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This PDF contains: Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic, and Pande’s Procedures
Abstract,
Keywords,
DOI,
1 Introduction,
2 A Glance of Three Mathematicians and Their Methods,
2.1 Bhaskaracarya and his Lilavati,
2.1.1 Bhaskaracarya’s method for finding cube roots,
2.2 Gopal Pande and his method,
2.2.1 Pande’s method to find cube roots,
2.3 Bharati Krishna Tirthaji and his Vedic methods,
2.3.1 Vedic method to find cube roots,
3 Findings,
4 Conclusion,
References.
1.
Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K. K. Parajuli
Three Classical Methods to find the Cube Roots: A
Connective Perspective on Lilavati, Vedic and Pande’s
Procedures
Krishna Kanta Parajuli
Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University, Nepal
Correspondence to: kknmparajuli@gmail.com
Abstract: South Asian region has made a glorious history of mathematics. This area is considered as fer-
tile land for the birth of pioneer mathematicians who developed various mathematical ideas and creations.
Among them, three innovative personalities are Bhaskaracarya, Gopal Pande and Bharati Krishna Tirthaji
and their specific methods to find cube root are mainly focused on this study. The article is trying to explore
the comparative study among the procedures they adopt.
Gopal Pande disagrees with the Bhaskaracarya’s verse. He used the unitary method against that method
mentioned in Bhaskaracarya’s famous book Lilavati to prove his procedures. However, the Vedic method
by Tirthaji was not influenced by the other two except for minor cases. In the case of practicality and
simplicity, the Vedic method is more practical and simpler to understand for all mathematical learners and
teachers in comparison to the other two methods.
Keywords: Cube roots, Lilavati, Nepali method, Vedic mathematics
DOI: https://doi.org/10.3126/jnms.v4i1.37110
1 Introduction
The history of the calculation of cube roots can be traced back to Babylonian civilization [1]. The calcula-
tions of cube roots also appeared in the Chinese mathematical system around the second century BCE [2].
Methods for extracting cube root was also devised by Greek mathematician Hero of Alexandria (10 AD –
70 AD) in the first century [3]. In the classical period of the South Asian region, important contributions
were made by scholars like Pingala (around the first century), Aryabhata (476 – 550), Varahamihira (505
– 587), Brahmagupta (598 – 668), Bhaskara-I (around the seventh century), Lalla (720 – 790), Sridhara
(870 – 930), Shreepati (1019 – 1066), Bhaskara-II (1114 – 1193) [4, 5, 10]. Among them, the method for
determining the cube root of the number having many digits was firstly provided by the great Mathemati-
cian or Astronomer Aryabhata in Aryabhatiya [5, 6]. Aryabhata has mentioned in his book Ganitpaad a
method to extract the cube root of any number. In the fifth sloka of Aryabhatiya which was mentioned in
Sanskrit as [5].
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2.
Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic . . .
(The meaning of Aryabhata’s and Brahmagupta’s stanza (Sloka) are more or less similar to the Bhaskaracarya’s
stanza [5]. Therefore, common meaning is mentioned after Bhaskara’s sloka)
After his time many mathematicians felt that it was complex to understand. Many great mathemati-
cians followed the ideas in Mathematics, even though most of the ideas were likely similar but sim-
plified than Aryabhata’s method to find the cube roots. Their methods are closely related to, i.e.,
(x + y)3 = x3 + 3x2 y + 3xy 2 + y 3 (where x and y are variables that they used) Brahmhagupta expressed
his method in this regard in his book Brahmsphutsiddhanta Ganitadhyaya in Sanskrit as [5].
In the twelfth century, Bhaskara-II was popularly known as Bhaskaracarya. He wrote his Siddhantasiro-
mani which contained four parts: Lilavati, Algebra, Planetary Motion and Astronomy. Before the time
of Bhaskaracarya, many contributions were made by mathematicians in the field of Mathematics includ-
ing finding the cube roots of the numbers. After the period of Bhaskaracarya , mathematician Narayan
(1340-1457) slightly revised in his book Ganitkaumudi to the method of Bhaskaracarya [5, 7]. Other mathe-
maticians Samanta Chandrasekhar (1835–1904) and Bapudev Shastri (1821–1900) differed somewhat from
those in the Aryabhata’s and Bhaskaracarya’s method for extracting cube roots [8].
