Basic Operations on Vedic Mathematics: A Study on Special Parts

Contributed by:
Harshdeep Singh
This PDF Contains :
1. Introduction,
2. Basic Operations on Vedic Mathematics,
2.1 Addition,
2.2. Subtraction,
2.3 Multiplication,
2.3.1 Antyayotdashakepi,
2.3.2 Vamanlyayoh Dashakepi,
2.3.3. Nikhilam Navatascharam Dasatah,
2.3.4. Ekanyunena Purvena,
2.4. Division,
3. Findings,
4. Conclusion,
5. Suggestions and Implications,
1. Nepal Journal of Mathematical Sciences (NJMS), Vol.1 ,2020 (October): 71-76
Basic Operations on Vedic Mathematics: A Study
on Special Parts
Krishna Kanta Parajuli
Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University
Email: [email protected]
Abstract: Vedic Mathematics was rediscovered and reconstructed by Sri Bharati Krishna Tirthaji from
ancient Sanskrit texts Veda early last century between 1911 – 1918 is popularly known today is Vedic
Mathematics. It is an extremely refined, independent and efficient mathematical system based on his 16
formulae and some sub-formulae with simple rules and principles.
The main purpose of this paper is to communicate a new approach to Mathematics, offering simple, direct,
one-line, mental solutions to mathematical problems. In the way of basic mathematical operations like
addition, subtraction, multiplication and division can be done in simple ways, and results are obtained
quickly and can be checked in a minute by using the Vedic techniques. In this system, for any problem, there
is always one general technique and also some special pattern problems. This paper especially concentrates
only on the specific pattern of elementary operation of Vedic Mathematics.
Keywords: Vedic Mathematics, Ekadhikena Purvena, Nikhilam Navatascharam Dasatah,
Urdhvatiryagbhyam, Antyayordashkepi.
1. Introduction
The most common meaning of Veda is knowledge [3]. It is considered as the oldest layer of Sanskrit
literature and the oldest scriptures of Hinduism. The Vedas are considered divine in origin and are assumed
to be direct revelations from God [7]. There are four Vedas: Rigveda, Yajurveda, Samaveda and
Atharvaveda [2]. The Vedas are ancient writings whose date is disputed but which date from at least several
centuries of B.C. The content of the Vedas was known long before writing was invented and was freely
available to everyone. It was passed on by word of mouth. The writings called the Vedas to consist of a huge
number of documents (there are said to be millions of such documents, many of which have not yet been
translated) and these have recently been shown to be highly structured, both within themselves and
concerning each other [2].
Bharati Krishna Tirthaji spent eight years between 1911 – 1918 at the Shringari forest near Shringari Moth
for the practice of Brahma-Sadhana to study the advanced Vedanta Philosophy [10]. According to him, the
rediscovery and reconstruction of Vedic Mathematics were one of the outputs of his devotion from stray
references within the appendix portions of the Atharvaveda. Vedic Mathematics is a system of reasoning
and mathematical working based on 16 formulas and 13 sub-formulas with simple rules and principles.
Vedic mathematical techniques are also based on the ancient mathematical system as well as the modern
system. Each formula provides a principle of mental working applicable to many diverse areas of
Mathematics [10].
The most significant quality of Vedic Mathematics is its consistency. It sharpens the mind, improve memory
power with concentration, speed up mathematical calculations, minimize careless mistakes and encourage
innovations [12]. The beautiful coherence between arithmetic and algebra is visible in the Vedic system. The
real beauty and effectiveness of Vedic Mathematics cannot be fully appreciated without actually practicing
the system [14]. It will be benefitted from these wonderful, logical, and systematic and faster methods of
2. Krishna Kanta Parajuli / Basic Operations on Vedic Mathematics: A Study on Special Parts
solving the most complex sums. It helps the students to remember that many big digit calculations can be
done much faster by Vedic methods than the calculator [5].
After going through the content presented in this paper, serious mathematical issues, higher-level
mathematical problems are not taken up in this paper, even though many aspects like four fundamental
operations: addition, subtraction, multiplication and division are operated with. In Vedic Mathematics, there
are two types of techniques: Specific and general. Those techniques which are fast and effective but can be
applied only to a particular combination of numbers are called specific and those which have a much wider
scope of application than specific as they deal with a wider range of numbers are general. Mathematical
calculations can be done much faster by Vedic methods than by calculators when we use the specific
methods. So, this article is concentrated only on the tiny glimpse of elementary operations on mathematics
by specific techniques of Vedic Mathematics.
