Posted 6 November 2008

Definition: A **rational
number** is a number that can be represented as a fraction ,
where *m* and *n* are integers and .

¨ The numbers 3/4 and 11/5 are rational.

¨ 6 is rational because 6 = 6/1.

¨ 0.33 is rational because . (See Approximations).

**Any integer is rational.** **Proof:**
The integer *n* is the same as the fraction .

Rational numbers may be referred to as
“rationals”. The name comes from the
fact that they represent **ratios ****and is not related to the meaning “able to reason” or
“sane”.**

Rational
numbers have two familiar representations, as **fractions** and as **decimals**. Decimals are discussed in the section on real
numbers.

The
definition of rational number says that it must be a number that **can be represented as a fraction
of integers.**** **

** **

“Can be represented” does not mean “is represented”!

The
number **can
be represented** as ,
but the expression “0.25” **is
not itself a fraction representation**.

** **

**An expression a / b does not
automatically denote a rational number. **

**You must check that a and b are
integers**

For
example if you see the expression you cannot conclude that it denotes a rational
number, because and *e*
are not integers. (I do not know if is rational.)

**The representation of a rational number as a fraction is not
unique**. For example,

Two representations and give the same rational number if and only if .

because .

Let
be the representation of a rational number with and .
The representation is in **lowest terms** (or is **reduced) **if there is no integer for which *d*
divides
*m* and *d *divides *n*. (See parenthetic assertion.)

The
symbols 3/4 and 6/8 are **two
representations of the same number. ****One of the representations, 3/4, is in lowest terms and
the other is not. **So
when someone says “3/4 is in lowest terms”, the symbol “3/4” refers to the **representation, ****not the ****rational number. ****See context
sensitive interpretation.**

¨
3/4 is in
lowest terms but 6/8 is not, because

¨
74/111 is
not in lowest terms because 74 and

To
calculate the lowest terms representation you **divide the numerator and denominator by the largest integer
that divides both of them.**
For example, the largest number dividing both 74 and 111 is 37. 74/37 = 2 and 111/37 = 3. So 74/111 = 2/3 and “2/3” is in lowest terms.

**Every nonzero rational number
has a representation in lowest terms. **

A proof of this will appear in the number theory section if I ever get around to writing it.

**Rational numbers are closed
under addition, subtraction, multiplication, as well as division by a nonzero
rational.**

These operations are carried out according to the familiar rules for operating with fractions. Thus for rational numbers and , we have

The expressions ,
and denote rational numbers because integers are
closed under addition and multiplication, so that *ac, bd, ad,* *bc *and* ad+bc *are integers.

**However, if** “ ”
and “ ” **are in lowest terms, **“ ” **and **“ ” **may nevertheless not be in lowest terms. ****For example, the formulas give **

** and **** **

In other words, “being in lowest terms” is **not closed under addition and
multiplication**.

Suppose
we have a line segment *L* that is 5/8
of a unit long. Then we can take a line
segment *M *that is 1 unit long and
divide it into ** exactly 8 segments of 1/8 unit each**, and we can divide

This
is the sense in which *rational numbers represent ratios**of
integers.*

The integers can be thought of as beads or points in a row going to infinity in both directions. The rationals go to infinity in both directions, too, but:

**The rationals must not be thought of as a row of
points. **

**That
is because ****between any two rational numbers there is another one. **** **

**In
general, ****if
r and s** ** are any distinct rational numbers, then is a rational number between them.** This number

**Theorem.** Let *r*
and *s* be distinct rational
numbers. Assume WLOG
that *r < s. *Then .

**Proof.** Let and ,
where *a, b, c *and *d *are integers. Then by
AMD,

which is rational because and are integers.

I recommend that you check using AMD that if and , then .

Warning: is
not the *only* rational number between *r *and *s. * In fact, **between any two distinct
rational numbers there are infinitely many other rational numbers. ****This means that if you are given a rational number ****r, ****there
is no “next largest” rational number ****(or next smallest, either).****
**

**These properties are discussed ****for all real numbers**** **here.