The Rhind papyrus; the first handbook of mathematics€¦ · The two volumes on the Rhind Papyrus by Dr. A. B. Chace are the basis of this thesis. Hie work is a detailed study of
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The Rhind papyrus; the first handbook of mathematics
Subm itted in p a r t i a l f u l f i l lm e n t o f the
requ irem en ts f o r the degree of
M aster o f Soienoee
in the G raduate C ollege
U n iv e rs ity o f A rizona
1 9 3 6
Approved: U eM ajor p ro fe s s o r
<w)/ * (&x— -----------LDate V 8 6
Table of C o n ten ts .
£ 9 ? f ;/ 9 3? 3
2-
S ec tio n Page
I n t r o d u c t io n ........................................................................... ...............
I The U nit F ra c t io n ............... ................... ........................................... 1
I I The Rhind P a p y ru s ............................... 3
I I I The Egyptian Number System .................... , f
IV The Four Fundamental O p e ra tio n s . . . ........................................ .9
E gyptian U nit F ra c t io n s .............. . . . . . . . . . . . . . . . . 1 3
VI The Table o f 2 over (2n f l ) . ................................................ .20
VII The Table o f 10th e ............................................................... 31
V III Problems 7 to 20. . . . . . . . . . 3 5
IX Com pletion Problem s. . . . . . . . . . . . . ... .......................... .39
X "Aha* P roblem s. ............. . . . . . . . . . . . . . . . . . . . . 4 2
XI Problem 3 3 . ................................. 48
XII The A rith m e tic P ro g re ss io n ........................... .......................... .54
B ib lio g rap h y . . ................ .. . . . . . . . . ........... .............................. 57
I l l u s t r a t i o n s .
E gyptian System o f Number Symbolism. ............................... .. ........... 5
E gyptian Numbers and F r a c t io n s . ..........................................................8
E gyptian A d d itio n , S u b tra c tio n , and M u l t ip l ic a t io n . . . . . . . . 12
Table of 2 D ivided by the Odd N u m b ers .... . . . . . . . . . . . . . 1 7 - 19
H ie ra t ic Symbols and Problem 3 3 . . . . . . . . . . . 4 7
105591
a
I n t ro d u c t io n .
D uring my te a c h in g e x p e r ie n c e , I have o f te n n o tic e d
th a t work in f r a c t io n s i s d i f f i c u l t f o r many s tu d e n ts in
e lem en tary a l g e b r a . . There seems to he an in b o rn f e a r o f
th e word " f r a c t io n " and a g e n e ra l d e s i r e to av o id work in
v o lv in g th e u se o f f r a c t i o n s . I have u s u a l ly l a id th e blame
f o r t h i s s t a t e o f a f f a i r s on e a r ly and f a u l ty t r a in in g in
th e s u b je c t . But my c u r io s i ty was a ro u se d . Do p eo p le o f te n
f in d work in f r a c t io n s d i f f i c u l t ?
In an O f f ic e r s * ,T ra in in g Gamp in 1918, I was a ss ig n ed
to te ach a gruop o f 50 men such elem ents o f m athem atics as
th ey would need b e fo re they began an in te n s iv e co u rse in
F ie ld A r t i l l e r y . Our f i r s t le s so n was on f r a c t i o n s . The
group in c lu d ed men from th e p ro fe s s io n s a s w e ll a s th o se
whose sc h o o lin g was o n ly th rough the g ra d e s . These men n o t
o n ly had l i t t l e r e a l co n cep tio n o f th e meaning o f a f r a c t io n
b u t a ls o had l i t t l e a b i l t y to u se a f r a c t io n in th e s im p le s t
o p e ra t io n s . T h is f u r th e r added to my i n t e r e s t in f r a c t io n s *
There seemed to be a rea so n back o f a l l t h i s d i f f i c u l t y . I
d ec id ed to f in d o u t what i t was i f I ev e r had an o p p o r tu n ity .