2. Basic Operations on Vedic Mathematics
In Sanskrit, the terms sutra means ‘Thread of Knowledge’. Vedic Mathematics consists of 16 sutras and a
similar number of sub-sutras. Only some of these formulae are used in the basic operation of arithmetic. The
meaning of these used sutras in this article is tabulated below [6], [9], [10].
Vedic Sutras (Formulae) Meanings
Ekadhikeana Purvena (Psflws]g k"j]{0f) By one more than the previous one
Nikhilam Navatascaramam dasatah All from nine and last from ten
(lglvn+ gjtZr/d+ bztM)
Urdhvatiryagbhyam (pWj{lto{uEofd,) Vertically and crosswise
Sankalana-Vyavakalanabhyam (;+sngAofjsngfEofd\) By addition and subtraction
Puranapuranabhyam (k"/0ff k"/0ffEofd\) By completion and non-completion
Paravartya Yojaayet (k/fjt{ of]ho]t) Transpose and apply
Ekanyunena Purvena (PsGo"g]g k"j]{0f) By one less than the previous one
Antyayordashkepi (cGToof]b{zs] ˜lk ) The total of the last digit is ten and the previous
part is the same
Antyayoshatakepi (cGToof]zts]˜lk) The total of the last digit is a hundred and the
previous part is the same
Vamanlyayoh Dashakepi (jfdgNofof]x bzs˜lk) The total of the last digit is ten and the previous
part is the same
Vamanlyayoh Dashake Gunijah Api (jfdgNofof]x˜zs The total of left two digits is a multiple of ten
Mu'l0fhM˜lk) and the unit digit is the same
2.1 Addition
The formula used for addition in Vedic Mathematics is Puranapuranabhyam, Sankalana-
Vyavakalanabhyam and Ekadhikeana Purvena. The whole procedure of adding can be summarized in the
following steps [8], [9]:
Add the digits column-wise and when the running total becomes greater than 10, put a dot or tick on that
number. Move ahead with the excess of 10 and add it to the next digit of the column. Lastly, count the
number of dots or ticks and add it to the next column.
Examples: 9' 8' 7 4 2 0 0
4 6' 6' 0 0 9' 8'
8' 4 7' 8' 7' 6' 5'
+248 +5 7 8 9
2548 18 8 52
3. Nepal Journal of Mathematical Sciences (NJMS), Vol.1 ,2020 (October): 71-76
2.2. Subtraction
For the subtraction process, we use the Vedic formula Nikhilam Navatascaramam dasatah. The meaning of
this formula is 'All from nine and the last from ten'. This method works faster when subtraction is done from
a multiple of 10 i.e. 10, 100, 1000, 10000, …. .While calculating is adopted by the conventional method,
several carry-overs are needed, which wastes time and confusion about accuracy remains, the Vedic method
helps us in this regard and saves our precious time [4], [13]. The concepts can be illustrated by taking
examples as following:
 Start moving from right to left. Replace every zero from the left with a 9 and the last zero with a 10.
The extreme left digit before zero will get reduced by 1 [8], [9].
For example, for subtracting 5472 from 400000
400000 399910
–5472 will become –5472
 When the digit at minuend (upper digit) > Subtrahend digit (lower digit), normal subtraction is done.
 In case the upper digit < lower digit, we take the complement of the difference (i.e. complement of 0 is
10. Complement of 1 is 9, the complement of 2 is 8 and so on). The complement of the last digit is
taken from 10 and the complements of the rest of the digits are taken from 9.
 When we arrive at a stage where there is no need to take the complement, subtract 1 extra from that column.
For example, to subtract, 89543 – 40597, we write
8 9 5 4 3  from 10
–4 0 5 9 7
4 8 9 4 6
2.3 Multiplication
For multiplication, we can use eight Vedic formulae Antyayordashkepi,, Nikhilam Navatascharam Dasatah,
Anurupyena, Ekanyunena Purvena, Antyayoshatakepi, Vamanlyayoh Dashakepi, Vamanlyayoh Dashake
Gunijah Api, Urdhvatiryagbhyam [9] [10].
