to wha t Leopol d Kronecke r late r calle d "genera l arithmetic."
1
In 21st centur y terms , Gauss' s subjec t i s "algorithmi c mathe -
matics," mathematic s i n whic h th e emphasi s i s o n algorithm s an d
computations. Instea d o f set-theoreti c abstraction s an d unrealizabl e
constructions, suc h mathematic s deal s wit h specifi c operation s tha t
arrive a t concret e answers . Regardles s o f wha t Gaus s migh t hav e
meant b y hi s titl e Disquisitiones Arithmeticae, wha t I mea n b y m y
title Higher Arithmetic i s a n algorithmi c approac h t o th e number -
theoretic topic s i n th e book , mos t o f whic h ar e draw n fro m Gauss' s
great work .
Mathematics i s abou t reasoning , bot h inductiv e an d deductive .
Computations ar e simpl y ver y articulat e deductiv e arguments . Th e
best theoretica l mathematic s i s a n inductiv e proces s b y whic h suc h
arguments ar e found , organized , motivated , an d explained . Tha t i s
why I think ampl e computational experienc e is indispensable t o math -
ematical education .
In teachin g th e numbe r theor y cours e a t Ne w Yor k Universit y
several time s i n recen t years , I hav e foun d tha t student s enjo y an d
feel the y profi t fro m doin g computationa l assignments . M y ow n ex -
perience i n readin g Gaus s ha s usuall y bee n tha t I don' t understan d
what h e i s doin g unti l h e give s a n example , s o I tr y t o ski p t o th e
example righ t away . Moreover , o n anothe r level , i n writin g thi s an d
previous books , I hav e ofte n foun d tha t creatin g exercise s lead s t o a
clearer understandin g o f th e materia l an d a muc h improve d versio n
of th e tex t tha t th e exercise s ha d bee n mean t t o illustrate . (Ver y
often, th e greates t enlightenmen t cam e whe n writin g answers t o th e
exercises. Fo r thi s reason , amon g others , answer s ar e give n fo r mos t
of th e exercises , beginnin g o n pag e 179.)
Fortunately, numbe r theor y i s a n idea l subjec t fro m th e poin t o f
view o f providin g illustrativ e example s o f al l order s o f difficulty . I n
this ag e o f computers , student s ca n tackl e problem s wit h rea l com -
putational substanc e withou t havin g t o d o a lo t o f tediou s work . I
xSee
Essa y 1.1o f m y boo k [E3] . Fo r th e relatio n o f genera l arithmeti c t o
Galois theory , se e Essa y 2.1.
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