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Posted 6 November 2008

RATIONAL NUMBERS

Introduction

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

Examples

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

¨  6 is rational because 6 = 6/1. 

¨  0.33 is rational because . (See Approximations).

Theorem

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

Terminology

 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”.

Representations of rational numbers

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 0.25 is a rational number because it 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.)

 

Properties of the fraction representation

Non-uniqueness

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 .

Example

 because .

Definition: lowest terms

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.)

Usage

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.

Examples

¨  3/4  is in lowest terms but 6/8 is not, because 6 and 8 have 2 as a common divisor. 

¨  74/111  is not in lowest terms because 74 and 111 have 37 as a common divisor.  In fact, 74/111 = 2/3.

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. 

Theorem

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. 

Closure properties of rationals

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

 

                   ,     and              [AMD]

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.

Images and metaphors for the rationals

Ratios

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 L into exactly 5 segments of the same length.  That is because the ratio of L to M is 5:8.

Text Box: Remarkably, we cannot take a line of length 1 and a line of length  and divide them into whole numbers of segments of the same length, no matter how small we make the segments.  This means that   is not rational.  This is discussed in the section on irrational numbers.

 

 

 

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

Density of the rationals

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  is the average (or mean) of r and s, so it makes sense that it is between them.  The preceding sentence is an example of using the rich view to see why something is true.  Here is a rigorous proof:

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.