Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.


 Laurence Lee
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1 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this type require numbers like 1. In generl, numbers of the form where nd b re integers with b 0 2 b re solutions to the eqution bx =. The set of ll such numbers is the set of rtionl numbers, denoted by Q : Q = { b :, b Z, b 0}. Tht is, the set of rtionl numbers consists of ll frctions with their opposites. In the nottion we cll the numertor nd b the denomintor. b Note tht every frction is rtionl number. Also, every integer is rtionl number for if is n integer then we cn write =. Thus, Z Q. 1 Exmple 2.1 Drw Venn digrm to show the reltionship between counting numbers, whole numbers, integers, nd rtionl numbers. The reltionship is shown in Figure 2.1 Figure 2.1 All properties tht hold for frctions pply s well for rtionl numbers. Equlity of Rtionl Numbers: Let nd c be ny two rtionl numbers. Then = c if nd only if d = bc (Crossmultipliction). 1
2 Exmple 2.2 Determine if the following pirs re equl. () 12 (b) nd nd. 215 () Since (144) = ( 12)( 6) then = (b) Since ( 21)(215) (86)( 51) then The Fundmentl Lw of Frctions: Let b nd n be nonzero integer then b = n bn = n b n. be ny rtionl number As n importnt ppliction of the Fundmentl Lw of Frctions we hve b = ( 1) ( 1)( b) = b. We lso use the nottion for either or. b b b Exmple 2. Write three rtionl numbers equl to 2. 5 By the Fundmentl Lw of Frctions we hve 2 5 = 4 10 = 6 15 = 8 20 Rtionl Numbers in Simplest Form: A rtionl number is in simplest form if nd b hve no common fctors greter thn 1. b The methods of reducing frctions into simplest form pply s well with rtionl numbers. Exmple 2.4 Find the simplest form of the rtionl number Using the prime fctoriztions of 294 nd 84 we find = 2 2 ( 2) 2 = 2 = 2 2
3 Prctice Problems Problem 2.1 Show tht ech of the following numbers is rtionl number. () (b) (c) 5.6 (d) 25% Problem 2.2 Which of the following re equl to? 1, 1, 1, 1, 1, 1, 1. Problem 2. Determine which of the following pirs of rtionl numbers re equl. () 5 (b) nd nd. 60 Problem 2.4 Rewrite ech of the following rtionl numbers in simplest form. () 5 (b) 21 (c) 8 (d) Problem 2.5 How mny different rtionl numbers re given in the following list? 2 4,, 5 10, 9 1, 4 Problem 2.6 Find the vlue of x to mke the sttement true one. () = x (b) 18 = x Problem 2. Find the prime fctoriztions of the numertor nd the denomintor nd use them to express the frction 24 in simplest form. Problem 2.8 () If =, wht must be true? b c (b) If = b, wht must be true? c c
4 Addition of Rtionl Numbers The definition of dding frctions extends to rtionl numbers. b + c b = + c b Exmple 2.5 Find ech of the following sums. () (b) (c) b + c d = d + bc bd () = 5 2 ( 5) + 1 = 10 + = ( 10)+ = ( ) ( 5) (b) = ( 2) ( 5) ( 5) ( ) = = ( 14)+( 20) 5 = 4 5. (c) + 5 = +( 5) = 2 Rtionl numbers hve the following properties for ddition. Theorem 2.1 Closure: + c is unique rtionl number. Commuttive: + c = c + d b Associtive: ( + ) ( ) c + e = + c + e f f Identity Element: + 0 = b b Additive inverse: + ( ) b b = 0 Exmple 2.6 Find the dditive inverse for ech of the following: () (b) 5 (c) 2 (d) () 5 = 5 = 5 (b) 5 11 (c) 2 (d) 2 5 4
5 Subtrction of Rtionl Numbers Subtrction of rtionl numbers like subtrction of frctions cn be defined in terms of ddition s follows. b c d = b + ( c d ). Using the bove result we obtin the following: ) Exmple 2. Compute Prctice Problems c = + ( c = + c = d+b( c) bd d bc = bd = = 206 ( 105) 48 = = Problem 2.9 Use number line model to illustrte ech of the following sums. () + 2 (b) + 2 (c) Problem 2.10 Perform the following dditions. Express your nswer in simplest form. () 6 25 (b) Problem 2.11 Perform the following subtrctions. Express your nswer in simplest form. () 1 1 (b) Problem 2.12 Compute the following differences. () 2 9 (b) ( )
6 Multipliction of Rtionl Numbers The multipliction of frctions is extended to rtionl numbers. Tht is, if re ny two rtionl numbers then b nd c d b c d = c bd. Multipliction of rtionl numbers hs properties nlogous to the properties of multipliction of frctions. These properties re summrized in the following theorem. Theorem 2.2 Let, c, nd e be ny rtionl numbers. Then we hve the following: f Closure: The product of two rtionl numbers is unique rtionl number. Commuttivity: c b ( = c d ). d b Associtivity: c e = ( ) c f e. f Identity: 1 = = 1. b b b Inverse: b = 1. We cll b the reciprocl of or the multiplictive b b inverse of. b ( ) Distributivity: c + e = c + e. f b f Exmple 2.8 Perform ech of the following multiplictions. Express your nswer in simplest form. () 5 (b) () We hve (b) 5 6 = ( 5) 6 = = = ( 1)( 5) 2(9) = 5 18 Exmple 2.9 Use the properties of multipliction of rtionl numbers to compute the following. 6
