ab = ba, (ab)c = a(bc), a(b + c ) = ab + ac
all follo w fro m th e meanin g o f additio n an d multiplicatio n i n term s
of countin g an d wil l b e take n fo r granted .
With thi s narro w meanin g o f "number, " subtractio n an d divisio n
are no t alway s possible .
The symbo l b a represent s "th e numbe r which , whe n adde d
to a, give s 6, " an d ther e i s obviousl y n o suc h numbe r whe n b a.
Therefore, thi s symbo l ca n onl y b e use d legitimatel y (i n vie w o f th e
meaning of "number " here ) afte r b a has been proved. Fo r example ,
the las t algorith m abov e coul d hav e sai d tha t t i s t o b e replace d b y
t k becaus e i t ha s determine d k i n suc h a wa y tha t t k.
Division require s a similar restriction . Th e symbo l b/a represent s
"the numbe r which , whe n multiplie d b y a, give s 6. " Fo r randoml y
chosen a an d b ther e i s ver y rarel y an y suc h number . Again , th e
notation b/a wil l b e used , bu t onl y whe n b has bee n show n t o b e a
multiple o f a .
However, division with remainder work s i n al l case s i n whic h a
is no t 0 : Give n tw o number s a an d b with a ^ O , ther e ar e number s
q an d r fo r whic h b = qa + r an d r b. Moreover , q an d r ar e
determined b y a an d b by mean s o f th e simpl e algorithm :
Input: Tw o number s a an d b with a ^ 0
Algorithm:
Let q = 0 an d r = b
While r a
Reduce r b y a an d ad d 1 to g
End
Output: Th e quotien t q and th e remainde r r o f th e division .
or, mor e efficiently ,
Input: Tw o number s a an d b with a ^ O
Algorithm:
Let q = 0 an d r = b
While r a
Set k = 1
While r 2ka
Multiply k b y 2
End
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