Math Foum A Mathematical Seductio Fist let me say that I was a omal math studet cetaily o geius. I got all the way though high school e-calculus ad did okay, I guess. Of couse just like may healthy teeages, I used to seak off ito the gaage whee I had foud a hidde stack of old Math Hoizos fom the mid 1970 s. While it was always excitig to look at the aticles ad illustatios, the best ats wee the lettes to the Math Foum. Those lettes got me ad my othe adolescet fieds though some tese tigoometic times. But eve i my wildest math deams did I thik I would eve wite a lette to Math Foum. That all chaged i 1992 whe I left home to atted a small, libeal ats Coecticut college (I do t wat to eveal its ame sice some of the math ofs might emembe me). I took a yea of calculus i college. Eve though I had geat ofs, I hated ifiite seies with a assio. How could ayoe like all those stuid tests: The Comaiso Test; the Limit Comaiso Test; the Ratio Test; the Alteatig Seies Test; the th Root Test; the Itegal Test; the Blah Blah Blah Test... the list wet o ad o. Beig a healthy college calculus studet, I had evey calculus studet s fatasy: If lim a = 0, the = 1 a must covege. It was icedibly fustatig fo me to face the fact that my fatasy was just that a fatasy. The hamoic seies 1 =1 actually diveges to ifiity eve though the idividual tems head to zeo. The whole thig was a huge tu off fo me ad I decided to abstai fom mathematics foeve. But eveythig chaged i the fall of 1993. That s whe I met this icedible math majo (I ll call he Emmy). It was a Fiday ight i late Octobe. Eveyoe I kew had take off to Hat- fod fo the Vailla Ice cocet. I had a eco midtem comig u o Moday so I was cammig away i the deseted libay. I always studied o the to floo of the libay which housed all the laguage books. I woked at a lage oak table i a secluded coe behid the Eseato sectio. Magial demad difted ito magial oesity to cosume ad I difted ito a dee slee. I must have bee aslee fo ove a hou whe I was statled by a loud thud. As my head lifted off the oak table, I saw Emmy. Oh, I m soy to have distubed you studyig, she whiseed as she eached dow to ick he Hewlett-Packad calculato off the floo. I looked at he falle calculato ad elied, No oblem. You kow, I eve udestood how to use those. You have to uch the keys i the wog ode ad hit Ete istead of =. Emmy giggled ad said, I guess you e ot a math majo But Emmy was she was a exchage studet fom Gemay ad at that momet I would have eve guessed that this exotic stage would soo, ight thee o that oak table, make all my fatasies come tue. Gettig Close Togethe I told he about my calculus classes ad how I hated ifiite seies ad all those dumb tests. As I summed u, Emmy smiled, looked ito my eyes, ad i a soft voice asked, Have you eve egaged i eal aalysis with someoe? Of couse, as I had stoed (ehas ematuely) at calculus, I just blushed ad asweed, No With a quick glace, she asked if I was u fo tyig it ight thee ad the fo the fist time. I was so evous that my alms wee sweatig. At that momet I could t thik staight. I just stammeed, Yeah but be getle. With that Emmy stood u ad slowly slid he chai to my side of the table. She oeed u he otebook to exose a eam of ue white (ulied) ae. With oe stoke of a ecil she bega. It was icedible. It was as though thee wee oly two mathematical objects i the etie wold: the atioal umbes Q, ad the absolute value o Q,. It felt so atual eveyoe loves the factios ad the absolute value allows us to measue how close thigs ae to each othe. Little did I kow that thigs wee about to get eally close. My Fist Real Exeiece Emmy showed me a ifiite list of atioal umbes a 1, a 2, a 3,. I immediately had the uge to sum them u as a ifiite seies, but Emmy would t let me. She teased me focig me to slow dow ad build u the sequece of atial sums, S 1 = a 1, S 2 = a 1 + a 2,, S N = a 1 + a 2 + + a N ad so o. The Emmy moved towad me ad whiseed i my ea, If those atioal umbes S 1, S 2, S 3, evetually all get eally eally close togethe, the let s say that the ifiite seies = 1 a coveges. She the taslated that itimate declaatio oto the age usig ovocative math symbols: Suose that give ay tiy umbe e > 0, thee exists a itege N so that fo all idices, t ³ N, we have S S t < e. The we say that the associated ifiite seies = 1 a coveges (to some sum). Emmy told me that if we wated to be withi e, the N is how log we would have to hold off util we wee destied to be that close ad stay that close foeve. It all made comlete sese ad thee was a ush to my head as I exeieced a awakeig. At that momet, Emmy evealed that the collectio of all ossible coveget sums of atioal umbes is othig moe o less tha the set of eal umbes R. I could t believe what I Math Foum has bee a oexistet, egula featue of Math Hoizos sice the late 1960 s cotaiig lettes fom eades deictig itimate mathematical exeieces. This stoy may ot be aoiate fo all eades esecially those who ae offeded by exlicit math acts. 12 FEBRUARY 2002
