11-16-2009Fractions and the Set of Rational NumbersWe introduced the whole numbers as a system for keeping track of the number of objectsin a set. Later on, we discovered that negative numbers could help us to, for example,track debits and credits in our bank account, and so we introduced the integer numbersystem. Our task in this section is to describe fractions which help us to count partsof whole objects. We then present the rational number system which is the set of allfractions.Definition. A fraction is an ordered pair of integers a and b with b 6= 0 which we write inthe formabor a/b. The a is called the numerator and the b is called the denominator.Models for Fractions. A good model for fractions must do the following:1. Specify the unit or “whole”.2. Describe how many equal parts the unit has been subdivided into. This gives thedenominator.3. Describe how many parts of the whole are present. This gives the numerator.Some possible models are:1. Colored regions. Choose a shape. Divide the shape into equal parts. The numberof parts is the denominator of your fraction. Color the number of parts for thenumerator. See page 346 for some example pictures.2. Set model. Draw your universe set U so that the number of objects in the universeis the number in your denominator. Collect all the objects that you are trying tocount in a set A. Then the corresponding fraction isn(A)n(U)see the diagram on page 347.3. Fraction strips. Give each student a strip of paper. To represent the fractionab,have the students use a pencil to subdivide their strips into b equal rectangles. Thenhave them color a of the rectangles.4. Number lines. To represent the fractionab, take the standard integer number lineand divide each of the intervals between the integers into b equal subintervals. Then,beginning at zero, count out a subintervals on the number line. See page 348.Basic Properties of Fractions.Definition. Two fractions are called equivalent if they represent the same quantity.Example. Make fractions strips for the following fractions:23,46,69,812, ...The nthfraction of this sequence is2·n3·n. Moreover by our fraction strips each of thesefractions are really equivalent. That is,23=46=69= ... =2 · n3 · n.In general, this is the fundamental property of fractions:Theorem. foraba fraction and n 6= 0 an integer we have thatab=anbn.In words this says that if I multiply the numerator and the denominator of a fraction byany non-zero integer I get an equivalent fraction back.Now notice that3521=5 · 73 · 7=53by the fundamental property of fractions. Thus, to reduce a fraction we can factor thenumerator and the denominator and then “cancel out” all the common factors.Suppose I give you two fractions. How can you tell if they are equivalent? The ruleis given by the following theorem.Theorem. The fractionsabandcdare equivalent if and only if ad = bc. That is,ab=cdif and only if ad = bc.So are312and28equivalent?A good model for these ideas are the pizza diagrams. See page 350.Definition. A fraction is in simplest form if a and b have no common divisor largerthan 1 and b is positive.Example. Write360600is simplest form.An easy way to do this is to successively divide out the common factors of the numeratorand denominator:360600=36 · 1060 · 10=3660=6 · 610 · 6=610=3 · 25 · 2=35.Another procedure is to divide the numerator and denominator by their GCD:360 = 23· 32· 5600 = 23· 3 · 52GCD(560, 600) = 23· 3 · 5 = 120360600=360120600120=35.Often the easiest trick is to just write the prime factorizations of the numerator anddenominator and cancel out the common factors:360600=23· 32· 523· 3 · 52=35.Common Denominators.When working with two fractions it is often useful to rewrite both fractions in an equiv-alent form with a common denominator.Example. Write58and710as equivalent fractions with a common denominator.One choice is to use 8 · 10 for the common denominator. Then we would write:58=5 · 108 · 10=5080and710=7 · 810 · 8=5610.Really any common multiple of 8 and 10 could serve as our common denominator,however, it is often useful to choice the least common multiple for our common denomi-nator. Since LCM(8, 10) = 40 another answer is:58=5 · 58 · 5and710=7 · 410 · 4=2840For more examples, see page 353-354.The Rational Numbers.Definition. The set of rational numbers is the set of numbers that can be expressed inthe form of a fractionabwhere the a and b are integers and b 6= 0. Two rational numbersare equal if and only if they can be represented by equivalent fractions.Notice that since the integer n can be written asn1the set of rational numbers includesall the integers. If we let Q denote the rational numbers, I the integers, W the wholenumbers, and Z the integers we have the following subset relationship for our numbersystems:Z ⊂ W ⊂ I ⊂ Q.Remember that with the relations <, ≤, >, and ≥ we were able to put the integers ina useful order. Of course we can do the same thing with fractions.Given two fractions, is is sometimes tricky to tell at first glance which is the largernumber. For example, which is bigger34or56? You can see the answer by drawing thenumber strips. Alternatively, rewrite34and56over a common denominator. Thus:34=912<1012=56.This leads us to the following test:Definition. Let two rational numbers be represented by the fractionsabandcdwhere band d are positive. Thenabis less thancd, writtenab<cd, if and only if ad < bc.This makes sense since if we get two fractionsabandcdwe can writeab=a · db · dandb · cb · dso thatab<cdis true if and only if ad <