The h i s to r y o f m athem atics b eg in s w ith th e E gyp tians
and t h e i r e a r l i e s t known w ork, th e Rhind P ap y ru s . S ince a
la rg e p a r t o f t h i s a n c ie n t work i s on a r i th m e t i c , and t h i s
b .
a r i th m e tic * 1b tu rn* i s la r g e ly on u n i t f r a c t i o n s , I dec ided
to s tu d y the Rhimd Papyrus* Here we f in d th e e a r l i e s t con
c e p t o f a f r a c t i o n and i t s f i r s t sy s te m a tic u s e . The s tu d y
h as been f a s c in a t i n g . There was d i f f i c u l t y w ith f r a c t io n s
3500 y e a rs ago - p le n ty o f i t .
The two volumes on the Rhind Papyrus by D r. A. B. Chace
a re the b a s is o f t h i s t h e s i s . Hie work i s a d e ta i l e d s tu d y
o f th e Papyrus to w hich a s tu d e n t could hope to add l i t t l e *
i f a n y th in g . In th e fo llo w in g pages I hope to p re s e n t a c a re
f u l s tu d y o f th e u n i t f r a c t io n and i t s u s e . There w i l l be a
s e c t io n devoted to the ta b le o f rt2 -4- 2 a + 1" which may w e ll be
c a l le d " T h e F i r s t Handbook o f M athem atics % I have con
f in e d oy I n v e s t ig a t io n to th a t p a r t o f the, Papyips w hich
d e a ls w ith a r i th m e t ic and a lg e b r a .
S ince a f r a c t i o n i s such a w e ll known concept* and
s in c e no v e ry advanced m athem atics i s involved* I have s e t
m yse lf the a d d i t io n a l aim o f making th i s th e s is e n t i r e ly
w ith in the g ra sp o f th e non-m athem atioal mind• So much o f
I n te r e s t in g m athem atics i s n e c e s s a r i ly a c lo sed book to the
laym an. The Rhind Papyrus need n e t b e .
1.
I . The U n it F ra c t io n .
B efo re b eg in n in g our s tu d y o f th e Rhind Papyrus i t
m ight be h e lp fu l to u n d ers tan d what a u n i t f r a c t io n i s and
what p la c e i t o ccu p ies in ou r own a r i th m e t i c .
Any s ta n d a rd a r i th m e t ic c l a s s i f i e s f r a c t io n s a s o f two
k in d s , common and d e c im a l. We f in d th a t ooBnson f r a c t io n s may
be s im p le , compound, o r com plex. Of th e s e , a sim p le f r a c
t io n i s one whose num era to r and denom inator a r e s in g le i n t e
g e r s , such as 6 /2 9 . A sim ple f r a c t io n i s p ro p e r o r im proper
a c c o rd in g a s the num era to r i s l e s s th a n , o r g r e a te r than th e
d en o m in ato r. Suppose we c o n s id e r on ly th o se sim ple f r a c t io n s
whose num era to rs a r e u n i ty , o r 1 . We m ight c a l l th e se f r a c
t io n s the r e c ip ro c a ls o f the I n te g e r s r b u t a s im p le r name
f o r them i s u n i t f r a c t i o n s .
T h is , a s we s h a l l s e e , was th e meaning o f th e word
f r a c t io n in e a r ly E gy p tian m athem atics and la r g e ly th rough
o u t th e a n c ie n t w o rld . O n e -h a lf , o n e - th i r d , o r o n e - f i f t h ,
f o r exam ple, could be g rasped a s a c o n c e p t, b u t r e a l d i f f i
c u l ty was encoun tered when i t became n e c e ssa ry to ex tend the
co n cep t to in c lu d e tw o - f i f th s o r th re e f o u r th s . Much o f the
2*
t ro u b le la y In the in a b l l ty to ex p ress th e se co n cep ts in
sym bolst o r even v e r b a l ly . The B abylon ians were a b le to
ex p ress some o f th e sim ple f r a c t io n s a s p a r t o f t h e i r u n i t
60; th u s , l / 2 was 30 , l / 3 was 2 0 , and so on* But th i s
method f a i l e d where 60 f a i l e d to d iv id e the d enom inato r.
