Contributed by:
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Abstract,
Introduction,
Historical sources,
Fractions,
Reduction to the lowest term and reduction to the common denominator,
Arithmetic operations,
Classes of fractions in combination,
Unit fractions,
Conclusion
1.
WDS'10 Proceedings of Contributed Papers, Part I, 133–138, 2010. ISBN 978-80-7378-139-2 © MATFYZPRESS
Fractions in Ancient Indian Mathematics
I. Sýkorová
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. Fractions have been used in Indian mathematics since ancient times.
In this article the notational system is described and ancient Indian terms are
explained. Basic arithmetic operations with fractions and their mathematical
properties are presented.
The knowledge of fractions in India can be traced to ancient times. The fractions one-half
(ardha) and three-fourths (tri-pāda) occured already in one of the oldest vedic works the R
. gveda
(circa 1000 BC). In mathematical works Śulba-sūtras (circa 500 BC)1 , fractions were not only
mentioned, but were used in statements and solutions of problems.
Unlike ancient Egyptians who used only unit fractions (i.e. fractions with unit numerators)
in ancient India composite fractions were used.2
Fractions were necessary for the expression of smaller units of weight, length, time, money,
etc. All works of mathematics began with definitions of the weights and measures employed in
them. Some of works contained a special rule for reduction of measures into a proper fraction.
The systems of weights and measures described in different works differed. It depended on
the time and the locality in which the book was composed, see [Kaye, 1933, Colebrooke, 1817,
Rangacarya, 1912].
Historical sources
The best known mathematical texts containing fractions are as follows. Fractions were
used in Bakhshālı̄ manuscript (circa 400 AD) – the anonymous mathematical work written on
birch–bark. The rules for arithmetic with fractions were described especially by Brahmagupta
(circa 598–670) in his work Brāhma-sphuta-siddhānta, Mahāvı̄ra (circa 800–870) in his work
Ganita-sāra-samgraha, Śrı̄dhara (circa 870–930) in his work Triśatika, Śrı̄pati (1019–1066) in
his work Ganita-tilaka and Bhāskara II (1114–1185) in his book Lı̄lāvatı̄.
The Sanskrit term for a fraction was bhinna which means “broken”. The other terms for
a fraction were bhāga and aṁśa meaning “part” or “portion”. The term kalā which in Vedic
times represented one-sixteenth was later used for a fraction too. Ganeśa, a commentator of
Lı̄lāvatı̄, called a numerator bhāga, aṁśa, vibhāga or laga and the terms hara, hāra and chheda
he used for a denominator.
In Śulba-sūtra, unit fractions were named by a number with the term bhāga or aṁśa, thus
pańca-bhāga (five parts) was the name of 51 . Sometimes fractions were denoted by an ordinal
number with the term bhāga or aṁśa, so pańcama-bhāga (fifth part) is also equivalent to 15 .
Even the word bhāga was occasionally omitted, probably for the sake of metrical convenience,
thus only pańcama (fifth) could be used for 15 . Composite fractions like 27 or 38 were called
dvi-saptama (two sevenths) and tri-as..tama (three eigths) respectively.
Fractions were written in the same way as we do now, the numerator above the denominator,
but without the line between them. Both the numerator and the denominator were expressed
1
Śulba-sūtras are works in which geometrical rules for constructions of sacrificial altars are given.
2 2
Apart from fractions with unit numerators Egyptians used also 3
, see [Bečvář et al., 2003].
133
2.
SÝKOROVÁ: FRACTIONS IN ANCIENT INDIAN MATHEMATICS
in the decimal place value system. When several fractions occured in the same problem, they
were separated from each other by a vertical and a horizontal line. When a mixed number has
2
to be written the integer was given above the fraction so 2 35 was witten as 3 .
5
The next figure shows folio 10 verso from the Bakhshālı̄ manuscript.
Figure 1. There is the mixed number 3 38 (in the middle), which was called trayastraya-asht. ha
(three three eigths), see [Kaye, 1933].
Due to the lack of proper symbolism, the Indian mathematicians divided combinations of
fractions into several classes and there existed rules for calculation with them. These classes
were called jāti and the word bhinna denoted such a class of fractions too.
Reduction to the lowest term and reduction to the common denominator
It was recommended to reduce a fraction to the lowest term before performing operations.
The process of reduction was called apavartana. This procedure was not included among ope-
rations and is not described in mathematical works. Probably it was taught by oral instruction.
The reduction to a common denominator was called kalā-savarn.ana, savarn.ana or sama-
chheda-vidhi. This operation was required when the operation addition or subtraction followed.
The process was generally mentioned together with these operations.
Mahāvı̄ra was the first who mentioned the lowest common multiple, he used the term
niruddha for it. Bhāskara II recommended the process for shortening, but didn’t apply the
word niruddha.
