The notion s o f countin g an d recordin g number s g o bac k t o th e
earliest period s o f human prehistory , bu t ou r decima l syste m o f writ -
ing number s an d computin g wit h the m o n pape r i s o f comparativel y
recent origin—i n Europe , a t an y rate , i t wa s stil l a novelt y five hun -
dred years ago—and effective machin e computations bega n to be done
only i n th e 20t h century .
The decima l system write s number s usin g just th e te n symbol s 0 ,
1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 fo r th e firs t te n number s an d describe s large r
numbers using a place system to represent powers often, s o that 12340
means 1
104
+ 2
103
+ 3
102
+ 4 10. Elementar y an d familia r a s
this syste m is , its powe r an d simplicit y ar e definitel y worth y o f som e
attention a t th e outse t o f a cours e i n numbe r theory . Fo r example ,
this syste m make s i t possibl e fo r schoolchildre n t o lear n t o perfor m
an additio n lik e 12340 + 56789 0 = 58023 0 with littl e difficulty , a tas k
that fiv e hundre d year s ag o require d a skille d professional .
Multiplication o f numbers i s a much more sophisticate d operatio n
than addition , an d i t i s harde r t o teac h t o schoolchildren . I f a an d b
are numbers, thei r product is the number ab of objects i n a rectangula r
array o f object s tha t contain s a row s an d b columns. Sinc e countin g
ab object s i s th e sam e a s addin g b to itsel f a times , th e proble m o f
computing the product o f a and b —the problem o f computing ab —can
be reduce d t o additio n b y th e algorithm :
Input: Tw o number s a an d b.
Algorithm:
Let p = 0 an d t = a.
While t 0
Reduce t b y 1 and ad d b to p
End
Output: p
The "while " loo p i s execute d a time s i n th e cours e o f reducin g t
from a t o 0 (i f a = 0 , th e loo p i s neve r execute d an d p remain s a t
zero) an d eac h executio n o f th e loo p add s b to p , s o tha t th e fina l
value o f p i s the product—th e numbe r b added t o itsel f a times .
This algorith m i s unusabl e fo r han d computatio n i f a i s a t al l
large. Amazingly , moder n computer s ar e s o lightnin g fas t tha t the y
can multiply number s with 4 or 5 digits rather quickly in this primitiv e
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