MAE 301 / 5 01 , FAL L 2006, L EC TURE NO TES BERNARD MASKIT 1 . I ntroduction 1 .1 . What is mat hematics? It is no t at al l easy to say what mat h e mat i c s is, but , in bro ad outline, it is a way of thought, or co llect io n of ways of thought. These often co nce rn pro bl e ms ar i s i n g in the re al world, whe re the mat h e mat i c al pro b l e m so l v i ng st art s wi t h the co nst ruct io n of a mat h e mat i c al mo d e l of the re al- wo rl d pro bl e m; then so l v i ng this mat h e - mat i c al pro b l e m; then translating the so l ut i o n back to the re al world; an d , nally, as k i n g the quest io n: how we l l do e s this mat h e mat i c al so l ut i o n ac t u al l y so l ve the original pro bl e m. Man y people have the mi s t ake n no t i o n that mat h e m at i c s c o nsi st s of a co llect io n of " mo d e l pro bl e ms" , an d that the go al of mat h e mat i c s educatio n is to teach these " mo d e l pro bl e ms" an d their so l ut i o ns, so that the st ude nts can so l ve mat h e mat i c al pro bl e ms that ar e exam p l e s of these " mo d e l pro bl e ms" . One of the go al s of MAE 301 /501 is to di sabuse people of this no t i o n. Wh i l e we can' t re all y say what mat h e mat i c s is, we can gi ve an ap p r o ximat e an s wer to the quest io n of what mat h e mat i c s is al l ab o u t . First of al l , there ar e mathematical objec t s, such as numbers, se t s , matrices, t riangles an d pro babi l i t i e s. Then there ar e func tio ns or proce sse s, such as ad d i n g two numbers, forming the co mplement of a se t , nd i ng the inve r s e of a no n- si ngular mat r i x, c o ns t ruc t i ng the me d i ans of a t riangle or writ i ng a formul a for the pro bability of a compound event. Finally, there ar e pro o fs or thoughts or so l ut i o ns to they mo r e or le ss c o mpri s e what mat h e m at i c i an s pro bl e ms; these ar e di - c ul t to de s c ri b e , an d do ; one imp o rt ant exam p l e wo u l d be to view the se t of al l func t i o ns fro m one se t to an o t h e r as a ne w mat hemat ic al o b j ect . Mat h e m at i c i a n s re all y do n' t li ke to go ar o u n d in ci rcle s, so we wi l l no t furt he r pu rs ue the quest io n of what mat h e mat i c s is, bu t rat h e r st art talking ab o u t mat h e mat i c al ob j ects, co nst ruct io ns an d ot her pro c e sse s, an d thoughts. We ar e go i n g to be pri mari l y mat h e mat i c al in this develo pment; that is, we' ll st art wi t h so me un de n e d ob j ects an d pro c e s s e s , an d then careful l y, an d lo gic ally, bu i l d up ot her ob j ects, an d pro c e sse s or o p erat io ns that wo r k wi t h them, even ot her kinds of o b j ect s an d ot her kinds of pro cesses. This de vel o pment will, to so me mi n o r extent, mi r r o r the hi s t o ri c al de vel o pment. 1 .2. Basic m at he m at i c al objects. First of al l , we ne e d so me kind of o b j ect , so met hi ng to talk ab o u t . The us u al ob j ects wi t h which mat h e mat i c i ans st art ar e the nat u ral numbers, N , an d there wi t h whi ch we co unt; that is, 1 ; 2 ; 3 ; 4 ; : : : . Not ice that we st art at 1 , there is no 0, ar e no n e gat i ve numbers. Associated wi t h the nat ural numbers, we have two pro c e sse s. 2 BERNARD MAS K I T First, we can co unt in clumps; that is, we can ad d numbers. Addition sat i se s the two rul e s : Co mmutativity of a ddit io n : For al l nat ural numbers a an d b , a + b = b + a ; an d Associativity of a ddit io n : For al l nat ural numbers a , b an d c , ( a + b ) + c = a + ( b + c ). Next we o b se rve that we can ad d in clumps, that is, mul t i pl y. Mul t i p l i c at i o n al s o sat i se s two ru l e s , which we call by the same name s : Co mmutativity of multiplica t io n: For al l nat ural numbers a an d b , ab = ba; an d Associativity of m u lt i p li c a t i o n : For al l nat ural numbers a; b; c , ( ab ) c = a ( bc). Question: Why ar e these ru l e s c o nc e rni n g di #erent op erations called by the same names? There is al s o a rul e , the di s t ri but i ve rul e , co ncerning the co nnect io n between these two operations: For al l nat ural numbers, a; b; c , a ( b + c ) = ( ab ) + ( ac). Problem 1.1. What ha ppens to this ru le if yo u in t e rchange a ddit io n and mu ltiplica t io n? 1 .3. Inve r s e o p er at io ns . Is there an inve r s e o p eratio n ( subt ract i o n) to ad d i t i o n ? Wh e n is it de ne d? That is, for whi ch a an d b can we so l ve the equatio n a x = b ? Problem 1 .2. Is the re a na t u ral numbe r a so that the eq ua t io n a x = b ca n always be so lved? Never be so lved? Problem 1 .3. Is the re a na t u ral nu m b e r b for whic h the eq ua t io n a x = b ca n always be so lved? Never be so lved? We do n' t have the to ols to prove it , bu t we know that subt ract i o n is al ways un i q ue ; that is, if a = b + x an d a = b + y , then x = y . We can l i kewi se as k the quest io n: Is there an inve r s e op eration ( divisio n) to mul t i pl i c at i o n ? That is, can we so l ve the equatio n ax = b for x . Wh e n can we so l ve this equatio n? Problem 1.4. Is the re a na t u ral nu m b e r a so that the eq u a t io n ax = b always ha s a so lutio n? Never ha s a so lutio n? Problem 1.5. Is the re a na t u ral numbe r b for whic h the eq u a t io n ax = b always ha s a so lut io n? Never ha s a so lut io n? We al s o do n' t have the to ols to prove that di v i si o n is al ways uniq ue ; that is, if a = bx an d a = by, then x = y .