abstractmath.org

help with abstract math

Produced by Charles Wells. Home Website TOC Website Index Blog

Posted 11 July 2011

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.

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 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.**

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:

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.

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 .

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 5, but the directed distance from 2 to 7 is 5.

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 .

By definition of absolute value, , since . But (13) = 13, which finishes the proof. This is a totally straightforward proof by rewriting the definitions. Return.