A MATHEMATICAL SEDUCTION was heaig. My mid was eelig. Wow! We just coceived the eal umbes with just Q ad! It was my fist eal exeiece with aalysis ad I was eve goig to foget it. I thought that would be it. But Emmy had othe ideas. As I wied a bead of sweat fom my foehead, Emmy leaed ove ad devilishly gied, What if we elace by aothe absolute value o Q? Ae you u fo it? Aothe absolute value o Q? Thee is o othe, that s why we call it the absolute value! I kew that I was out of my league ad I should tu away, but I could t esist so I beathlessly elied, I do t kow Absolute Pleasue Afte Emmy comosed heself, she evealed that a fuctio : Q [0, + ) is a absolute value if 1) x ³ 0 fo all x Î Q ad x = 0 if ad oly if x = 0. 2) xy = x y fo all x, y Î Q. 3) x + y x + y fo all x, y Î Q. (Affectioately kow as the tiagle iequality.) Of couse oe of this shocked me sice I d had a lot of exeiece with the absolute value. But the Emmy laid a umbe o me: she let be a fixed ime umbe ad I felt a stage sesatio. She itoduced me to a ew move a ma [0,+ ) ad she declaed that 0 = 0. She took a ozeo s, ad, befoe ayoe could have said The Fudametal Theoem of Aithmetic she had ied off the ime to eveal = s s, whee Î Z ad both ad s wee stied of ay factos of. She the tedely aouced that s =. I was cofused but Emmy quickly tued my uzzlemet ito leasue by goig all the way to illustate he techique with 140 / = 2 2 3 3 5 7 11 1 : 140 2 ad eve 2 140 3 140 1 = 2, = 3, = 7, 140 I ealy flied ove whe I got it. So 140 1 = 5! I shouted. Emmy 5 smiled kowigly ad odded. It was the best five-lay I eve had. I ealized I was exeiecig somethig icedible. I fact, if you lug i ay itege the value caot exceed 1! I excitedly exclaimed. You e absolutely ight, she said as she wote: úay itegeú 1. Emmy looked aoud to make sue we wee still aloe ad asked i a low voice, Now, ae you eady to go wild? I did t kow what to say but she sue did. Let be a ime umbe. The is a absolute value o Q. She the said that they call the - adic absolute value. Well it was easy fo me to see that x ³ 0 fo all x Î Q, x = 0 if ad oly if x = 0, ad eve that xy = x y fo all x, y Î Q. But the tiagle iequality, x + y x + y fo all x, y Î Q, was, fo me, taboo. I the ed, its allue was too much ad I leaded with Emmy, Give me the easo Why does the tiagle iequality hold? Okay, I ll give it to you but oly fo iteges x ad y. I figued that if I exeieced Emmy s atioale fo the iteges, I could do it with the atioals by myself. She wated to ole lay. She said we should eted that x = x ad y = m y, whee the iteges x ad y had o factos of ad that 0 m. I eadily ageed to lay alog ad added that i this sceaio, y = m = x ad so x = max{ x, y }. I could see Emmy was excited that I was ito it. She the showed me the light: x+ y = x + y = ( x + y ) = 19 3 = 19 = 1. 0 m m 7 some itege 1 = x = max{ x, y } x + y! I was i absolute -adic ecstasy. But she ushed me ove the edge whe she showed me that we wet eve futhe tha I had thought. Togethe we actually eached a heighteed awaeess: x + y max{ x, y }. While I was comletely suised, it was clea that this was ot the fist time fo Emmy. I a wisy voice she calmed me dow, It s okay. Just elax ow, it s always itese whe you fist ecoute the stog tiagle iequality. It was tue Emmy offeed me a ew ad wodefully stage measue of closeess i the wold of atioal umbes. I fact, Emmy ad I had goe the distace: the oly otivial absolute values o the atioal umbes ae the - adics, fo each ime, ad the usual oe. My heat was acig as I tied to catch my beath. The oak table i fot of us was ow comletely coveed with tousled sheets of ae. I closed my eyes to take it all i. I wated to emembe that momet foeve. Livig Out My Fatasy