U.U.D.M. Project Report 2018:27
Examensarbete i matematik, 15 hp
Handledare: Veronica Crispin Quinonez
Examinator: Martin Herschend
Juni 2018
Department of Mathematics
Uppsala University
Equation Solving in Indian Mathematics
Rania Al Homsi
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U.U.D.M. Project Report 2018:
Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018
Department of Mathematics
Uppsala University
Equation Solving in Indian Mathematics
Rania Al Homsi
Sammanfattning
Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi-
kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft
djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de
områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate-
matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska
matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en
eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för
dem.
Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma-
tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för
att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt:
- Kvadratisk och kubisk extraktion av Aryabhata
- Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 𝑐 = 𝑎𝑥 + 𝑏𝑦
- Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 𝐷𝑥^2 + 1 = 𝑦^2
- Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk
ekvation 𝐷𝑥^2 + 1 = 𝑦^2.
Content
- 1 - Introduction.....................................................................................................................................
- 2 - Purpose
- 3 - Historical background
- 3.1 The history of mathematics in Ancient India
- 3.2 The history of mathematics in Medieval India
- 4 - Arithmetic algorithms
- 4.1 The square root extraction method.............................................................................................
- 5 - Equation Solving
- 5.1 Quadratic equations..................................................................................................................
- 5.2 Bhaskara’s procedure for a linear equation..............................................................................
- 6 - Linear congruences
- 6.1 Kuttaka
- 6.2 Brahmagupta algorithm............................................................................................................
- 7 - Pell’s equation
- 7.1 Historical background
- 7.2 Solution methods......................................................................................................................
- 7.2.1 Method of composition (Bhavana)
- 7.2.2 Method of chakravala........................................................................................................
- 7.2.3 Interpretation of the chakravala rules................................................................................
- 7.2.4 The rationale of chakravala
- 8 - Diskussion.....................................................................................................................................
- 9 - Bibliografi
and cube roots (the cube root of 1444), arithmetic progressions, and finding solutions of equations of
the first degree, such as finding a solution of equations of the form 𝑎𝑥 + 𝑐 = 𝑏𝑦 where a, b, c are
unknown integers. Aryabhata found such solution by using the kuttaka method. This method will be
useful in finding the roots of equations of the second degree, as we will see later.
The various arithmetical rules for calculating the four operations: addition, subtraction, multiplication
and division on both positive and negative numbers, were mentioned in the major Indian work the
Brahmasphutasiddhanta, by Brahmagupta (598 - 668 CE). Moreover, the roots of the general quad-
ratic equations 𝑎𝑥^2 + 𝑏𝑥 = 𝑐 where a, b, c are integers and x is unknown. Brahmagupta went on to
solve equations with multiple unknowns of the form 𝐷𝑥^2 + 1 = 𝑦^2 (called Pell’s equation) by using
the pulveriser method. The Kuttaka and Chakravala methods for solving equations, were illustrated
clearly with help of examples in the Brahmasphutasiddhanta.
Siddhantasiromani written by Bhāskara II (1114 - 1185 CE), was the work that dealt with algebra,
geometry, permutations and combinations, and even quadratic equations with multiple roots. It also
handled linear equation system in several variables and there is a proof that these equations have
multiple solutions. He solved Pell’s equation by using the chakravala method. Mahavira, the ninth
century mathematician, composed the first Sanskrit textbook devoted completely to mathematics,
rather than being an adjunct to astronomy.
The Kerala is the Indian mathematical school that succeeded in giving advanced contributions to
mathematics, such as the developments of infinite series especially those associated to trigonometric
functions.
2 - Purpose
This study focuses on Indian mathematical contributions, specifically solving equations. The Indian
mathematicians contributed to very important achievements in mathematics, over a period of some
thousands of years. Though the Indian mathematical development is very significant, the knowledge
and awareness of it, is not spread, and even to that extent that some mathematical contributions are
attributed to other mathematicians. We will write about the Indian mathematics mostly to highlight
it, and this thesis concentrates on the ancient Indian discoveries in solving equations, especially Pell’s
equation which has significantly influenced the mathematics of current ages.
3.2 The history of mathematics in Medieval India
One of the most noteworthy features of the first century CE was the commercial prosperity in the
period of the Kushan empire in the north of India, while it was the culture which took the first place
of importance under the rule of the Guptas in the early fourth century CE.^5
Aryabhatya, the important work written by Aryabhata (b. 476 CE), consists of 123 verses divided
into four chapters, and it was centered a lot of astronomy and many mathematical themes without any
proofs.
Bhaskara I and Brahmagupta were the famous mathematicians in the early seventh century, who were
known under the rule of Harsha. As Sanskrit was the common language in the Indian subcontinent
that enabled the scholars to discuss both astronomy and mathematics. Along all these thousands of
years the mathematics had been associated with astronomy so much that most the mathematicians
were astronomers at the same time.