Even a f t e r th e Greeks had developed a q u i te s a t i s f a c to r y
system o f f r a c t i o n s , t r a d i t i o n le d them back o f te n to the
u se o f the u n i t f r a c t i o n . Theodorus (c .4 0 0 B .C .) used u n i t ■ ■ - - . y ' :
f r a c t io n s In ap p ro x im atin g th e s q u a re - ro o t o f 3 .
1 jr-frit 3i tv? > YTy> 1 i: Vt i l
Where no symbol i s used betw een th e f r a c t i o n s , a d d i t io n i s
u n d e rs to o d . T his was th e custom ary p r a c t ic e w herever u n i t, . . . . ■ '
f r a c t io n s were added . In 50 B.C . we f in d Heron u s in g them
and as l a t e a s the 10 th cen tu ry A.D. they ap p ea r in a Hebrew
w r i t in g . D uring th e R ennaisance Buteo favo red them . I t i s
i n t e r e s t i n g to n o te th a t con tinued f r a c t io n s o f the formI 1 - ___ ■
w hich seem to have a co n n e c tio n w ith u n i t f r a c t io n s were n o t
su g g ested u n t i l 1613 in a work by O a ta ld i .
Mow th a t we u n d ers tan d what i s meant by a u n i t f r a c
t io n , we a re ready to c o n s id e r what the E g y p tian s could do
w ith th e se "most e lem en ta ry , p ro p e r , s im p le , common f r a c t io n s ."
L D .E .Sm ith : H is to ry o f M athem atics, Vol I I . page 212.
3 .
I I . The Rhlnd P apyrus•
The Ahmes Papyrus sms found a t Thebes in the ru in s o f a
sm a ll b u ild in g n e a r th e Harnesseura* Purchased in 1858 by
A. Henry Rhlnd, i t became known as th e Rhlnd P apyrus« I t
i s now in p o sse ss io n o f the B r i t i s h Museum. I t i s a copy
o f an e a r l i e r work, o r w orks, made by a a o t ib e A,Hoos6, o r
Ahmes. The d a te i s ap p ro x im ate ly 1650 B.C. I t i s prob
a b le th a t i t r e p re s e n ts the knowledge o f E gyptian a r i t h -
m etio c e n tu r ie s e a r l i e r than 1650 B.C . O ther E gyptian
p ap y ri and fragm en ts g iv e ev idence th a t th e work o f Ahmes
on f r a c t io n s was n o t u n iq u e .
W ritten in h i e r a t i c ( a c u rs iv e form and n o t a s fo rm al
a s th e h ie ro g ly p h ic ) , i t was o r ig in a l ly on a s in g le r o l l of
papyrus n e a r ly 18 f e e t long and 13 inches h ig h . I t s t i t l e :
" D ire c tio n fo r O b ta in in g the Knowledge o f A ll Dark T h ings",
has a c e r t a in ap p ea l to th e average s tu d e n t o f m athem atics.
I t was t r u ly the " F i r s t Handbook o f M athem atics". I t gave
answ ers to type problem s, showed no s o lu t io n s excep t one,
and even co n ta in ed a ta b le o f v a lu e s to h e lp in w orking
p rob lem s. What more do we a sk o f a good handbook today? No
th in g , excep t th a t i t be a l i t t l e le s s awkward to c a rry
around w ith u s .
