1 Fractions from an advanced point of view

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1 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning will e why the rule: invert n multiply is orret, when we write. Let s strt with the prolem of unerstning wht the frtion 3 mens. Often, we 4 teh this y introuing the ie of pie whih we ivie into 4 equl prts, whih we ll qurters of the pie. Then the frtion 3 mens three qurters (of the pie), whih we 4 n represent symolilly s This eqution is not n empty phrse! We re sying tht the frtion three-fourths is three of something, the something is tht numer whih when you hve four of them, you hve 1, or si nother wy, is tht numer x whih solves the eqution 4x 1. Just extly how we solve this eqution will e isusse in some etil, elow. Of ourse, we lrey know how to solve it, just ivie oth sies y 4. But tht egs the question, euse if we knew how to ivie oth sies y 4, we woul know how to ivie 3 y 4, n thus wht the mening of 3/4 is. Tht s wht we re going to ress. The set of rel numers forms fiel. Thertionl numers re suset of the rel numers, whih suset itself is fiel. It is the fiel properties whih llow ivision to tke ple, n tell us (vi the xioms for fiel) wht ivision mens, n thus wht 3/4 mens. The xioms for fiel F re liste t the en of this setion. There re too mny to tlk out right here. Tht s etter left for grute ourse in Moern Alger, or Astrt Alger. We ll just tlk out the xioms we nee for our work now. 1

2 Axiom of Commuttivity When we multiply two (or severl) numers together, the orer in whih we multiply them oesn t mtter. Tht is for every x n y in F. Axiom of Multiplitive Ientity xy yx There exists n element e in F suh tht for ll x in F, ex x. We ll suh n element multiplitive ientity. This element n e proven to e unique 1, tht is, there is only one suh element in fiel F. We usully enote the multiplitive ientity y the symol 1. Axiom of Multiplitive Inverse For every x in F, ifx 0, then there exists y in F suh tht xy 1. We ll suh n element y multiplitive inverse of x, n usully enote it y x 1 or 1 x. It is importnt to note tht -1 here is not n exponent. It is merely symol tht is onveniently hosen. We oul hve hosen x s the symol for inverse, inste. When we write x 1 or 1 we unerstn the symol stns for tht numer whih when x multiplie y x, yiels 1. To return to our erlier onsiertion, we were intereste in the x whih solve the eqution 4x 1. Now we know tht suh x is the multiplitive inverse of 4, nmely 4 1,or1/4. So, then, the frtion 3 4 stns for 3 4 1, where y the xioms, there is numer 4 1 whih is the multiplitive inverse of 4. In short, ( ) A proof ppers in the setion eling with the Axioms. 2

3 Now we n unerstn wht we men y ivision. multiplitive inverse. It is relly multiplition y the ( ) 1 ( ) Let s see wht x 1 is, when x is frtion, sy (/): ( ) 1 is tht numer whih when multiplie y equls 1. But ( ) ( ) is suh numer: ( 1 )( 1 ) ( 1)( 1) (1) ( 1 ) 1 (2) ( 1) 1 (3) (1) 1 (4) whih mens tht 1 1, (5) ( ) ( is the 2 multiplitive inverse of, tht is, ) ( ) 1 1. To justify the steps in equtions (1) to (5), ove: In eqution (1), we use wht frtion / stns for. In going from (1) to (2), we use the ssoitive property of multiplition, nmely, tht we n re-group the ftors s we like. This is Multiplition Axiom 3, elow. In going from (2) to (3) we use the ommuttive property of multiplition, nmely tht, Multiplition Axiom 2. multiplitive ientity, tht 1 Then going from (3) to (4) we use the efinition of 1, n finlly in going from (4) to (5), we use the property of multiplitive ientity, n the ssoitivity of multiplition twie. Rell, we si tht ivision ws relly multiplition y the multiplitive inverse: The sme hols true for frtions s well: 2 Atully, even this nees to e prove, tht there is only one multiplitive inverse for n element in fiel. Tht requires tht the group multiplition e ommuttive. Otherwise, it is possile to hve mny ifferent left-inverses, n mny ifferent right-inverses. Tht s nother topi for nother time. See Setion 1.3 elow for proof. 3