Arithmetic operations
The terms for addition and subtraction of fractions were bhinna-sam . kalita and bhinna-
vyutkalita respectively. The method of performing operations with fractions was the same as
now. Addition and subtraction were performed after the fractions were reduced to a common
denominator. When fractions were added or subtracted together with integers, the integer was
seen as a fraction with a unit denominator.
a c ad ± cb a z a zb ± a
± = or z± = ± =
b d bd b 1 b b
Multiplication of fractions was called bhinna-gunana. Brahmagupta described multiplica-
tion as the product of the numerators divided by the product of the denominators, see [Cole-
brooke, 1817].
a c a·c
· =
b d b·d
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3.
SÝKOROVÁ: FRACTIONS IN ANCIENT INDIAN MATHEMATICS
Mahāvı̄ra moreover reminded cross reduction in order to shorten the work, see [Rangacarya,
1912]. The process of cross reduction was called vajrāpavartana-vidhi and the numerator of
the first fraction was abbreviated with the denominator of the second one and vice versa. So
3 2 1 1
· is reduced to · .
4 9 2 3
The operation of division was called bhinna-bhāgahāra and was performed in the same way
as today, first the numerator and the denominator of the divisor were interchanged and then
the operation of multiplication was performed.
a c a d a·d
: = · =
b d b c b·c
Square and square-root, cube and cube-root were included among basic arithmetic ope-
rations. Brahmagupta expressed the square of a fraction as the square of the numerator of
a proper fraction divided by the square of the denominator. He used the similar description for
the square-root of a fraction: the square-root of the numerator divided by the square-root of
the denominator, see [Colebrooke, 1817]. The rules for cube and cube-root were analogical.
a 2 a2 r √ a 3 a3 r √3
a a 3
a a
= 2, = √ , = 3, = √ 3
b b b b b b b b
Classes of fractions in combination
For the sake of shortage of suitable symbolism the expressions with fraction were divided
into several classes, see [Datta, Singh, 1935].
a c
(1) The classbhāga (“simple
fractions”), i.e. the form
with two fractions
b ± d , with three
fractions ab ± dc ± fe or with more fractions ab11 ± ab22 ± . . . ± abnn was usually written as
a c a •c
or , where the dot denotes subtraction. This form with three fractions
b d b d
a c e a •c •e
was written as or .
b d f b d f
a c a c e
(2) The class prabhāga (“fractions of fractions”), i.e. the form b · d or b · d · f which was
a c a c e
written as or .
b d b d f
(3) The class bhāganubandha (“fractions in association”) included form
a) rūpa-bhāganubandha (“fractions containing associated integers”) meant
a
a + bc written as b
c
a c a
b) bhāga-bhāganubandha (“fractions containing associated fractions”), i.e. b + d · b
a
a b
b c
or ab + dc · ab + fe · ab + dc · ab
in notation or .
c d
d e
f
135
4.
SÝKOROVÁ: FRACTIONS IN ANCIENT INDIAN MATHEMATICS
(4) The class bhāgapavāha (“fractions in dissociation”) included form
a) rūpa-bhāgapavāha (“fractions containing dissociated integers”) meant
a
b
a − c written as • b
c
a c a
b) bhāga-bhāgapavāha (“fractions containing dissociated fractions”), i.e. b − d · b or
a
a b
a c a e a c a
b • c
b − d · b − f · b − d · b written as or .
•c d
d •e
f
(5) The class bhāga-bhāga (“complex fractions”) denoted expressions a : cb or a c
b : d which
a
a
b
were written as b or .
c
c
d
There didn’t appear any graphic symbol for division, the written form was the same as for
bhāganubandha. The fact that division was required followed from the formulation of problems.
(6) Some authors meant extra class bhāga-mātr, i.e. combinations of forms enumerated above.
Mahāvı̄ra remarked that the number of such combinations was 26. As there were five
primary classes, he enumerated the total number of combinations
5 5 5 5
+ + + = 10 + 10 + 5 + 1 = 26.
2 3 4 5
The rules for reduction in the first and the second class are the same as the rules for
addition, subtraction and multiplication, the rule for reduction in the fifth class corresponds to
the rule for division of fractions. The rule for reduction in the class bhāga-bhāganubandha and
bhāga-bhāgapavāha could be written as
a c a a · (d ± c) a d±c
± · = = · .
b d b b·d b d
The following example was given by Śrı̄dhara (circa 870–930) from [Datta, Singh, 1935].
What is the result when half, one-fourth of one-fourth, one divided by one-third, half plus half
of itself, and one-third diminished by half of itself, are added together?
In today’s notation it is
1 1 1 1 1 1 1 1 1 1
+ · + 1: + + · + − ·
2 4 4 3 2 2 2 3 2 3
corresponding old Indian notation was
1 1 1 1 1 1
2 4 4 1 2 3
.
