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Posted 29 April 2017
Introduction
1
Images and metaphors for real numbers 1
Properties of real numbers
2
Decimal representation of real numbers 3
Algebra of the real numbers
Irrational numbers Error! Bookmark not defined.
Density Error! Bookmark not defined.
Addenda 3
REAL LINE (NOT INCLUDED
IN ABMATH)
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.
2
 A real number is any positive or negative number 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
8
2
measure it top to bottom or right to left you get a negative number. For example
2
the length of the diagonal of a square whose sides have length 2 is
8 . If you
regard “left to right” as the positive direction, then if you measure it from right to
left you get -
8.
 A real number has a decimal representation.
zeroes) and the decimal representation of
For example, 2 is 2.000… (infinite number of
8 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
8.
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 cm 3 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).
Quantum jumps and individual molecules are ignored in largeSome authors refer to the custom
of treating physical quantities as
scale physical applications because the scale at which they occur is so
continuous as the “continuum
hypothesis”. This is not what
tiny it doesn’t matter. For such applications, physicists and chemists
“continuum hypothesis” means to
(and cooks and traffic policemen!) think of the quantities they are
most mathematicians.
measuring as continuous, even though at tiny scales they are not.
The real line
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:
.
..
...
–
–-2
-1
0
3/2
1
2
3
2
4
p
.
..
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 + t = s .
 r > s means that s < r.
 r £ s means that either r < s or r = s. Fine point: This means that both the statements
“ 42 £ 42 ” and “ 42 £ 65 ” are correct! (See inclusive or).
 s ³ r means that r £ s .
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 r ³ 0 .
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 {x | r £ x £ s}and the open interval (r,s) is the set
{r | r < x < s}.
Distance
For a real number r, the absolute value of r, denoted by r , is defined this way:
ïì r if r ³ 0
r = ïí
ïïî - r if r < 0
Thus 13 = 13 , 0 = 0 and - 13 = 13 (proof).
If r and s are real numbers, the distance from r to s is r - s , which of course is the same as
s- r .
Example: The distance from 7 to 2, which is the same as the distance from 2 to 7, is
7 - 2 = 2 - 7 = 5 (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 –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 0.9 = 1.0 .
Algebra of the real numbers
Decimal representation of real numbers
Irrational numbers
Density of the reals
Addenda
Proof that | –13 | = 13
By definition of absolute value, - 13 = - (- 13) , since - 13 < 0 . But –(–13) = 13, which finishes
the proof. This is a totally straightforward proof by rewriting the definitions. Return.