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Extended Essay: Mathematics
An Investigation of Vedic Multiplication
By: Zwelakhe Bhengu
American International School of Johannesburg
Candidate: 000756-008
Advisor: Johan Kriel
Word Count: 2625
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Table of Contents
Heading Page
Abstract
Introduction
Vertically & Crosswise
Simple Multiplication of Numbers Above 10
By One More than the Number Before
Multiplication of Number Just Above 100
Multiplying by 11
Long Multiplication Sutra
When the Samuccaya is the Same, that
Samuccaya is Zero
Conclusion
Acknowledgments, Bibliography and References
2
3
3
4
6
7
8
9
10
11
12
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Abstract
Vedic Mathematics has found its recent popularity amongst a number of maths students and
teachers as it has provided an enticing array of methods for solving challenging problems
easily using patterned algorithms. Although some of its findings being of a somewhat
elementary standard of mathematics, the patterns used to solve a series of mathematical
problems can prove to be confusing to even the most qualified maths educationalists.
However, after an adequate introduction to the sixteen aphorisms provided by the roots of
Vedic Mathematics, students are enticed to enjoy maths whilst learning shortcuts and patterns
that aid towards solving the most seemingly difficult and strenuous maths problems.
The investigation of Vedic multiplication has led to contesting of the question: to what extent
can Vedic multiplication shorten and ease common: easy, intermediate and complex
multiplication demanding mathematical problems for maths students using numerical
patterns. To fulfil the requirements of the investigation, the basic multiplication methods
were obtained from Kevin O’Connor’s Vedic Mathematics EBook. The more complex
multiplication problems such as that of the long multiplication sutra and the deriving of
algorithms for simplifying exponential functions were extracted from the referenced sources
as well as the Vedic Mathematics Hinduism website. Thus it is conclusive to state that the
Vedic Mathematics aphorisms can shorten and ease a variety of mathematical problems,
although some do require a bit of innovation to work around getting the correct solution for a
problem that does not follow the prerequisites for successful solutions.
Word Count: 244
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1. Introduction
Vedic Mathematics, originally being of Indian roots, is a system of mathematics often
considered to be a set of strategies or methods of solving various algebraic and other
mathematical problems using patterns and creative numeric connections. Although its main
developers consider the entire discipline of mathematics to be composed of the sixteen Vedic
sutras, most current mathematicians merely consider it as the aforementioned collection of
methods to solve problems. The original notes containing the sixteen sutras
(aphorisms/methods) are believed to have been “rediscovered from the Vedas between 1911
and 1918 by Sri Bharati Krsna Tirthaji (1884-1960)” (Hinduism). Nowadays, Vedic methods
of calculation are used to teach school pupils different and more enticing methods of solving
mathematical problems, with the hopes of making it fun and interesting vis-à-vis the “modern
system”.
“Vedic Mathematics manifests the coherent and unified structure of mathematics, and the
methods are complementary, direct and easy” (Hinduism) . The majority of all the sutras that
involve multiplication have an interdependence on other mathematical operations such as
addition, division etc. The sutras involving the mathematical operation of multiplication, in
particular, have introduced very mind-boggling patterns in solving most multiplication
problems even up to the extent of converting numerical units from one form to another and
solving complex exponential problems. An exploration of the main multiplication sutras :
“Vertically and Crosswise; By the Deficiency; All The Multipliers; The Product and Sum and
The Ultimate and Twice the Penultimate and If the Samuccaya is the Same that Samuccaya is
Zero” which extensively engage the usage of multiplication, even when it is not the sole
mathematical operation used, will lead to the provision and contesting of problems ranging
from intermediate algebraic multiplication to seemingly complex and tedious exponential
problems. Throughout the discussion and mathematical presentation of the investigation, the
question of: To what extent can Vedic multiplication shorten and ease common: easy,
intermediate and complex multiplication demanding mathematical problems for maths
students using numerical patterns will be explored.