The first mathematical work that separated astronomy from mathematics was written in Sanskrit by
Mahavira in the ninth century and it was quite dedicated to the mathematics.
In the twelfth century, Bhāskara II (1114 - 1185 CE) made a number of contributions to mathematics
by writing his two-intelligent works, the Lilavati (it is believed without definite evidence, that Lilavati
was the daughter of Bhaskara), and the Bijaganit (1150 CE), on arithmetic and algebra, respectively.
Ancient and medieval mathematical works consist generally of rules or problems without any proofs.
They composed in verse in order to aid memorisation by the students and were explained later with
more details by other mathematicians.
Under two centuries, the fourteenth and the sixteenth centuries, the Kerala school of astronomy and
mathematics in southern India which was founded by the Indian mathematician Madhava (1350 -
1425 CE), managed through oral transmission from teachers to students to prove many of the results
that have been handed down in India for long times.^6
4 - Arithmetic algorithms
The discovery of the decimal place value system, was the basis for most of the excellence and skills
achieved by Indian mathematicians in arithmetic. For instance, the two ingenious algorithms for cal-
culating square and cube roots were mainly based on using the decimal value system in a technique
(^5) V. J. Katz, A history of mathematics, p. 232. (^6) Ibid, p. 233.
similar to the long division method. These two methods were described digit by digit by Aryabhata,
in his work Aryabhatiya.
4.1 The square root extraction method
Aryabhata represents the rule of square root extraction method as follows:
“ 2.4. One should divide, constantly, the none-square [place] by twice the square-root. When the
square has been subtracted from the square [place], the quotient is the root in a different place.”
Plofker interprets this algorithm by six steps:
- Consider that the square places refer to the even power of 10, while the non-square places refer
to the odd powers of 10.
- Determine the greatest possible perfect square. The root of this perfect square, gives the first
and the highest digit of the desired square root.
- Subtract the perfect square from the digit(s) in the highest square place and obtain a remainder
- Divide the digit in the non-square place by twice the preceding root.
- The remainder from the division is substitute into the current non-square place.
- Repeat the process as long as there are still digits on the right.^7
Example: This rule for extracting the square root of a number was illustrated with examples in order
to be understood. K. Plofker presents the technique by computing the square root of 1444 by using
Aryabhata’s method of square root extraction: The first square place, that is in the hundred places,
containing 14 , is the highest square place.
1 4 4 4
As 3 < √ 14 < 4 the square of 4 is too big and not possible to take in this calculation, and then the
greatest possible perfect square is 9 = 32. According to Plofker’s interpretation, 3 is the significant
digit in the square root that is determined by trial and error. Subtract the current square from 14 (14 -
9 = 5 ) and then moving the next digit 4 down next to the difference gives 54 in the non-square root.
(^7) K. Plofker, Mathematics in India, p. 403.
__________
Thus 38 is the square root of the number 1444. Similarly, Aryabhata presents cube root extraction
procedures as follows:
“ 2.5. One should divide the second none-cube [place] by three times the square of the root of the
cube. The square [of the quotient] multiplied by three and the former [quantity]should be subtracted
from the first [non-cube place] and the cube from the cube [place]”.^8
Example: Find the cube root to number 12977875.
Solution: By Aryabhata`s procedure the unit is always a cube place. Let define the first, the fourth
and the seventh places are names as the cube places. The second, the fifth and so on are named as the
first non-cube place. The third, the eighth and so on are named as the second-non-cube place. The
first step is to determine the last cube place and find the number that holds this position which is 12.
Now find the nearest lesser or equal cube to 12 , that is 2. The significant digit in the cube root is 2.
We subtract the current cube 8 from 12
1 2 9 7 7 8 7 5 Root result = 2
- 8 ______ 4 9
Dividing 49 by 3 times the square of the current root 2, we approximate (^349) ⋅ 22 to 3 (4 it is too large).
Thus 3 is the second digit in the cube root. We subtract 3 times the square of the current root times
the last quotient, that is, 3 ⋅ 22 ∙ 3 ,from 49. Then we take the next digit of the number to the right
down to get 137.
- 3 6 Root result = 23
1 3 7
(^8) K. Plofker, Mathematics in India, p. 403.