4 ( ) 1 ( tht is, the rule for frtions is invert n multiply! ) 1 ( While we re t it, let s justify nother rule for evluting ) ( ), 1 1. The reson this works is euse we re multiplying oth the numertor n enomintor of the ig frtion y the multiplitive inverse of the enomintor, so utomtilly the enomintor eomes 1. : ( ) 1. The justifition for this step is quite it of work: First let s oserve tht the multiplitive inverse of is 1 1. Tht s not given! To see this, we hve to show tht when we multiply y 1 1,weget1: ( 1 1) () 1 ( 1 ) (6) 1 (1) (7) 1 (8) 1. (9) You might wnt to tke moment n see if you n think of the justifitions neessry in Eqution (6), n then in going from (6) to (7), to (8), et. The nswers re t the en of this hnout. Consequently, from the efinition of frtions, ()() 1 (10) () ( 1 1) (11) ( 1)( 1) (12) ( ) ( ), (13) where we use ssoitivity n ommuttivity severl times in going from (11) to (12). 4

5 1.1 Axioms for Fiel F We re given set F, together with two opertors on F whih we shll think of s ition n multiplition, n so we shll lel them s + n. The elements of F will stisfy the following xioms: Axioms for ition There is inry opertion on the set F, lle +, with the following properties: 1. For every x, y in F, x + y is lso n element of F. (Axiom of Closure.) 2. For every x, y in F, x + y y + x. (Commuttivity of ition). 3. For every x, y, z in F, (x + y)+z x +(y + z). (Assoitivity of ition). 4. There exists n element e in F with the property tht e + x x for every x in F. We ll this element n itive ientity for F. (Notie tht we hven t ssume tht there is only one suh itive ientity. Tht will follow from the xioms 3.) We shll ll the itive ientity 0. 3 Here is proof: Suppose tht there were two itive ientities, tht is, two elements, 0 n 0,suh tht for ll x, 0+x x, n0 + x x. Then sine 0 is n itive ientity, ut sine 0 is lso n itive ientity. Sine ition is ommuttive, from whih it then follows tht 00. 5

6 5. For every element x in F there exists n element y in F suh tht x + y 0. y is lle n itive inverse for x, n it will lso e shown lter, s with ientity, tht the itive inverse of n element is unique (tht is, there re not two ifferent itive inverses for n element.) We shll enote the itive inverse of the element x y x, where we unerstn this symol - s the so-lle unry minus sign. Nowys it is fshionle to ll x negtive x rther thn minus x euse we unerstn minus to stn for sutrtion. Any set F together with n opertor + whih stisfies the ove 5 xioms is lle group. Being n itive ommuttive group is prt of the requirement for eing fiel Axioms for multiplition Now we ontinue on to the reminer of the efinition of fiel. In ition to eing group uner ition, we require tht the elements of F whih re non-zero form group uner multiplition, tht is, the xioms (1)-(5) ove re repete, this time with the opertion eing multiplition rther thn ition, fter we exlue 0: Let F e the suset of F whih is ll of F exept For every x, y in F, x y is lso n element of F. (Axiom of Closure.) 2. For every x, y in F, x y y x. (Commuttivity of multiplition). 3. For every x, y, z in F, (x y) z x (y z). (Assoitivity of multiplition). 4. There exists n element e in F with the property tht e x x 6

7 for every x in F. We ll this element multiplitive ientity for F. (Notie tht we hven t ssume tht there is only one suh multiplitive ientity. Tht will follow from the xioms 4.) We shll ll the multiplitive ientity For every element x in F there exists n element y in F suh tht x y 1. y is lle multiplitive inverse for x, n it is lso shown lter, s with ientity, tht the multiplitive inverse of n element is unique (tht is, there re not two ifferent inverses for n element.) We shll enote the multiplitive inverse of the element x y x 1, or lso y 1/x. Further, we shll write x y to stn for x y 1. We hve een enoting multiplition y ut now we will gree to inite multiplition in the usul wy, y writing xy to stn for x y Axiom of Distriutivity (of Multiplition over Aition.) For every x, y, z in F, x(y + z) xy + xz. 4 Here is proof tht the multiplitive ientity is unique. It is essentilly ientil to the proof tht we gve erlier tht the itive ientity is unique: Suppose there were two multiplitive ientities, 1 n 1. Then sine 1 is multiplitive ientity. But 1 11 sine 1 is multiplitive ientity. But multiplition is ommuttive, so , whihthenshowstht 11. 7