Except for the last formula Urdhvatiryagbhyam, all are the specific formula of multiplication, which has
limited application. Many special formulae help us to find the answer to a special type of multiplication even
in seconds and the Urdhvatiryagbhyam method helps us to encounter all types of multiplication. The special
types of formulae are:
2.3.1 Antyayotdashakepi
This formula has limited application and is valid as long as the sum of the unit digit at multiplicand and
multiplier is 10 and the remaining digits are the same. The final product will consist of two parts LHS and
RHS [10], [12].
Multiply the unit digits and write it in the RHS part. In the LHS part, write the product of (Remaining digit
at Ten's/hundred place)  (Remaining digit at Ten's/hundred place +1).
For example, to multiply 75 by 75, RHS = 5  5 = 25 and LHS = 7  (7 + 1) = 72 (by using the
formula Ekadhikena Purvena). So, result = 7225.
2.3.2 Vamanlyayoh Dashakepi
This formula is applicable when the sum of digits placed at the ten's place in the multiplicand and multiplier
is 10 and the unit digit of both multiplicand and multiplier is the same. For method, see, [8], [1].
The answer to such questions consists of two parts.
4. Krishna Kanta Parajuli / Basic Operations on Vedic Mathematics: A Study on Special Parts
LHS = Product of two left digits whose sum is 10 + unit digit; and RHS = Square of the unit digit.
For example, to multiply 98 by 18, LHS = 9  1 + 8 = 17 and RHS = 8  8 = 64. The result = 1764.
2.3.3. Nikhilam Navatascharam Dasatah
This formula works better when both the multiplicand and multiplier are very close to the base. The base
should be in the form of 10n, where n is a natural number. The ideas can be illustrated in stepwise as below [10].
 Write the two numbers one below the other and write the deviations of the two numbers from the base.
 There are two parts
(a) the left-hand part will be obtained by cross operation of two numbers written diagonally.
(b) the right side of the answer will be obtained by multiplying the deviations.
 The number of digits in the right-hand part will be in accordance with the number of zeros in the base number.
Under this formula, there are many cases
(i) When both the numbers are above the base:
For examples, the multiplication of 104 by 103 and 16 by 12 are expressed as:
(where the (where the base
104 + 4 base is 16 + 6 is 10)
103 + 3 100) 12 + 2  192
107 / 12 18 / 12
So, result = 10712 So, result = 192
(ii) When both the numbers are (iii) When one number is above the base and
below the base another is less than the base.
For example, to multiply 97 by 98, To multiply 15 by 9, it can be expressed as:
it is expressed as
15 + 5
97 – 3 9 – 1
98 – 2
14 / –5
95 / 06 14/10 – 5  13/5 (where the base is 10)
(where the base is 100) So, result = 135. It is noted that when there is a (–)
So, result = 9506 sign at the right-hand side, we use the Nikhilam
formula i.e. subtracting the right-hand digit (–5)
from 10 and the left-hand part will get diminished by 1.
2.3.4. Ekanyunena Purvena [9], [10]
The meaning of the Formula is "By one less than the previous one". This formula is used when the
multiplier is 9 or 99 or 999 or 9999 etc. The method is divided into two cases
(i) When the number of digits in the multiplicand is equal or less to the number of nines:
The method is as: Subtract 1 from the multiplicand and write the result in LHS and subtract the
multiplicand by applying the Nikhilam formula and write the result in RHS.
For example, to multiply 3785 by 999999, we have
LHS = 3785 – 1 = 3784, RHS = 999999 – 3784 = 996215, Result = 3784996215.
5. Nepal Journal of Mathematical Sciences (NJMS), Vol.1 ,2020 (October): 71-76
(ii) When the number of digits in the multiplicand is higher than the number of nines:
It is a little different from (i). To get the result, we have to
 Add as many zero as the numbers of nines to the multiplicand.
 Subtract the original multiplicand from the figure obtained in the 1st step.
For example, to multiply 23758 by 999, there are 3 nines, so for this, subtracting original multiplicand:
23758000 – 23758 = 23734242.
In the above discussion of case (ii), the formula Ekanyunena Purvena is not seen in use, but can also be used.