It was a while befoe eithe of us soke. I felt exhausted ad my clothes wee still moist with esiatio. I olled my head to the side to look at Emmy she was glowig ad bight. Hey thee, I said with some awkwadess; I did t kow if I should stay thee with he o leave so that she could do he homewok. But Emmy kew what she wated. She smiled at me ad asked, Ae you eady to make you fatasy a eality? I felt a tigle ad I jumed. What was she goig to do ext? I was o loge evous. I ealized that Emmy kew what she was doig ad whateve it was she wated to do, it was goig to ock my wold. Emmy wated me to do some of the wok ad I was eady. So what would it mea fo a ifiite seies of atioal WWW.MAA.ORG 13
MATH HORIZONS umbes to covege -adically? she osed. I said exactly what she wated to hea. Suose that a 1, a 2, Î Q ad N SN = = 1a. If give ay tiy umbe e > 0, thee exists a itege N so that fo all idices, t ³ N, we have S S t < e, the we say that the associated ifiite seies = 1 a coveges -adically (to some sum). Emmy was tickled ad laughed with delight. She the added, We call the collectio of all ossible coveget sums the set of -adic umbes Q. It was, at the same time, comfotably familia ad excitigly ew. But suddely I felt a uge withi me. It was so atual I could ot hold back, How do we detemie if a ifiite seies coveges -adically? Emmy just staed at me ad beamed. I yeaed fo he mathematical touch but she just ket lookig ito my eyes ad giig. It was as though she could ead evey thought iside my head. He silece made me feel ueasy. Slowly she oeed he lis ad softly whiseed wods that cotiue to sig i my eas. If lim a = 0, the the ifiite seies coveges -adically. = 1 a I was ovecome. Emmy made evey calculus studet s fatasy a eality fo me! Hee we eed oly oe test to detemie covegece If the tems -adically go to 0, the seies coveges! With teas of leasue ollig dow my cheeks, I looked ito he eyes ad declaed my love fo all that is -adic. I the offeed he a utouched, ue white iece of comute ae. She took it i he had ad thee was a suge of excitemet as we both held that same iece of ae. She looked at me logigly ad said, Let s do it togethe. I just wated to satisfy he as best as I could. So we got dow to busiess. We let e > 0. Emmy the said that sice lim a = 0, thee must exist a itege N so that fo all ³ N, we have a < e. I was comletely with he; fo that istat, we wee oe. I jumed o the fact that fo ay idices ³ t ³ N, the stog tiagle iequality bigs us to S S t t = a a = a = 1 = 1 = t+ 1 = at+ 1+ at+ 2 + L+ a max{ at 1, a + 2,, a }. + t Emmy was goig absolutely cazy. She fiished the deed by shoutig out that each of those idices is at least as lage as N, ad so the maximum -adic value of all those tems is less tha e. So fo all, t ³ N, S S t < e, ad we comleted the act of makig my fatasy come tue. Sum Echated Eveig The mood quickly chaged as we bega to folic i Emmy s amazig gade of coveget delights. Emmy wote 3 + 3 2 + 3 3 + 1 3 = =. She asked if the seies coveged. I laughed; befoe I met Emmy I would have thought, Of couse it did t those tems ae diftig off to ifiity! But Emmy had tued this calculus kid ito a mathematical ma. I could tell she wated a aswe. I a dee voice I elied, It deeds: With esect to what absolute value? At that oit, we both lost cotol of ou seses. Fo ay ime ¹ 3, lim 3 = lim 1 = 1. Sice the tems ae ot eve aoachig zeo, thee is o way fo the seies to covege -adically fo ¹ 3. But, lim 3 3 = lim 3 = 0. So 3-adically, the seies coveges! That seies coveges with esect to 3 but diveges with esect to all othe absolute values o Q! Sum it! Sum it! Sum it! Emmy cied out. I could t hel myself. I was swet u i the momet. She wated me to sum it 3-adically ad I was t goig to let he dow. I took my am ad focefully thusted the aes off the oak table oto the floo ad bega to sum. It was just like Good Will Hutig but a millio times bette I was doig it with Emmy athe tha Matt Damo. I used a tick I emembeed fom calc class: I wote S = 3 + 3 2 + 3 3 + 3 4 +, which, afte multilyig by 3, gave ise to 3S = 3 2 + 3 3 + 3 4 +. I subtacted the two idetities to get 2S = 3, ad I summed it: S = 3 / 2! Petty wild (ad way wog) if you e just livig life o the eal lie R. But 3-adically I had all the ight moves! Emmy loved it. 14 FEBRUARY 2002