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2.1 Vertically and Crosswise
Simple Multiplication of Numbers Below 10
One of the most prominent sutras the “Vertically and Crosswise” has a variety of uses in
multiplication involved problems. Nowadays it has been used in many SAT (standardised
testing for college entrance) prep courses as well as elementary maths classes as a shortcut to
standard multiplication. The basic algorithm for the application of the Vertically and
Crosswise sutra is that the following are equal to each other:
(10a + b)(10c + d) = 100ac + 10(ad +bc) + bd (O’Connor)
Problem: 8 x 9
For most students, figuring out the above problem without the use of a calculator would
involve a memorisation of times table whereas the Vedic Sutra “Vertically and Crosswise”
presents a more interesting and easy method to solve any multiplication problem involving
numbers that are not greater than 10.
8 2
9 1
= 72
As presented above, the two sub 10 numbers involved in the problem are written one below
the other and then the difference between the number on the left and 10 is written to the right
of the respective number,10 – 8 = 2 and 10 – 9= 1. Thereafter, the student can choose any of
the two methods represented by the arrows in which they can choose to obtain the first digit
of the answer by subtracting 2 from 9 or 1 from 8 which yield to the answer of 7. Then to find
the second digit of the number, the student vertically multiplies the two numbers on the right
which produce the second digit of the answer 2, therefore producing an answer of 72 for the
initial problem.
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2.2 Simple Multiplication of Numbers Above 10
The Vertically and Crosswise sutra goes as far as to provide methods for solving numbers
greater than 10, 100, 1000, ad infinitum. For the multiplication of numbers above 10 and
close to 100 a different method under the same sutra is used to produce the correct answer.
Problem: 97 x 88
9 7 3
8 8 12
85 36
100 – 97
= 3
100 – 88
=12
Similar to the standard method of multiplication, the sutra states to place the numbers to be
multiplied one on top of the other as shown in the above figure. To begin the multiplication
both the numbers are subtracted from 100 and the resulting differences written next to their
respective numbers as shown in red. The two differences are then multiplied together and the
product written below as shown in red. Thereafter you subtract either 97 or 88 by the
opposing difference (red number) and write the number on the left of the previously written
number and then the answer is produced.
Furthermore, by virtue of the same sutra, another method is used to multiply numbers that
have the same first digit and are greater than 10, using a rather elegant pattern. However this
method will only apply if the first digits are the same and the last digits do not yield to a
product greater than 10 as shown below.
Problem: 21 x 23
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2 1
2 3
483
2 x 2 = 4
(3 x 2) + (2 x 1) = 8
3 x 1 = 3
The tens digits are multiplied by each other and the resulting product of 4 serves as the first
digit of the answer written right below. The middle digit is obtained by adding the crosswise
multiplications of the problem as shown in brackets above and the answer written right next
to the 4. To finalise the answer and figure out the last digit of the solution the units digits of
the respective numbers are multiplied producing an answer of 3 and a final product of 483 for
the entire problem.
The letdown about this particular sutra is that some problems that fall under the listed
requisites for a correct solution to be found are not correctly solved. For example the product
of 24 and 21 cannot be solved using this sutra as an incorrect solution is found using all the
correct steps listed as shown below:
2 4
2 1
4104
2x 2 = 4
(2 x 1) + (4 x 2) = 10
4 x 1
However if place holders are used in the calculation, which is not mentioned in the steps for
solving the problem, integrating the usage of “normal long multiplication” can lead to the
correct answer as shown below
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2 4
2 1
504
2x 2 = 4
(4 x 2) + (2 x 1) = 10
If the ten is carried over to the 4 it produces a value of 50 which serves as the first two digits
of the answer, the last digit is found using the same aforementioned method of multiplying
the numbers in the units place therefore producing a correct final solution of 504. The quick
solving of such problems can be particularly useful when solving very complex long division
problems that require the multiplication of very large numbers or even when a student is
faced with the task of having to multiply very large numbers in an assessment or exam where
calculators are not allowed.
3.1 By One More than the Number Before
Squares of Numbers Ending with 5
Vedic Multiplication also encompasses squaring certain numbers with certain digit endings.