3 1 •1
2 2
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5.
SÝKOROVÁ: FRACTIONS IN ANCIENT INDIAN MATHEMATICS
1 1 1 1
1 1
The unclarity of notation is obvious, could be read as 4 · 4 or as 4 + 4 ,
4 4
1
1 could mean 1 : 13 as well as 1 13 . The right meaning of the notation could be understood
3
only from the formulation of the problem.
Unit fractions
In old India, there didn’t exist a special term for unit fraction. The term used was
rūpāṁśaka-rāśi (“quantity with one as numerator”).
Mahāvı̄ra gave several rules for expressing any fraction as the sum of unit fractions. These
rules didn’t occur in any other work, probably the other authors didn’t consider them important,
see [Rangacarya, 1912].
(a) To express 1 as the sum of n unit fractions. The rule which was given in words can be
expressed by the formula
1 1 1 1 1
1= + + 2 + . . . + n−2 + 2
2·1 3 3 3 3 · 3n−1
After leaving out the first and the last fractions, there are (n − 2) terms in the geometric
progression with 13 as the first term and 13 as the common ratio. The sum of these (n − 2) terms
1 1 − ( 31 )n−2 3n−2 − 1
sn−2 = · =
3 1 − 13 2 · 3n−2
and together with the first and the last term
1 1 3n−2 − 1 3n−2 + 1 + 3n−2 − 1 2 · 3n−2
+ + = = =1
2 2 · 3n−2 2 · 3n−2 2 · 3n−2 2 · 3n−2
(b) To express 1 as the sum of an odd number of unit fractions. The rule can be algebraically
represented as
1 1 1 1 1
1= 1 + 1 + 1 + ... + 1 + 1
2·3· 2 3·4· 2 4·5· 2 (2n − 1) · 2n · 2 2n · 2
The validity of this formula is evident
1 1 1 1 1
2 + + + ... + + =
2·3 3·4 4·5 (2n − 1) · 2n 2n
1 1 1 1 1 1 1 1
= 2 − + − + ... + − + =2· =1
2 3 3 4 2n − 1 2n 2n 2
(c) To express a unit fraction as the sum of a number of other fractions, the numerators
being given. This rule gives
1 a1 a2 a3
= + + + ... +
n n(n + a1 ) (n + a1 )(n + a1 + a2 ) (n + a1 + a2 )(n + a1 + a2 + a3 )
ap−1
+ +
(n + a1 + a2 + . . . + ap−2 )(n + a1 + a2 + . . . + ap−1 )
ap
+
(n + a1 + a2 + . . . + ap−1 )ap
1
When a1 = a2 = . . . = ap = 1, in this way, we can express the unit fraction n as the sum
of p unit fractions.
137
6.
SÝKOROVÁ: FRACTIONS IN ANCIENT INDIAN MATHEMATICS
(d) To express any fraction as the sum of unit fractions
If we denote the given fraction as pq and i is so chosen that q+i
p = m, where m ∈ N, then
p 1 i
= +
q m m·q
The first summand is a unit fraction and a similar process can be used to the second one
to get other unit fractions. Since i < p, the process ends after a finite number of steps. The
result depends on the optionally chosen quantities.
(e) To express a unit fraction as the sum of two other unit fractions. Mahāvı̄ra describes
two rules which can be algebraically expressed as
1 1 1
= + p·n
n p·n p−1
where the natural p is so chosen that n is divisible by (p − 1).
The other way according to the second rule is
1 1 1 1
= = + .
n a·b a(a + b) b(a + b)
The use of fractions was common in medieval India, Indian mathematicians gave a lot
of rules for arithmetic operations with fractions. The present method of fraction notation is
derived from Indian sources.
The Indian way of number notation including fractions was transmitted into the Islamic
world. The Arabs added the line which we now use to separate the numerator and the denom-
inator. From Arab countries fractions spread to medieval Europe.
Bečvář, J., M. Bečvářová, and H. Vymazalová, Matematika ve starověku. Egypt a Mezopotámie, Dějiny
matematiky, svazek 23, Prometheus, Praha, 2003.
Colebrooke, H. T., Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and
Bhascara, John Murray, London, 1817.
Datta, B., and A. N. Singh, History of Hindu Mathematics (part I), Molital Banarsidass, Lahore, 1935,
1938.
Kaye, G. R., The Bakhshali Manuscript: A Study in Medieval Mathematics (parts 1–2, part 3), Calcutta:
Government of India Central Publication Branch, 1927–1933.
Rangacarya, M.: Ganita-sara-sangraha of Mahaviracarya with English Translation and Notes, Govern-
ment Press, Madras, 1912.
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