The Sanskrit stanzas specified by Brahmagupta, Aryabhata and Bhaskaracarya-II are more alike similar to
each other. Many other dignified mathematicians also accepted these methods and with some minor twists
and turns, they have slightly revised these methods. However, their slight revision has not been mentioned
in this study for it being not relevant to the topic. Many classical mathematicians prior and of a much
later period than Bhaskaracarya-II followed the same method as Bhaskaracarya with some minor modi-
fications in his method to find the cube root of a number. Above mentioned mathematicians who opted
for classical procedures has made remarkable contributions. Among them, this article mainly emphasizes
on three mathematicians who significantly differed or provided the elementary structure for finding the
cube root. Furthermore, only the comparison of methods of Bhaskaracarya, Gopal Pande and Bharati
Krishna Tirthaji has been designated for this study and a neutral analysis with connections, comparison
and contrast has been conducted for their respective methods to find the cube roots.
Pandit Gopal Pande (1847 - 1920) was the Nepalese student of Bapudev Shastri. He was the first Mathe-
matics teacher of the first school of Nepal, Sanskrit Pathshala (currently Durbar high school) [5]. Another
renowned mathematician Bharati Krishna Tirthaji (1884 – 1960) reinstated a new form of mathematics in
the South Asian region namely Vedic Mathematics. Tirthaji is known as the father of Vedic Mathematics
[9]. The contributions of Pande’s and Tirthaji’s to find cube roots are remarkable.
The cube roots of a real number x is a number y such that y 3 = x. In Mathematics, each real number
(except 0) has exactly one real cube root and other two imaginary cube roots. But for non-zero complex
numbers have three distinct imaginary cube roots. Finding the cube roots of a number by the conventional
method is time consuming and tedious job with more complexity. However, using a proper technique it
becomes interesting and fast too. We can find the cube root of any number within a very short period by
just looking at the number with perfect practice. This article only focuses on extracting the real part of
the cube roots not for imaginary parts of the numbers.
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Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K. K. Parajuli
2 A Glance of Three Mathematicians and Their Methods
Many mathematicians contributed a significant role in making the glorious history of Mathematics. Among
them, three mathematicians Bhascaracharya-II, Gopal Pande and Bharati Krishna Tirthaji, and their math-
ematics are more relevant for this article.
2.1 Bhaskaracarya and his Lilavati
Bhaskaracarya or Bhaskara-II was the famous leading Mathematician, Poet, Astronomer of ancient Indian
during the twelfth century. Bhaskaracarya wrote his Siddhantasiromani when he was 36 [10]. The first
part of Siddhantasiromani, Lilavati mainly deals with Arithmetic but also contains portions of Geometry,
Trigonometry and Algebra. Bhaskaracarya’s Lilavati was considered as the popular book in mathematics
at that time. It has been used as a text-book for the last eight hundred years in many south Asian countries
and is still in practice in many Indian provinces [10].
2.1.1 Bhaskaracarya’s method for finding cube roots [5, 10, 11]
Bhaskaracharya also found a method to find the cube root of a number. However, the method that on
finding the cube root, Bhaskaracarya had his words as follows:
Meaning: Draw a vertical bar above the digit in the unit’s place of the number whose cube root is wanted.
Then put horizontal bars on the two digits to its left, vertical bar on the next and repeat until the extreme
left-hand digit is reached. From the extreme left-hand section, deduct the highest cube possible and write
to the left side the number ’a’ whose cube was subtracted. Write to the right of the remainder, the first
digit of the next section to get a new sub-dividend. Now divide by 3a2 and write the quotient b next to a.
Then write the next digit from the section to the right of the remainder obtained above. The next divisor
is 3ab2 . In the next step take b3 as the divisor. Continue this procedure until the digits in the given number
are exhausted [10].
For illustration, consider the perfect cube number 50653.
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4.
Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic . . .
Table No.1
Root Description Given Number Remarks
5̄ 0̇ 6̄ 5̄ 3̇ The largest perfect cube
a=3 a3 -2 7 below 50 is 27
2 36
3a2 b = 3 × 32 × 7 -1 89 When b = 8, 236 − 3a2 b is
positive but 236 − 3ab2 is
negative. Therefore, b=8
is not acceptable. So, we
proceed to the next small
number b=7.
475
b=7 3ab2 = 3 × 3 × 72 441
343
b3 = 73 343
000
√
3
Therefore, 50653 = 37.
Bhaskaracharya’s method of finding the cube root was similar to the method used by Aryabhata, Sridhara,
Brahmagupta and Sripati [5].
2.2 Gopal Pande and his method
Gopal Pande (1847–1920) was the first Nepalese mathematician to write Mathematics in the Nepali lan-
guage. He wrote Wyaktachandrika with consecutively four editions within the period 1883 to 1914 in the
Nepali language [5]. Among them his third edition was written in Hindi [5, 12]. He expressed his mathe-
matical techniques in his book from a simple understandable primary level to a higher-level realization [5].