4.A fte r a s h o r t In tro d u c to ry p a rag rap h , the work i s d iv id ed
In to th re e g e n e ra l g ro u p s ;
1 . Problem s in a r i th m e tic * la rg e ly devoted to u n i t f r a c t io n s *
2 . P r a c t i c a l problem s in geometry*
3* Problems o f measure®, the d iv is io n o f p ro p e r ty , and even
a fe n sim ple p ro g re ss io n s* :
v.e s h a l l co n fin e ou r in v e s t ig a t io n la rg e ly to th e f i r s t
p o r tio n on u n i t f r a c t io n s * To u n d erstan d th e d i f f i c u l t i e s
th e se f r a c t io n s p re se n ted to the S o rtb e Ahoes we must lo o k
f o r a moment a t th e E gyptian number sy stem .
I x 3 H ______ s b 7 ? 9
U n i t S l i i i n l l t ln ri i
//////
1 1 1 1I I I
m i
u u
I I I I I I 1 1 1
T e n s n n n nnn nnnn nnnnn non000
nnnnnnn
nnnnnnnnnnnnnnnnn
Hundreds e e e. cec eeeccceec
eeecce
ceeccec
ccccccec
ccccccccc
■ :
Thousands I II n? WISIIII m s
I f fI f f
T ens
T h o u s a n d s 1 )) r )))) 1 1 ) m 1111
6 .
I l l • The E gyp tian Humber System*
Of th e two forma o f E gyp tian n o ta t io n , th e h ie ro g ly p h ic
i s much e a s ie r to read than th e more r a p id ly w r i t te n h i e r
a t i c , o r c u r s iv e . B efo re 2000 B.C . th e num eral system was
l a r g e ly a d d it iv e * By a method o f r e v e r s in g th e sym bols,
numbers could be w r i t te n from r i g h t to l e f t o r from l e f t to
r i g h t a t w i l l . The l a t t e r method i s " e a s i e r f o r us to read as
i t co rresp o n d s to o u r own o rd e r o f w ritin g * The l&ok o f a
symbol f o r ze ro h in d e red the developm ent o f a p la c e -v a lu e
system s im i la r to ou r A rab ic system . The n e c e ssa ry symbols
w ere j / f o r on®, /^) f o r 10, (3» f o r 100,
f o r 1000,
f o r 1 ,000 ,000 and _ P — f o r 10,000,000*
The symbol f o r one m il l io n ap p ea rs to be a k n e e lin g man
h o ld in g up h is hands in asto n ish m en t* There i s no symbol fo r
one b i l l io n * The a d d i t iv e n a tu re o f E gyptian numbers i s
shown by a c h a r t on page [ S ) • Examples of s e v e r a l numbers
w r i t t e n in th e two methods ap p ea r on page ( ? ) . A l l th e se
exam ples a r e in h ie ro g ly p h ic . A few numbers w r i t te n in h i e r
a t i c a r e shown on page { HJ )•
Save f o r 2 /3 , a l l f r a c t io n s were w r i t te n a s u n i t f r a c
t io n s and th e s e , by u sa g e , were alw ays w r i t te n w ith no denom
in a to r r e p e a te d . The f r a c t i o n 2 /7 could n o t , in E g y p tia n ,
be w r i t te n 1 /7 1/7 b u t became l / 4 1 /2 8 . When two f r a c t io n s
f o r 10,000 f o r 100#000
7 .
were w r i t te n s id e by s i d e , a d d i t io n was u n d e rs to o d . The
symbol f o r 2 /3 was^j> w h ile a l l o th e r f r a c t io n s were w r i t
ten w ith C o v er th e number r e p re s e n t in g th e d en o m in ato r.
O c c a s io n a lly a d o t rep la ced th e f r a c t io n s ig n •
x iIn h e r H is to ry o f M athem atics, H iss V era S anford s a y s :
* I t i s i n t e r e s t i n g to s p e c u la te w hether th e sym bolism p re
v en ted th e use o f f r a c t io n s w ith num erato rs o th e r than one o r
' w hether the e x c lu s iv e use o f u n i t num era to rs was th e reaso n
f o r the sym bolism ."
S an fo rd i A S h o rt H is to ry o f M athem atics. Page 103•
A few exam ples o f u n i t f r a c t io n s w i l l be found on page
e ig h t* The f r a c t i o n l / 2 .was o f te n w r i t te n . ;
8 .