2.4. Division
In division operation, we shall deal with three Vedic formulae: Nikhilam, Paravartya Yojaayet and
Urdhvatiryagbhyam [10]. Here, Nikhilam and Paravartya Yojaayet are specific rules whereas
Urdhvatiryagbhyam is a general rule which is also known as the Dhwajanka method of division and is based
on the long-established Vedic process of mathematical calculations [10], [12].
In the Nikhilam formula, it has limited application and is useful when every digit of the divisor is greater
than 5. The best part of this formula is that there is no subtraction process to be carried out at all [1], [9].
Paravartya Yojaayet formula is Transpose and Apply, which is slightly different from the Nikhilam
formula. Paravartya Yojaayet formula works effectively when the first digit of the divisor is 1. In the
Paravartya formula, the complement obtained from the Nikhilam formula will be revised by changing the
sign separately. For example, if the divisor is 86, the nearest base = 100 and complement = 100 – 86 = 14
and revised complement = –1 – 4.
For the Nikhilam method, the working rules are [8], [9], [10].
 Take a base (in the power of 10) nearest to the divisor and write its complement below the original divisor.
 Separate the extreme right digit of the divided by drawing a slash equal to the number of digits in the
divisor. This block is known as the remainder block and the left block is known as the quotient block.
 The number of digits to be placed in the remainder column should be equal to the number of zeros in the base.
 Carry down the first digit of the divisor, which will be the first digit of the quotient, multiply this
quotient by the complement and place it in the dividend column; next to the first digit of the dividend.
 Write mechanically the sum of the digits of the 2nd column to get the 2nd digit of the quotient.
 Repeat the process until we get a number in the remainder column. If the remainder is greater than the
divisor, continue the same process in the remainder block until the digit in the remainder column is
less than that of the original divisor.
For example, to divide 1221340 by 8987,
Divisor Column Quotient Column Remainder Column
8987 1 2 2 1 3 4 0
Complement = 1013 1 0 1 3
3 0 3 9
5 0 6 5
1 3 5 8 0 9 5
Here, quotient = 135 and remainder = 8095.
3. Findings
The documents for the special parts of basic operation on Vedic Mathematics may not be sufficient for the
reader. But the presented documents and its representative examples are clear enough to obtain some
6. Krishna Kanta Parajuli / Basic Operations on Vedic Mathematics: A Study on Special Parts
findings with its techniques.There is a contrast between Vedic and Conventional Methods for calculating
basic operations. Vedic Mathematics itself is the easiest, enjoyable and one-line mental form of
mathematics and some of the calculations are faster than the calculator. Modern methods have just one way
of doing, say, division and this is so cumbrous and tedious that the students are now encouraged to use a
calculating device whereas the Vedic method can be done without devices. While calculating is adopted by
the conventional method, several carry-overs in subtractions, large multiplication tables in multiplications
and many hits and trial methods are needed, which wastes time and confusion about accuracy remains, the
Vedic method helps us in this regard and saves our precious time.
4. Conclusion
Vedic Mathematics is considered as mental Mathematics. It develops accuracy, exactness and precision. The
real beauty and effectiveness of Vedic Mathematics cannot be fully appreciated without actually practicing
the system. Hence the methods discussed, and organization of the content of the paper shows the basic
operation of Vedic Mathematics is an extremely refined and efficient mathematical system. The technique
involved in basic operations of Vedic Mathematics is highly efficient, and certainly needs an explanatory
approach for further development. This article hence is an explanatory approach for the basic mathematical
operations using Vedic Mathematics; its special parts. So, a stepwise, procedural and algorithmic framework
of Vedic Mathematics in basic operations can be drawn from this article.
5. Suggestions and Implications
With the knowledge of Vedic Mathematics at primary level classes, mathematics would become a favorite
subject of all, as they would be able to perform calculations accurately with speedily. To realize this
objective, the assimilation of Vedic Mathematics should be given prime importance. The methods and
principles can be integrated into an existing school curriculum. The incorporation of Vedic Mathematics into
the present issue-based approach makes the system both conceptual and calculation based. Both Vedic
Mathematics and Conventional Mathematics give the same result on calculation. Students should be trained
on both methods and they should be given to choosing between the methods which they find convenient.
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