He wrote the book Wyaktachandrika by restructuring the concepts of Lilavati, the idea from his teacher
Bapudev Shastri and idea from other books.
Even though, Pande’s most of the work was based on Lilavati of Bhaskaracarya [5, 7]. He has published his
book and made it the text-book of mathematics. His book was considered as the easy book for teaching
mathematics at that time [13]. In 1884, the first Nepali calendar (Panchanga) was published in Nepal.
In this calendar, Pande found an error in the forecasting of the Lunar eclipse of that year. After a long
discussion, it was found that Gopal Pande’s result was true. Due to this achievement, he was honored with
Royal Astrologer [12, 13].
Being a neighboring country with the same cultural and social-behavioral, there was a direct influence on
the teaching-learning process between Nepal and India. Most of the Nepalese scholars were joined with the
Indian scholars and shared the ideas in every field of education. Every mathematical text-books of Nepal
was purchased from India, which were written either in Sanskrit or in Hindi. Moreover, India was the main
source of Nepalese scholars in every aspect of the subjects at that time [5]. For finding the cube roots of
the number, Nepalese mathematician Gopal Pande has disagreed with the Bhaskaracarya’s verse. This new
approach is considered as a remarkable contribution of Gopal Pande in Nepali mathematics.
Bhaskaracharya expressed in his book Siddhantasiromani that the unitary method or rule of three cannot
be applied to find the square root and cube roots. In this regard, he expressed his idea in Sanskrit as [5]:
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Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K. K. Parajuli
Meaning: The meaning of the above Sanskrit verse is that the whole mathematical problem can be solved
by the rule of three or by unitary method except for finding squares, square roots, cubes and cube roots
Pande’s mathematical contribution with innovative techniques was more illustrated by Professor Naya Raj
Pant (1913 – 2002) in 1980 on his book Pandit Gopal Pande and his method of getting Cube Root. Gopal
Pande claimed that the rule of three can be used for finding squares, square roots, cubes and cube roots of
a positive integer [5].
Figure 1: Images of three books: Lilavati of Bhaskaracarya (English Translation), Vedic Mathematics of
Bharati Krishna Tirthaji and Pandit Gopal Pande by Nayaraj Pant (Pant’s book on Gopal Pande is rarely
available in the market. It can be obtained in the central library, Kathmandu.)
2.2.1 Pande’s method to find cube roots [5, 14, 15]
Most of the basic mathematical problems can be solved by using a unitary method, which is an alternative
technique of solving problems that can be solved by direct and inverse proportion method. In the unitary
method, we first find the value of unit quantity by division in the case of direct proportion, and by multi-
plication in the case of inverse proportion. Then, we find the value of any other quantity by multiplication
in the case of direct proportion, and by division in the case of inverse proportion.
For calculation of the cost of 90 books when it was given that the cost of 25 books was $20, we use the
unitary method. If the cost of 25 books was $20 then the cost of 1 book is $20
25 , so the cost of 90 books
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6.
Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic . . .
= $20
25 × 90 = $72. According to Bhaskaracarya, the process is not useful for finding squares, square
roots, cubes and cube roots. For example, If the square root of 49 is 7 then the square root of 5776 is
7
49 × 5776 = 825.1428. Here, the result obtained by the unitary method is wrong.
Gopal Pande was not agreed with the Bhaskaracarya’s verse about to find the squares, square roots, cubes
and cube roots. Pande expressed his techniques differently to find these by taking an example of cube root.
Pande wrote his method in his book Wyktachandrika about cube roots. A piece of method of Pande’s in
his own language in Nepali for finding cube roots looks like:
Illustration of Pande’s method: To perceive the ideas of Pande’s method for finding cube roots, it is
convenient by taking an example to find the cube root of the number 13824. For this purpose, he found
the correct result by expressed his ideas by using the methods stepwise as follows: [5, 16]
• Place dots above the first digit (Unit digit) then above every fourth digit (Thousandth place) then
above the seventh digit and so on from the right of the given number which has to be found the cube
root, i.e., making the group of three from the right side of the number.
• The number of dots is equivalent to the digits of required cube roots.
• For the number 13824, it will have a 2-digits cube root being two dots 13̇824̇.
• From the left of the number, first, it will be taken cube root of 13 which is greater than 2 but
less than 3. Then it will be confirmed that the required cube root will be greater than 20 and less
than 30. Therefore, for the difference in cube root by 30–20 = 10, the difference of their cubes is
303 –203 = 19000. That is, when the difference between two cubes is 19000 then the difference between
two cube roots would be 10.