8 This importnt xiom unerlies ll our usul lgorithms for multiplition with rries: ( ) The Aitive Inverse of n Element is Unique Suppose tht oth y n z were itive inverses to x: x + y x + z 0. By the efinition of itive ientity, the sssoitive property of ition, n the ommuttivity of ition, y 0+y (14) (x + z)+y (15) (z + x)+y (16) z +(x + y) (17) z + 0 (18) z. (19) So y z, n hene there is only one itive inverse to x. Exerise: For eh of the steps in (14) to (19) ove, stte whih xiom justifies the step. 1.3The Multiplitive Inverse of n Element is Unique Suppose tht oth y n z were multiplitive inverses to x: xy xz 1. Then y 1y (xz)y (zx)y z(xy) z1 z, (20) 8

9 y the existene of multiplitive ientity, ommuttivity n ssoitivity of multiplition. But then y z. So, there is only one inverse to x. Exerise In (20) ove, justify every step. 1.4 Why is negtive numer times negtive numer positive? In orer to nswer this question, we nee first to gree tht wht negtive mens is eing the itive inverse of positive numer, n the positive numers we shll tke s the ounting numers: 1,2,3,..., n the quotients p/q of suh ounting numers. If x is positive numer, we gree to ll x negtive numer, relling tht y x we men the itive inverse of x: x +( x) 0. Let s tke moment to prove n ovious ft: For every x in F, 0x 0. Sine 0 is the itive ientity, But then 0x 0x +0x. When we the itive inverse of 0x to oth sies, we get 00x +( 0x) (0x +0x)+( 0x) 0x +(0x +( 0x)) 0x +0 0x, i.e. 0x 0. Now let us onsier the prout of two negtive numers: ( x)( y), 9

10 where eh of these is, respetively, the itive inverse of positive numer x n y. Oserve tht 00y (x +( x))y xy +( x)y, whih mens tht ( x)y is the itive inverse of xy. Also 00( x) ( x)0 ( x)(y +( y)) ( x)y +( x)( y), whih mkes ( x)y the itive inverse of ( x)( y). But we si tht ( x)y ws the itive inverse of xy, nitive inverses re unique! Therefore, ( x)( y) xy, i.e. the prout of two negtive numers is positive! 1.5 In ition of frtions, why o we nee ommon enomintor? + + Do you rell the ol sying, You n t pples n ornges? Well, it s true, in this ontext, t lest. If we sk, how muh (volume) is 2 gllons n 2 ups, we nee to fin wy to express ups n gllons in ommon terms efore we n them. While it is true tht we oul use ups s the ommon term, let s preten for moment tht we on t know how mny ups there re in gllon. In this se, wht n we o? We n fin ommon unit of mesure, in whih we n express oth ups n gllons. The ovious hoie (ut y no mens only hoie) is ounes. A up is 8 ounes, n gllon is 128 ounes. So, 2 gllons n 2 ups is ounes. (Of ourse nother solution woul e /8 gllons.) The ie here, s with frtions, is tht in orer to rry out ition, the items hve to e in ommon terms. Ifwewishto/ n /, relling tht / mens things, eh of whih is -inverse, in orer to perform the ition, just 10

11 s with ups n gllons, we nee to fin ommon term in whih to express -inverse n -inverse. When n re integers, so 1 is multiple of 1, n, similrly, Our ommon term eomes 1. Now ition of frtions eomes esy to unerstn: 1.6 Justifitions for Equtions (6) to (9) (21) + (22) + (23) +. (24) We use the xiom of ssoitivity of multiplition to re-write the prentheses in Eqution (6) n gin in Eqution (7), then the Axiom of Ientity in going from (7) to (8) n the efinition of multiplitive inverse to go from (8) to (9). 11