S y m hols>
Ohe. h u n d r e d "th < v $ a n
one ml Ilf on
o n e fen million . O d
fy*so- t h i r d s <̂ >
one -ha I f Zi:
E g y p t i i n N u m b e r s * .
'Tfeod Lcf Tfcic/'ffi'lkt "h L e f t ,
n n n 37/Vi n n n
C Hu / 0 ‘j
' 6 § | | r , n n i" / 7 3 6
u
' l l f f i e e . n 1/ ■2 3 , 3 1 a.
i r f y / > t i a n f r a c t r o n s .
nnnnnnnnn
CCnnm a a i
IV . The Four Ifondam ental O pera tions#
A d d itio n : T his p ro c e ss "Kas e s s e n t i a l l y th e same a s o u r
p re s e n t dec im al method# The numbers w ere p laced in colum ns»
u n i t s under u n i t s , te n s under te n s , and so on# Then th e num
b e r o f sym bols in each column were counted# F or every te n
gyribols in any one colum n, a s in g le symbol o f th e n e x t h ig h e r
column was added# F or exam ple, i f th e re w ere 12 te n s , one C-»
would he added to the hundreds and two te n s would ap p ear in
th e answ er. An i l l u s t r a t i o n o f a d d i t io n may he found on
page ( 72. )# .
S u b tra c tio n : In s u b t r a c t io n the p r in c ip le o f borrow
in g was used in th e sane way we employ i t to d ay . T his i s
a l s o i l l u s t r a t e d on page ( / SL ) .
M u lt ip l ic a t io n : D ir e c t m u l t ip l ic a t io n o f an in te g e r
by an in te g e r was aosom pllshed by an in g en io u s d ev ice o f
d o u b lin g and red o u b lin g # Suppose we w ish to m u lt ip ly 432
by 19 . Our work c a r r ie d o u t in ou r own decim al n o ta t io n
b u t a rran g ed a c c o rd in g to th e E gyptian method would ap p ea r
a s fo llo w s :
♦I 432
*2 864
4 1728
8 3456
♦16 6912
1 0 .
The m u l t ip l i e r s I , 2 , and 16 a re checked ( ’ )• These
th re e nunfcera add up to o u r m u l t ip l i e r 19. I f we now add
th e numbers 432, 864, 6912, w hich ap p ea r o p p o s ite th o se
we have checked , t h e i r sum 8208 i s our c o r r e c t r e s u l t .
I t I s e v id e n t th a t th i s p ro ce ss would be lo n g i f o u r m u lti
p l i e r were la rg e and we were l im ite d to d o u b lin g . To speed
up th e work th e E gyp tians used 10 a s a m u l t i p l i e r . L e t us
se e how they would m u ltip ly 569 by 78 .
I 569
*10 5690
*26 II380
*40 22760
2 1138
4 2276
*8 4552
The sum o f th e numbers o p p o s ite th e checked m u l t ip l i e r s w i l l
g iv e us the r e s u l t 4 4 ,3 8 2 . I t w i l l be n o tic e d th a t in th e
work we have taken on ly doub les o r a m u lt ip le o f te n .
D iv is io n : D iv is io n o f an In te g e r by an in te g e r was
perform ed by su c c e s s iv e m u l t ip l i c a t io n o f th e d iv i s o r u n t i l
th e d iv id en d was o b ta in e d , o r u n t i l a number had been reached
w hich was l e s s than th e d iv id en d by an amount n o t g r e a te r
th an the d iv i s o r , thus le a v in g a rem a in d er. T his p ro c e ss
i s based on th e f a c t th a t
D ividend = Q u o tien t X D iv is o r -f- Rem ainder.
1 1 .
