6 V. S . VARADARAJA N
As a wa y t o recal l wha t wa s learne d i n schoo l abou t geometr y tr y t o wor k ou t th e proof s o f
exercises 1-3 belo w i n Euclidea n geometr y notin g carefull y al l th e earlie r proposition s o n whic h
these proof s ar e based . Ca n yo u locat e i n which o f these an d wher e the paralle l postulat e i s used ?
1. Th e diagonal s o f a rectangl e ar e equal .
2. Le t AB b e a lin e an d le t BC an d AD b e perpendicula r t o AB o n th e sam e sid e o f AB
such tha t BC = AD. The n
(a) Th e angle s DCB an d BC A ar e equal .
(b) Th e abov e angle s ar e bot h righ t angles .
3. Th e su m o f th e angle s o f a triangl e i s two righ t angles .
4. Thi s an d th e nex t exercis e lead t o a proof o f the infinitenes s o f the sequence of primes. Prov e
first tha t i f 71 i s an y positiv e integer , eithe r i t i s a prime , o r els e i t i s divisibl e b y a prime .
(Sketch of argument : I f 71 i s no t a prime , the n w e ca n writ e ft = 7l\7l2 wher e 7l\ an d Tii
are bot h differen t fro m 1 an d 71. S o 1 7l\ 71 an d argu e agai n th e sam e wa y fo r 7l\.
This procedur e wil l en d i n a t mos t 71 steps , usuall y muc h less. )
5. Th e numbe r o f prime s i s infinite .
(Sketch of Euclid's argument: Otherwise , let£i,p2 Pn
D e
all the primes. Conside r
the numbe r
P l P 2 - . . P n + l
By exercis e 4 , i t i s eithe r a prim e o r i s divisibl e b y a prime . Thi s mean s i t i s eithe r
one o f th e p^' s o r i s divisible b y on e o f them . Bu t thi s numbe r leave s th e remainde r 1
when divide d b y an y p^.)
6. I f n i s no t a prime , sho w tha t i t ha s a prim e facto r yjn. (Thi s i s ver y usefu l i n testin g
the primalit y o f number s o f moderat e size. )
7. Verif y tha t th e number s 2
2"1+
1 are , fo r 0 7 1 5 ,
3,5,17,257,65537,429496729 7
Try t o verif y tha t al l except th e las t on e ar e prime s an d sho w tha t 64 1 divide s th e las t one .
A R C H I M E D E S (28 7 B . C.-21 2 B . C. )
Archimedes wa s one o f the greates t mathematician s o f al l time, an d certainl y
one of the most celebrated . H e was a native of the Gree k town of Syracuse situate d
on the island of Sicily. Hi s genius was universal, allowin g him to make fundamenta l
discoveries i n physic s a s wel l a s mathematics . H e wa s als o a grea t invento r an d
there ar e man y legend s surroundin g hi s achievements . H e wa s suppose d t o hav e
devised powerfu l catapult s whic h raine d heav y object s o n invadin g armies . H e
was reporte d t o hav e burne d ship s threatenin g hi s countr y b y focusin g th e sun' s
rays on them. Everyon e know s about th e stor y of his discovery o f the fundamenta l
principle of hydrostatics that a body when immersed in a liquid displaces an amoun t
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