• Now the number 13–23 = 5 with the number 824 (last or next three numbers) becomes 5824.
• Hence, in this case, Gopal Pande uses the unitary method or rule of three as:
Table No. 2
Difference of cubes The difference be- Cube root (By uni- Remarks
tween cube roots tary method)
10
19000 10 19000 × 5824 = 3.06 The next number of
required cube root
would be 3 + 1=4
Therefore, the cube root of 13824 = 24.
2.3 Bharati Krishna Tirthaji and his vedic methods
A very renowned cultured personality of India, Jagadguru Sankaracharya Sri Bharti Krishna Tirthaji (1884-
1960) was a glorious and divine person. He is well known for his formulation and reconstruction of Vedic
Mathematics from certain Sanskrit text Veda. According to him, the Vedic system is based on his Sixteen
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Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K. K. Parajuli
sutras and an equivalent number of sub-sutras cover all branches of mathematics, pure and applied. Vedic
Mathematics was formulated and reconstructed by Tirthaji in the time of 1911-1918, was only published
in 1965 after his death in 1960 [9, 17, 18]. Vedic Mathematics also relates to mental calculations.
2.3.1 Vedic method to find cube roots [9, 17, 18]
There is a very quick method in Vedic Mathematics to find cube roots of perfect cube numbers, for this we
need to remember that:
13 = 1; 23 = 8; 33 = 27; 43 = 64; 53 = 125; 63 = 216; 73 = 343; 83 = 512; 93 = 729; 103 = 1000
From the above cubes of 1 to 10, we need to remember the following facts:
i. If the last digit of perfect cubes is 1, 4, 5, 6, 9, or 0 then the last digit of cube roots is similar.
ii. If the last digit of perfect cubes is 2, 3, 7, or 8 then the last digit of cube roots is 8, 7, 3, or 2
respectively.
To be a perfect cube, the digital sum of the cubes is necessary to found 0 or 1 or 8. Observe the table:
Table No.3
Numbers 1 2 3 4 5 6 7 8 9
Cubes 1 8 27 64 125 216 343 512 729
Digital Sum 1 8 0 1 8 0 1 8 0
The left digit of a cube root can be extracted with the help of the following table:
Table No.4
Left-most pair of 1 8 27 64 125 216 343 512 729
cube root To To To To To To To To To
7 26 63 124 215 342 511 728 1000
Nearest cube roots 1 2 3 4 5 6 7 8 9
There are three cases in the Vedic method to find out Cube Roots of Perfect Cubes [9, 17, 18,
1. Cube roots of a number having less than 7-digits
2. Cube roots of a number having more than 7-digits but less than 10-digits
3. Cube roots of a number greater than 7-digits but ending with even numbers.
Make the groups of 3-digits, starting from the right then
a. The number having less than 4 digits will have a 1-digit cube root.
b. The numbers having 4, 5 to 6 digits will have a 2-digits cube root.
c. The numbers having 7, 8 or 9 digits will have a 3-digits cube root.
d. The number having 10, 11 or 12 digits will have a 4-digits cube root.
Illustration for Case I [9, 17, 18]
In Vedic mathematics, Vilokanam (by mere observation) method is used to extract the cube root of a
number having less than 7 digits. It is to be noted that, the Vilokanam is one of the formulae among Vedic
To find the cube root of 21952, making a group of three from the right as 21 952. So, it will have a 2-digits
cube root. From the above facts and table No. 1, the unit√digit of the cube root is 8. From the table No.2,
it determined the left-digit of the cube root is 2. Hence, 3 21952= 28.
Illustration for Case II [9, 17, 18]
The cube root of more than 7-digits and less than 10-digits number will contain 3-digits. Let it be denoted
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Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic . . .
by the unit digit (R), left digit (L) and middle digit (M). L and R can be determined by the Vilokanam
method, whereas M can be determined by the help of digital roots.
Suppose we have to find the cube root of 12977875. Making groups of three from the right side as 12 977
875. As in the case I, R = 5 and L = 2. Subtracting R3 from the given number 12977875, i.e., 12977875
– 125 = 12977750 and eliminating always last zero will be 1297775. The middle digit of the cube root is
obtained by 3R2 M = 75 M. We should be looking for a suitable value of M so that the unit digit of 75M
becomes equal to the unit digit of 1297775. If we obtain more than one value of M, we use the digital root
method and test which value of M is best suited in this case.