L et us c o n s id e r th e d iv is io n o f 923 "by 24 . The work
would be a rran g ed a s fo llo w s :
I 24
.*10 240
*20 460
2 48
4 ::: 96
*8 192
Yfe see th a t th e sum of the numbers o p p o s ite th e m u l t ip l i e r s
th a t a r e checked i s 912 . T his i s l e s s than th e number we a r e
d iv id in g (923) and the d i f f e r e n c e between 923 and 912 i s I I
w hich i s l e s s than the d iv i s o r (2 4 ) . th e r e fo r e ou r r e s u l t
i s 38 , th e sum of th e checked num bers,as a q u o tie n t w ith a
rem ainder o f I I . T his method i s sometimes c a l le d M u lt ip l i
c a t io n o f the Second K ind. ■
With t h i s b r i e f c o n s id e ra t io n o f th e fundam ental o p e r-
a t i o n s , we a re now ready to u n d e rtak e the d iv is io n o f 2 by
th e odd in te g e r s . I t i s h e re th a t we w i l l f in d th e work o f
th e S o rib e Ahmes v e ry e le g a n t in d e ed . -
12*
£ x a m H c s .
)) l CC non »'
cce r)f\ i
A d d i t i o n *
Ml
) u m ^ IIIi l lIII
2 I . ,Z 3 sr
2, 7 * 3
3 2 1
1H, 3 O <J
S u b T r a c t l o n .
1 ee %3ii; , , ^ , k
2 . 5 - , I l f))?? zo n n n i ny . nnn ii
H v ihpiy e nn in b y n III 15. 3 x n
'1 C nzi inin
'J 12-3' l l ii ii
e c nnnn j j1, c e c e 11% I,
5, 4 4
1 V 9 1
minn
^ 111
n n nH I nnnn mil e e n 0 00
1 III M S IV,1
» 9 M
' i t 11 4 i
Bm i 3.3 3 7
1 3 .
V. E gyptian U nit F ra c t io n s .
In a d d i t io n and s u b tr a c t io n o f f r a c t i o n s , we en co u n te r •
a p e c u l ia r d i f f i c u l t y . Suppose we w ish to add 1 /3 to 1 /2 1 .
The r e s u l t i s e a s i ly expressed in u n i t f r a c t io n s a s 1 /3 1 /2 1 .
Ho d i f f i c u l t y a r i s e s s in c e the denom inators a re n o t a l ik e
and th e re fo re w r i t in g them s id e by s id e v io la te s no r u l e . I f
we were to add l/B to l / 8 , we would double the 1 /8 and c a l l
the r e s u l t l / 4 . The w r i t in g o f two l ik e f r a c t io n s s id e by
s id e was s t r i c t l y a g a in s t good u sa g e . The E gyp tians were no
le s s bound to p re fe r re d m athem atica l forms than we a r e today .
We have a lre a d y seen th a t when the denom inator o f the two
f r a c t io n s was the same even number th e re was no d i f f i c u l t y .
R eal t r o u b le , however, a ro se when the denom inator was odd.
I f 1/5 were to be added to l / 5 , the r e s u l t 1 /5 1/5 was n o t
to be used and th e re was no symbol f o r 2 /5 . So h e re we s t r i k e
an am using snag le a d in g to p le n ty o f m ental g y m n astic s .
L et us b e , f o r - the moment, e a r ly E gyptians faced w ith
the problem to add l / 3 1/5 to l / 5 1 /2 1 . We have n ev er heard
o f a consnon denom inato r, l e a s t o r o th e rw ise ; we must n o t. . . . \w r i te l / 3 1/5 l / 5 1 /2 1 ; and we canno t w r i te 1 /3 2 /5 1 /2 1 .
