case of integers, a few authors say that r is positive if .

If r and s are distinct real numbers, then either r > s  or r < s.  This means the integers are totally ordered.  This is the property that makes the real line look like a line  either r is to the left of s or r is to the right of s.

If r < s, the closed interval [r,s] is the set and the open interval (r,s) is the set . 

Distance

For a real number r, the absolute value of r, denoted by , is defined this way:

                                                  

Thus ,  and (proof).

  If r and s are real numbers, the distance from r to s is , which of course is the same as .   

Example:  The distance from 7 to 2, which is the same as the distance from 2 to 7, is  (see parenthetic assertions). 

The directed distance (or displacement) takes into account the direction.  Left to right is positive and right to left is negative.   

Example: The directed distance from 7 to 2 is Description: [!DSTAG SYMBOL ID=ch0001 DIMS='[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,3,-9,0,0]]' CHARID=ch0 INSUBSUPER=0/]5, but the directed distance from 2 to 7 is 5. 

Archimedean property

The set of integers is a subset of the set of real numbers (but see integers and reals in computer languages).  A basic fact about these subsets is that they are spread out all over the reals.  Precisely:

(Archimedean Property)  If r is any real number, there is an integer n such that r < n.

This means for example that there is an integer bigger than 43,221,678.93456.  Of course there is!  One of them is the integer 44,000,000.   (Have you fallen into the “unnecessarily weak assertion” trap?)

The Archimedean Property is used here to prove that .

Algebra of the real numbers

Decimal representation of real numbers

Irrational numbers

Density of the reals

Addenda

Proof that | Description: [!DSTAG SYMBOL ID=ch0002 DIMS='[[7,1,-3,0,0],[9,2,-4,0,0],[12,2,-4,0,0],[],[],[],[30,7,-11,0,0]]' CHARID=ch1 INSUBSUPER=0/]13 | = 13

By definition of absolute value, , since .  But Description: [!DSTAG SYMBOL ID=ch0003 DIMS='[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,3,-9,0,0]]' CHARID=ch0 INSUBSUPER=0/](Description: [!DSTAG SYMBOL ID=ch0004 DIMS='[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,3,-9,0,0]]' CHARID=ch0 INSUBSUPER=0/]13) = 13, which finishes the proof.  This is a totally straightforward proof by rewriting the definitions.   Return.