The squaring of any number that ends with 25 involves the employment of the “By One More
than the Number before Sutra” used for various complex multiplication problems. The
products solved when following the sutra is based on the fact that the number being squared
has a particular deficit or surplus from the base of 100. Mathematically, this algorithm is
represented by naming the number to be squared as a, and the deficit or surplus as b
therefore:
a2= (a + b) (a - b) + b2 (O’Connor)
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To square any multiplication problem ending with the number five, the following method is
used:
Problem: 952 = 9025
52= 25
9 + 1 = 10
9 x 10 = 90
As represented by the sutra above, the first step to towards solving the above problem is
squaring the unit value as shown above, 5 squared as 25 which represents the last two digits
of the solution. To obtain the first two digits, the first digit of the initial problem is multiplied
by one more than itself, meaning that 9 is multiplied with 10 to get a product of 90, which
then becomes the first two digits of the solution, consequently, leading to a final solution of
the initial problem of 952 being 9025. Without a calculator, finding the squares of such large
numbers will prove to be impossible without the employment of a pattern like the used sub
sutra even to the most gifted maths students.
3.2. Multiplication of Numbers Just Above 100
The brilliance of the Vedic Sutras of multiplication is that they employ a pattern in the
solving of various mathematical problems which presents the opportunity for seemingly
challenging problems to be easily solved mentally using Vedic arithmetic. For multiplying
numbers just above 100, the following sub sutra of the “By One More than the One Before”
is used.
Problem: 107 x 104 = 11128
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107 + 4 = 111 104 + 7 = 111
7 x 4 = 28
To begin solving the problem, the sutra states that any of the two numbers can be added with
the last digit of its multiplicative partner as shown twice above with a sum of 111; this
represents the first three digits of the solution. The last digits are found by multiplying the
last digits of both numbers together as shown above 7 times 4 is 28 which then represents the
last digits of the solution, giving a final product of 11128.
3.3. Multiplying by 11
Most maths students have already memorised that whenever multiplying a single digit by 11,
simply changing the value of the numbers of 11 in the same form will produce the solution.
For example:
11 x 8 = 88
11 x 5 = 55
Therefore 11 x a = aa
However when multiplying 11 by a number with more than one digit, the above algorithm no
longer works. Fortunately, when multiplying 11 by any two digit number a Vedic pattern can
be used to easily find the solution as shown below.
Problem: 81 x 11 = 891
8 + 1 = 9
Problem: 15 x 11 = 165
5 + 1 = 6
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Thus the basic algorithm that represents the multiplication of 11 by any two digit number is
as follows:
a1a2 x 11 = a1 [a1 + a2]a2
Where a1 represents the first digit of the two digit number being multiplied by 11 and a2
representing the second digit. Thus to find the solution as solved above the digits of the
number are separated and written down and then the sum of those same digit is written in
between ( shown in red) the separated digits which then produces the correct solution.
4.1. Long Multiplication Sutra
It is quite often that many complicated maths problems in various mathematical disciplines
such as calculus and calculation of rates where large numbers have to be multiplied in order
to solve the solution of a plethora of problems. Using normal long multiplication it can be
quite complication to find the product of a problem such as 23958233 times 5830. However
with the long division algorithm to quickly lead to the solution, it can become very easy to
solve such a long and strenuous problem in a matter of seconds.
Problem: 23958233 x 5830
23958233
5830 ×
------------------------------------------
00000000 (= 23,958,233 × 0)
71874699 (= 23,958,233 × 30)
191665864 (= 23,958,233 × 800)
119791165 (= 23,958,233 × 5,000)
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----------------------------------------------
139676498390 (= 139,676,498,390) (Dutta)
The above algorithm can provide the solution for any pair of numbers, no matter the number
of digits, provided that the problem has a definite solution. As shown above the top number
multiplied by the digit in the units place then tens, hundreds etc.