Now it is required to subtract 3 times the first digit in the current root times the square of the last
quotient (which is the second digit in the root) 3 ⋅ 2 ⋅ 32 = 54. Then we take the next digit 7 of the
number down
1 2 9 7 7 8 7 5 …………………. 1 3 7
- 5 4
8 3 7
We subtract the cube of the second root which is 3 from the previous deference and then take the next
digit down
1 2 9 7 7 8 7 5 …………………… 8 3 7
- 2 7
8 1 0 8
We divide the last result by 3 times the square of the current root 23 and get 5 as an approximation 8108 3 ⋅ 232 =^
8108 1587 =^5 Now 5 is the next digit in the cube root. Subtracting 3 times the square of the current square times the
last quotient 3 ⋅ 232 ⋅ 5 = 7935 , from 8108 and then taking the next digit 7 down gives
1 2 9 7 7 8 7 5 ………………….. 8 1 0 8 Root result = 235
- 7 9 3 5
1 7 3 7
We subtract 3 times the current root times the square of the last quotient (the last digit in the root)
3 ⋅ 23 ⋅ 52 = 1725 and then take the last digit in the number down, which gives
1 2 9 7 7 8 7 5 ………………….. 1 7 3 7
- 1 7 2 5
The formula for Sn leads us to the quadratic equation 𝑑𝑛^2 + ( 2 𝑎 − 𝑑) ⋅ 𝑛 − 2 𝑆𝑛 = 0 ⇔ 𝑛^2 +
(^2 𝑎 𝑑− 𝑑)𝑛 − 2 𝑆 𝑑𝑛 = 0. Finding the roots of this equation, namely the value for n , easily gives
𝑛 = − 2 𝑎 2 −𝑑𝑑 ± √(^2 𝑎 2 −𝑑 𝑑)^2 + 2 𝑑𝑆𝑛 ⇔ 𝑛 = 𝑑− 2 𝑑^2 𝑎 + ±
√ 4 𝑎^2 − 4 𝑎𝑑+𝑑^2 + 8 𝑑𝑆𝑛 2 𝑑.^ Comparing^ Aryabhata’s^ formula
for finding the number of terms n with our result using the formula for general quadratic equations
applied on Sn to find n , we see that they are equivalent in the positive case. Thus, Aryabhata basically
succeeded in solving the quadratic equations but he did not give a general quadrant formula.
Brahmagupta was one of the early Indian mathematicians who gave a clear and general formula for
solving quadratic equations in one unknown. He gave examples and presented solutions, depending
on his formula. Brahmagupta’s quadratic formula has the same form as we know it except that he did
not refer to the part of solutions which results from the negative square roots. Brahmagupta came to
his ingenious result in solving quadratic equations a century and a quarter after Aryabhata did.^10
Bhaskara II was among the first Indian mathematician to deal with two positive roots of quadratic
equations. According to Katz (2009) the method used by Bhaskara II is different from the method
used by Aryabhata or Brahmagupta. Bhaskara II used the technique of completing the square, by
combining the square of half the coefficient of x on both sides of the equation 𝑎𝑥^2 + 𝑏𝑥 = 𝑐 in order
to get a complete square in the form of ( r x + s ) 2 = d and to get then the value of x in the form 𝑥 =
𝑠+√𝑑 𝑟.^ He^ continued^ that^ if^ √𝑑^ <^ 𝑠^ then^ he^ took^ the^ other^ solution^ 𝑥^ =^
𝑠−√𝑑 𝑟.^ Bhaskara^ II^ mentioned
also, that some roots are acceptable and others incongruous, and he presented different examples with
a view to clarify that. However, all the quadratic equations examples given by him never handled
negative or irrational solutions.^11
5.2 Bhaskara’s procedure for a linear equation
Linear equations in several variables were one of the equations that Bhaskara II was interested in,
and he presented the next example to illustrate it.
“ Doves are sold at the rate of 5 for 3 coins, cranes at the rate of 7 for 5, swans at the rate of 9 for 7,
and peacocks at the rate of 3 for 9. A certain man was told to bring at these rates 100 birds for 100
coins for the amusement of the king`s son and was sent to do so. What amount does he give for each?”
(^10) V. J. Katz, A history of mathematics, p. 243. (^11) Ibid, p. 244.
To solve this example, Bhaskara II began with formulating of the problem in mathematical models,
and reduced his problem to a system of linear equations in four unknowns. He named the sets of
doves, cranes, swans and peacocks as d, c, s and p respectively. Looking at the part of the problem,
describing the total cost of the birds as 100 coins (a set of dove’s costs 3coins, etc). The first equation
was formed as 3 𝑑 + 5 𝑐 + 7 𝑠 + 9 𝑝 = 100 Then the total of 100 birds, (where every dove set con-
sists of 5 doves, etc) where the second equation as 5 𝑑 + 7 𝑐 + 9 𝑠 + 3 𝑝 = 100. After getting the two
equations Bhaskara did some algebraical operations to determinate the unknowns. He solved the first
equation for d :
(1) 𝑑 = ( 100 − ( 5 𝑐 + 7 𝑠 + 9 𝑝))/ 3
and then he solved the second equation in the same way:
(2) 𝑑 = 100 −(^7 𝑐 5 + 9 𝑠+^3 𝑝).