A MATHEMATICAL SEDUCTION We did t sto thee. We thew covegece to the wid if it felt ight, we did it. Togethe we soo foud that 1 =1 diveged with esect to evey 1 absolute value o, ad that = 0! coveged to e i the eyes of but diveged like exlodig fiewoks i the midight sky fo evey -adic absolute value. We the switched who s o to ad saw that = 0! coveged with esect to fo evey ime, but obviously aced off like mad to ifiity whe viewed i R. Seemed as if we did it all: we had seies that coveged with esect to some absolute values but ot othes, ad seies that coveged owhee. But somethig was missig ad we both kew it. We looked at each othe ad wodeed. Could we ceate a ifiite seies of ositive atioal umbes that coveged with esect to evey sigle absolute value o Q? We had to ty. Emmy wote dow = 0! sice that seies coveged eveywhee excet with esect to the usual absolute value. Ude he beath I head he moa, Make it covege i R. I did t kow what to do, so I modified he sum to:!! 2 =0 sice I kew that coveged with esect to. Ufotuately I also kew I had let Emmy dow sice that seies diveged eveywhee else. Much to my amazemet, Emmy yelled, Yes! Yes! Yes! ad with oe fial stoke, she added 1:!.! 2 + = 0 1 It was ubelievable. We had doe it: that sum coveges with esect to each ad evey absolute value o Q! My body was tiglig fom head to toe. I had eve i my life exeieced leasue like that befoe. I just fell back i my chai ad collased ito a dee dee slee. I have o idea how log I slet. I was awakeed by a jaito who yelled at me fo makig a huge mess ad also fo beig i the libay afte it had closed. I looked aoud but Emmy was owhee to be foud. The Poit of No Retu That ecoute chaged my life. The followig semeste I stated takig moe mathematics couses ad eded u becomig a math majo. I wet o to a ea a PhD i -adic aalysis. I late leaed that the wold of -adic umbes holds moe suises tha just my old seies-test fatasy. Fo istace, all - adic tiagles ae isosceles ad evey oit iside a -adic cicle is its cete. Eve close to my ight with Emmy, I ow ealize that! 2! + = 0 1 3! coveges to a tascedetal umbe i R ad also i Q fo each ad evey ime. I wish I could have shaed all these othe discoveies with Emmy. But ehas she aleady kows them. I ve had may mathematical exeieces sice that ight, some have eve ivolved covex bodies. But oe of them have comaed to that fist ight of steamy mathematics. I ll always emembe Emmy that exotic stage who fist exosed me to a wold of -adic leasues. DREW ADERBURG Not eally at Williams College [Edito s Note: Fo detailed oofs of the - adic covegece of Dew ad Emmy s 1 seies see Does = 0! Really Covege? Ifiite Seies ad -adic Aalysis by Edwad Buge ad Tom Stueck i the August 1996 issue of The Ameica Mathematical Mothly. To lea moe about -adic aalysis i geeal cosult Exloig the Numbe Jugle: A Jouey ito Diohatie Aalysis by Edwad Buge (whose ame, cuiously eough, is a emutatio of Dew Adebug.)] WWW.MAA.ORG 15
MATH HORIZONS The oof cosists of two coies of the tiagle, with thei legs laced o a commo staight lie, as show i the figue. The taezoid is the comleted by joiig the emaiig vetices of the tiagles, comutig the aea of the taezoid, ad comaig it to the sum of the aeas of the thee tiagles ito which it is decomosed. Adjoiig the mio eflectio of this taezoid, oe obtais the squae foud i the aciet Chiese text, the Aithmetic Classic of the Gomo ad the Cicula Paths of Heave; howeve, thee is o idicatio that Gafield o ay of his colleagues i Cogess had ay kowledge of this o ay simila histoical oofs; ideed, what we have see of Gafield s biogahy shows that his mathematical taiig, esecially i geomety, was limited ad lagely selftaught. Thus he ca be cedited with some iate istict ad a wothy degee of geometical cuiosity, ehas ot ofte foud i the halls of legislatues. Ackowledgmets The autho ackowledges the kid assistace of Amy Ruet ad Lida Hall of the Williams College Secial Collectios, ad of his eseach assistat, Victoia C. H. Resick of Idiaa Uivesity. Fo Futhe Readig Read Alle Peski s Gafield: a Biogahy, Ket State Uivesity Pess, fo moe details of Gafield s life. The Pythagoea Poositio by Elisha S. Loomis, ublished by the Natioal Coucil of Teaches of Mathematics, cotais hudeds of oofs of the theoem: Gafield s is umbe 231. You ca ead about the Aithmetic Classic of the Gomo ad the Cicula Paths of Heave i David Buto s The Histoy of Mathematics: A Itoductio. 16 FEBRUARY 2002