What a re we to do ? L et us do r a th e r a n a tu r a l th in g : l e t us
th in k o f a number o r a group o f th in g s to which we may r e f e r
14«,
o u r f r a c t io n s , such th a t each f r a c t io n w i l l be a v,hole num
b e r o f ou r g roup . Suppose v/e tak e 105 lo av es o f bread as
o u r g roup . T his id e a o f r e f e r r in g our f r a c t io n s to a group
would ap p ear to be the same th in g a s ta k in g a common denom
i n a to r . The p ro cess has been much d iscu ssed by commentators
o f th e P apy rus. W hile R odet, H u ltsh , and P o st see i t a s
common denom inato r. D r. Chace has th i s to say o f i t : "The
id e a o f ta k in g a number, so lv in g the problem f o r th i s num
b e r , and assum ing the r e s u l t so o b ta in ed h o ld s tru e f o r any
number, i s e x a c tly what th e boy In sch o o l i s in c lin e d to do
w ith a l l p rob lem s, and what th e a u th o r o f our handbook
does in ranch o f h i s w ork."
Chaoes The Rhlnd M athem atical P apyrus, page ( 10 )
We had chosen 105 le av es as our re fe re n c e number. How our
1 /3 1/5 has th e meaning 35 lo a v e s , 21 loaves w h ile 1 /5 I / 2 I
means 21 lo a v e s , 5 lo av es making a t o t a l o f 82 lo a v e s . V/e
oannot say we w i l l have 82/105 o f the whole a s v/o cannot
w r i te th i s f r a c t i o n . The b e s t we oan do i s to ex p ress the 82
a s an ag g reg a te o f d i f f e r e n t p a r ts o f ou r w hole, 105.
v/e must now ta k e f r a c t io n a l m u l t ip l ie r s o f 105 and seek
to m u ltip ly 105 so a s to g e t our 8 2 . We w i l l have to u se a
l i t t l e I n tu i t i o n u n le s s we want to be a lo n g tim e ab o u t i t .
As 2 /3 i s u s u a l ly the f i r s t m u l t i p l i e r , we w i l l s t a r t
w ith t h a t . We remember th a t 2 /3 was the on ly f r a c t io n f o r
which the E gyp tians had a s e p a ra te sym bol. T his may acco u n t
f o r the f a c t th a t Ahmes u ses i t so f r e q u e n t ly .
1 105
*2/3 70
Ve n o t ic e th a t 70 i s somewhere n e a r ou r 82 . L et u s now tak e
h a l f o f our l a s t l i n e above.
1 /3 35
But 35 i s more than we need to add to the 70 to g ive us 8 2 .
S ince 1/10 i s o f te n used to reduce numbers q u ic k ly , we w i l l
tak e 1/10 o f our 1 /3 .
1/30 0 1/2
Suppose we double t h i s l a s t r e s u l t .
' 1/15 7
T his i s u s e f u l . Even i f we add th i s 7 to our 70 above we w i l l
s t i l l la ck 5 to com plete our 82 . Vr’e may have to s t r u g g le a
b i t to g e t i t . L e t us go hack to th e 105 and t r y a new s t a r t
by f i r s t ta k in g 1 /1 0 .
1/10 16 1/2
(doub le) 1 /5 21
H ere i s a 5 b u t i t i s in the wrong lo c a t io n . V.'e want 5 in th e
n u m era to r, n o t in th e denom inato r. L u c k ily , a s E gyp tians we
1 6 .
u n d ers tan d r e c ip ro c a l r e l a t io n s and see th a t i f 1/5 g ives
2 1 , then 1 /21 w i l l g iv e 5• That v/as a l l we needed .
' 1 /21 5
We check th e term s needed to make up our 82 (2 /3 , 1 /1 5 , 1 /21)
and we have o u r answ er.
1 /3 1/5 -f 1/5 1/Sn = 2 /3 1/15 1 /2 1 .
We r e c a l l th a t we r e f e r r e d our o r ig in a l f r a c t io n s to 105
lo av es o f b re a d . As th i s was sim ply an a b s t r a c t id e a , we
may conclude th a t th e above r e s u l t alw ays i s t r u e .
I f we had had a c ce ss to Ahmes Handbook, l e t us see i f
we could have made our work s h o r t e r . Our tro u b le a ro s e from