In this problem, here is 5 options for M, i.e. 1, 3, 5, 7 or 9 because digital roots of (215)3 = 8, (235)3 =
1, (255)3 = 0, (275)3 = 8, (295)3 = 1 and digital root of the given number 12977875 = 1. Here, digital root
3
of (235)√ = (295)3 = 12977875 = 1. Again, by Vilokanam rule of Vedic method, 235 is the best suited.
Hence, 3 12977875= 235.
Illustration for Case III [9, 17, 18]
The cube root of a number greater than 7 digits but ending with an even number can be obtained by
dividing the number whose cube root has to be extracted by 8 until odd cubs to be obtained and can be
used digital root method to ascertain the cube roots as in case II.
Consider the number 10-digits even number 2674043072 such that which is a perfect cube. We need to
divide the given number by 8 until the odd cube to be obtained.
8 2674043072
8 334255384
41781923
√
3
Here, the
√ cube root of 41781923 can be obtained as in case II. i.e. 41781923= 347.
Hence, 3 2674043072 = 2 × 2 × 347 = 1388.
3 Findings
• Except for the scholars Pande and Tirthaji, many classical mathematicians prior and of a much later
period than Bhaskaracarya followed the same method with some minor twist and turns to find the
cube roots of a number.
• Bhaskaracarya takes one-digit at a time whereas Pande and Tirthji take groups of three in their
procedures.
• Bhaskaracarya’s method is direct related to the formula (x + y)3 = x3 + 3x2 y + 3xy 2 + y 3
• Inall procedures, if the number has n-digits then the cube roots will have
n
digits, if n is multiple of 3.
= n3
3 + 1 digits, if n is not multiple of 3.
(It should be careful that the decimal part of the result would be neglected.)
• In all procedures, the first digit of the cube root can be found by observing the first part of the
number (it is either one-digit or two-digit or three-digit numbers).
• Pande’s idea for finding cube roots by the unitary method in each step of operation can be formulated
as follows:
Each number of the required cube roots from the left can be obtained (for each step) by using the
formula
The number formed by (x − a3 ) with the number of next three digits of given dividend
= +1
100{(a + 1)3 − a3 }
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Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K. K. Parajuli
where x and a are varying in each step and hence varying the cube roots. x = The number (taken
from the left side of the given dividend) to be taken at that time. In the above worked out example,
for the first step x = 13, for the second step, x = (13–23 = 5) with the number 824 (which is the next
three digits of the given number from the left side) becomes 5824.
a = digit of the cube root of previous steps.
• In Pande’s method, a3 = maximum cube to be taken from left. It should be noted that x must
be greater than a3 . If there occurred negative, there must readjust by deducting in the value of a
(deducting by an adjustable number from a).
• In Pande’s method, the numbers which are not a perfect cube can be obtained from the cube roots
of a decimal number by introducing three zeros to the remainder obtained in the previous steps and
proceed as above.
• Cube roots of non-exact cube numbers can also be extracted from all three scholar’s methods.
• In the Vedic method, there should be remembered the cube table from 1 to 10, the provided facts
and its left-most pair with the nearest cube roots to determine the required cube roots.
• In the Vedic method, Vilokanam is a unique one, which determines the result of a number having less
than 7 digits quickly.
• As mentioned in this paper, the case I and III are similar except dividing the number by 8 in the
Vedic method.
• On the basis of this paper, the case II is comparatively more difficult to ascertain the exact value of M
because both the sutras Vilokanam and Digital Roots are used simultaneously in the Vedic method.
4 Conclusion
From the pieces of evidence and findings, it can be concluded that Bhaskaracarya’s method for determining
cube roots represents all the methods of classical scholars before and after his period. Nepali mathemati-
cian Gopal Pande has followed all the ideas of Bhaskaracarya in his teaching profession. And this attests
that they are interrelated in some major aspects. But Pande did not agree with the Bhaskaracarya’s verse
concerning find the cube roots by using his unitary method. Pande expressed his techniques innovatively
and differently by breaking the verse of Bhaskaracarya. After the period of Pande, another emerging
mathematician Bharati Krishna Tirthaji known as the father of Vedic mathematics in India developed
an independent and constructive method. His idea of determining cube roots is less influenced by the
previously developed methods. To study the importance and effectiveness of the methods it is better to
compare it with conventional methods. The real beauty and effectiveness of these methods cannot be
fully appreciated without actually practicing the system. Although in comparison among the methods, the
Vedic method is interesting and fast, people must pay attention to some facts and basic cubing table. In a
modern mathematical system, people require a calculator for their numerical calculations. The practice of
Vedic Mathematics may be very fruitful for the teaching-learning process of basic mathematical operations
without using a calculator because Various investigation shows that calculator dependency deteriorates the
logical thinking of the learner.
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