7 () ( 11 ) (b) 2 ( + ) 5 2 (c) () ( 11 ) = 11 5 = 55 = 1 11 = (b) 2 ( + ) 5 2 = 2 1 = 1 1 = (c) = = ( 5 + ) = 2 Division of Rtionl Numbers We define the division of rtionl numbers s n extension of the division of frctions. Let nd c be ny rtionl numbers with c 0. Then d b c d = b d c. Using words, to find b c d multiply b by the reciprocl of c d. By the bove definition one gets the following two results. nd b c b = c b c d = c b d. Remrk 2.1 After inverting, it is often simplest to cncel before doing the multipliction. Cncelling is dividing one fctor of the numertor nd one fctor of the 2 denomintor by the sme number. For exmple: = 2 12 = 2 12 = = 8. 9 Remrk 2.2 Exponents nd their properties re extended to rtionl numbers in nturl wy. For exmple, if is ny rtionl number nd n is positive integer then n = } {{ } n fctors nd n = 1 n
8 Exmple 2.10 Compute the following nd express the nswers in simplest form. () 2 (b) 1 4 (c) () 4 2 = 4 2 = ( )() (4)(2) = (b) = = ( 4) = (c) = = = Prctice Problems Problem 2.1 Multiply the following rtionl numbers. form. Write your nswers in simplest () 10 (b) 6 (c) Problem 2.14 Find the following quotients. Write your nswers in simplest form. () 8 2 (b) 12 4 (c) Problem 2.15 Stte the property tht justifies ech sttement. () ( 5 ) 8 8 = 5 ( ) ( 8 8 (b) ) = Problem 2.16 Compute the following nd write your nswers in simplest form. () (b) 21 (c) Problem 2.1 Find the reciprocls of the following rtionl numbers. () 4 (b) 0 (c) (d)
9 Problem 2.18 Compute: ( 4 ) Problem 2.19 If b 4 = 2 wht is b? Problem 2.20 Compute Problem 2.21 Compute Problem 2.22 Compute ech of the following: () ( 2 4) (b) ( ) 2 (c) ( 2 ( 4 4) ) 4 Compring nd Ordering Rtionl Numbers In this section we extend the notion of less thn to the set of ll rtionls. We describe two equivlent wys for viewing the mening of less thn: number line pproch nd n ddition (or lgebric) pproch. In wht follows, nd c denote ny two rtionls. NumberLine Approch We sy tht is less thn c, nd we write < c, if the point representing on the numberline is to the left of c. For exmple, Figure 2.2 shows tht 1 < 2. 2 Figure 2.2 Exmple 2.11 Use the number line pproch to order the pir of numbers nd 5 2. When the two numbers hve unlike denomintors then we find the lest common denomintor nd then we order the numbers. Thus, = 6 nd 5 =
10 Hence, on number line, is to the left of 5 2. Addition Approch As in the cse of ordering integers, we sy tht b frction e such tht + e = c. f b f d Exmple 2.12 Use the ddition pproch to show tht < 5 2. Since 5 = then < 5 2 < c d if there is unique Notions similr to less thn re included in the following tble. Inequlity Symbol Mening < less thn > greter thn less thn or equl greter thn or equl The following rules re vlid for ny of the inequlity listed in the bove tble. Rules for Inequlities Trichotomy Lw: For ny rtionls nd c exctly one of the following is true: b < c d, b > c d, b = c d. Trnsitivity: For ny rtionls, c, nd e if < c nd c < e then < e. f d f b f Addition Property: For ny rtionls, c, nd e if < c then + e < f b f c + e. d f Multipliction Property: For ny rtionls, c, nd e if < c then f e < c e if e > 0 nd e > c e if e < 0 b f d f f b f d f f Density Property: For ny rtionls nd c, if < c then Prctice Problems b < 1 ( 2 b + c ) < c d d. Problem 2.2 True or flse: () 2 < (b)
11 Problem 2.24 Show tht < Problem 2.25 Show tht < using the ddition pproch. by using number line. Problem 2.26 Put the pproprite symbol, <, =, > between ech pir of numbers to mke true sttement. () (b) 1 (c) (d) Problem 2.2 Find three rtionl numbers between 1 4 nd 2 5. Problem 2.28 The properties of rtionl numbers re used to solve inequlities. For exmple, x + < 5 10 x + + ( 5 5) < + ( ) 10 5 Solve the inequlity x < x > 1. Problem 2.29 Solve ech of the following inequlities. () x 6 5 < 12 (b) 2 5 x < 8 (c) x >
12 Problem 2.0 Verify the following inequlities. () 4 < (b) 1 < 1 (c) 19 > Problem 2.1 Use the numberline pproch to rrnge the following rtionl numbers in incresing order: () 4, 1, (b), 2, 12 4 Problem 2.2 Find rtionl number between 5 12 nd 8. Problem 2. Complete the following, nd nme the property tht is used s justifiction. () If 2 < 4 nd 4 < 5 then 2. 5 (b) If < 6 then ( ) ( ) ( 6 ) ( 11 2 ) (c) If 4 < 4 then < 4 + (d) If > 11 then ( ) ( ) ( 11 ) ( 5 ) (e) There is rtionl number ny two unequl rtionl numbers. 12
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