By equaling both equations 100 −(^5 𝑐 3 + 7 𝑠+^9 𝑝)= 100 −(^7 𝑐 5 + 9 𝑠+^3 𝑝)⇔
Bhaskara II solved the last equation easily by assuming 𝑝 = 4 and then substituting 50 − 𝑐 − 2 𝑠 −
36 = 0 ⇔ 𝑐 = 14 − 2 𝑡. The next step was to let 𝑠 = 𝑡 (where t is an arbitrary number) and insert it
in (1): 𝑑 = −^6 + 3 3 𝑡= − 2 𝑡 + 𝑡. Since t is an arbitrary number, Bhaskara II considered 𝑡 = 3. Depend-
ing on the value of t , he found the corresponding values of the other unknowns.
𝑑 = 1 One set of doves means 1 ⋅ 5 = 5 doves.
𝑐 = 8 8 sets of cranes mean 8 ⋅ 7 = 56 cranes.
𝑠 = 3 3 sets of swans give 3 ⋅ 9 = 27 swans.
𝑝 = 4 and 4 sets of peacocks mean 4 ⋅ 3 = 12 peacocks.
Through the past procedure, it is remarkable that Bhaskara II found the solutions for the starting linear
equation so simply, by assuming arbitrary numbers at some steps. It can be said that he found actually
more than one solution, because one set of a possible roots is obtained and others can be obtained by
changing the value of those arbitrary numbers.
Then he considered 60 = 0 ⋅ 137 + 60 the first division, contrary to the method, and he arranged the
quotients above each other by considering 0 as the first quotient. The series which results is:
“ Multiply the remainder by an arbitrary number such that, when increased by the difference between
the two remainders (agras), it is eliminated. The multiplier is to be set down as the quotient.”
Continuing the procedure, the last non-vanishing remainder is the greatest common divisor, in this
case it is equal to 1. Hence by choosing some multiplier v , such that 1 ⋅ 𝑣 − 10 is divisible by the last
divisor 8 ( 1 ⋅ 𝑣 − 10 is a multiple of 8), an equivalent equation with smaller coefficients and an ob-
vious solution 1 ⋅ 𝑣 − 10 = 8 𝑤 is formed. Brahmagupta required increasing the difference between
the two remainders 10 − 0 = 10 , instead of decreasing, depending on whether the number of the
quotient is even or not. He chose later 𝑣 = 18 and 𝑤 = 1. The new resulting series is:
0 2 3 1 1 18 1
“Beginning from the last, multiply the next to the last by the one above it; the product, increased by
the last, is the end of the remainders (agranta). [ continue to the top of the column.]”
From this obvious solution Brahmagupta began determining the desired solution for the initial equa-
tion. By applying the last algorithm, he got the term ( 18 ⋅ 1 + 1 = 19 ) and then ( 19 ⋅ 1 + 18 = 37 ).
Repeating this procedure gave ( 37 ⋅ 3 + 19 = 130 ) then ( 130 ⋅ 2 + 37 = 297 ) and at last ( 290 ⋅
0 + 130 = 130 ). The process is over and the value of both the unknowns x, y in the initial equation
are 𝑥 = 130 (the top term that has the greatest remainder) and 𝑦 = 297.^14 To determine N in order
to get a smaller solution Brahmagupta said:
“ Divide it (the agranta), by the divisor having the least remainder. Increase the product by the great-
est remainder. The result is the remainder of the product of the divisors.”
The last procedure to be applied is as follows 130 60 =^2 and^ the^ remainder^ is^ equal^ to^10 10 ⋅ 137 = 1370 1370 + 10 = 1380
The underlying idea of this procedure can be summed up in three steps.
Assume the existence of the integer solution to this linear equation
Work out to get an equivalent equation with smaller coefficients and an obvious solution
Work backwards from this solution to get the required solution to the original solution.
Brahmagupta achieved the value for the required number N modulo the product of x and y ,
𝑥 ⋅ 𝑦 = 137 ⋅ 60 = 8220_._ Then 𝑁 ≡ 1380 (mod 8220 ) and y can be calculated simply to get the
new solution x = 10 and y = 23_._
Verifying by the solution 𝑁 = 137 𝑥 + 10 = 60 𝑦
gives 1380 = 137 ⋅ 10 + 10 = 60 ⋅ 23
Kuttaka or “pulverizer” is considered to be the most significant topical mathematical contribution of
geniuses Indians. The kuttaka method of solving linear equations is a method to obtain the greatest
common divisor. Furthermore, it plays an important role in the solution of the much more difficult
second-degree equations by Brahmagupta, as it will be discussed later in the next chapter.^15
(^14) V. J. Katz, A history of mathematics, p. 246. (^15) Ibid, p.247.