4.2. When the Samuccaya is the same, that Samuccaya is zero
After having memorised and practiced the above sutras for completing simple multiplication
that can even be done mentally as well as some of the complex multiplication such as long
multiplication that seems impossible without a calculator, the sutras of Vedic maths can be
applied to even more classroom related problems where multiplication plays a key role
towards solving the problem. When being introduced to IB material functions in the Standard
Level and Higher Level courses, using the samuccaya can help students visually solve
problems that seem very complex and strenuous. Furthermore, the usage of the samuccaya
can be applied in differential calculus, when a lot of multiplication is required to find the
derivative of a specific function. The samuccaya can have several meanings depending on
the nature of the mathematical problem at hand. For example, the samuccaya can represent a
common term in a set of functions such as the one below:
12x + 3x = 4x + 5x
x = 0
In this instance, as x occurs in both functions then it can be deduced that x = 0 is the solution
as the samuccaya is the same on both sides without having to employ the “normal” method of
solving the problem as shown below which takes more time and allows for more errors to be
committed.
12x + 3x = 4x +5x
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15x = 9x
6x = 0
x =0
For more complex multiplication problems involving multiplication in exponents such as
having to “FOIL” (a form of multiplying several terms of a problem) the Vedic Samuccaya
can provide a quick an easy solution as shown below.
(x + 3)3 = x + 1
(x + 5)3 x + 7
2x + 8 = 0
2x = -8
x = -4
If the samuccaya is the same, then the other samuccaya is zero is perfectly elucidated in the
solution of the problem above. Instead of having to cross multiply and then try to solve for x
by multiplying out the terms that are raised to the power of three, the algorithm of a N1 + D1
= N2 + D2 which shows that the samuccaya is the same as the terms N1 + D1, N2 + D2 are
equal to each other. Adding the x terms with their coefficients on the left produces the term
2x whilst ignoring the x terms on the right hand side because the samuccaya is zero, adding 1
and 7 produces 8. Thereafter simple linear equation methods of solving for x are used in order
to find the solution of x as -4.
5. Conclusion
Throughout the investigation of the application of Vedic Mathematics Sutras on the
investigation of the applicability and time efficiency of Vedic multiplication methods, it has
become apparent that the presented sutras can actually provide easier methods of solving a
vast array of problems. When dealing with simple multiplication that can happen anywhere
ranging from manipulating functions, solving exponential equations to solving vectors and
differentiating functions in calculus, the sutras: “Vertically and Crosswise; By One More than
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the One Before; and their respective sub-sutras can provide shortened methods of solving
multiplication problems that might be intended to trick or use the time allotted in a particular
assignment. Similarly, the sutras: “Long Multiplication; if the Samuccaya is the same, that
Samuccaya is Zero” and their respective sutras can provide shortened, patterned and easier
methods of solving very complex multiplication problems such as multiplying a 10 digit
number or simplifying an equation raised to a high nth power. Although controversially the
16 Vedic Mathematics Sutras and their respective sub-sutras are believed to be the main
pillars of current mathematics, through the investigation of the multiplication relate sutras it
has come to conclusion that the patterns and algorithms provided by Vedic Mathematics can
provide correct solutions to a range of problems in difficulty and nature with minimal
chances for mistakes as well essential time efficiency especially for IB students who
constantly taking mathematical assignments where time management is key to success.
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Acknowledgements
The general algorithms for the used sutras in the investigation were all adapted from the
referenced authors below. I have constructed my own problems related to the sutras
discussed; however the general derivations of the algorithms are all referenced work from
Kevin O’Connor’s Vedic Maths EBook.
Bibliography
Hinduism, B. Dutta, R. P. Kulkarni, and G. Kumari. "Vedic Mathematics." Vedic
Mathematics. N.p., n.d. Web. 10 Jan. 2013.
O'Connor, Kevin. "Vedic Multiplication." Vedic Maths Ebook. N.p.: E, n.d. N. pag. Digital.
References
Dutta, . (2002). Mathematics in Ancient India. Seattle, Wash.: Resonance Media.
Glover, J. (2002, Vedic Mathematics Today (Only a Matter of 16 Sutras). Education Times.
Williams, K. (2000). The sūtras of Vedic mathematics. Baroda: Oriental Institute.
