Definitions

# Swami Bharati Krishna Tirtha's Vedic mathematics

''For the actual mathematics of the Vedic period, see the articles on Sulba Sūtras and Indian mathematics.

Swami Bharati Krishna Tirtha's Vedic mathematics is a system of mathematics consisting of a list of 16 basic sūtras, or aphorisms. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century (Trivedi, 1965).

Tirthaji claims that he found the sūtras after years of studying the Vedas, a set of sacred ancient Hindu texts (Agrawala, 1992). However, labeling the mathematics he presented as ‘Vedic’ has provoked great controversy amongst Indian mathematicians who question both the Vedic origin of the mathematics, and whether the sūtras can fulfill the claim of encompassing all mathematics. Nonetheless, the calculation strategies provided by Vedic mathematics are creative and useful, and can be applied in a number of ways to calculation methods in arithmetic and algebra, most notably within the education system.

Vedic math and the Trachtenberg system are very similar to each other. Many of the arithmetic computational strategies are based on the same concepts.

## Tirthaji and the discovery of Vedic mathematics

Vedic mathematics was presented by Jagadguru Swami Sri Bharati Krishna Tirthaji, who is described as having the “rare combination of the probing insight and revealing intuition of a Yogi with the analytical acumen and synthetic talent of a mathematician” (Pratyagatmananda, 1965). Born in India in 1884, Tirthaji was an exceptional scholar; by age twenty he had studied at a number of colleges and universities throughout the country, been awarded the title of ‘Saraswati’ by the Madras Sanskrit Association for his remarkable proficiency in Sanskrit, and had completed seven masters degrees, including Sanskrit, Philosophy, English, Mathematics, History and Science, with the American College of Sciences (Trivedi, 1965).

Around 1911, Tirthaji resolved to study several sections of the Atharva-veda that had been dismissed by Orientalists, Indologists and antiquarian scholars as nonsensical (Tirthaji, 1992). He was part of a shrinking group of Indian scholars who believed that the Vedas represented an “inexhaustible mine of profound wisdom” both spiritual and secular (Pratyagatmananda, 1965). Tirthaji claimed that there were sections of the Atharva-veda labeled “ganita sūtras” or “mathematical formulae” that mysteriously made no obvious reference to mathematics (Kansara, 2000). Tirthaji explains that he was determined to understand the “ganita sūtra” references, and began studying ancient lexicons and lexicography in more detail (Kansara, 2000). With this resolve, Tirthaji went to Sringeri, Karnataka, where he began years of solitary study and meditation (Trivedi, 1965).

Eight years later, Tirthaji emerged claiming to have deciphered 16 fundamental mathematical sūtras in the Vedas, which today have become the foundation of Vedic mathematics (Tirthaji, 1992). According to Tirthaji, the sūtras cover every branch of mathematics, from arithmetic to spherical conics, and that “there is no mathematics beyond their jurisdiction” (Tirthaji, 1992).

After discovering the sūtras, Tirthaji traveled around India presenting Vedic mathematics, and even lectured in the United States and England in 1958 (Trivedi, 1965). In addition to lecturing, Tirthaji also wrote sixteen volumes, one for each basic sūtra, explaining their applications (Trivedi, 1965). Unfortunately, before they were published, the manuscripts were lost irretrievably (Kansara, 2000). Before falling ill and passing away in 1960, Tirthaji was able to rewrite the first of the sixteen volumes he had composed (Trivedi, 1965). This text — simply titled Vedic Mathematics, ISBN 81-208-0164-4 and published in 1965 — has become the basis for all study in the area (Glover, 2002).

## The sūtras (formulas or aphorisms)

Vedic mathematics is based on sixteen sūtras which serve as somewhat cryptic instructions for dealing with different mathematical problems. Below is a list of the sūtras, translated from Sanskrit into English:

• "By one more than the previous one"
• "All from 9 and the last from 10"
• "Vertically and crosswise (multiplications)
• "Transpose and apply
• "Transpose and adjust (the coefficient)
• "If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero
• By the Parāvartya rule
• "If one is in ratio, the other one is zero."
• "By addition and by subtraction.
• By the completion or non-completion (of the square, the cube, the fourth power, etc.)
• Differential calculus
• By the deficiency
• Specific and general
• The remainders by the last digit
• "The ultimate (binomial) and twice the penultimate (binomial) (equals zero),
• "Only the last terms,
• By one less than the one before
• The product of the sum
• All the multipliers

### Subsūtras or corollaries

• "Proportionately
• The remainder remains constant
• "The first by the first and the last by the last"
• For 7 the multiplicand is 143
• By osculation
• Lessen by the deficiency
• "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of the deficiency)".
• "By one more than the previous one
• "Last totaling ten
• The sum of the products
• "By (alternative) elimination and retention (of the highest and lowest powers)
• By mere observation,
• The product of the sum is the sum of the products
• On the flag

Editor's list of 16 Sūtras and 13 Sub-sūtras or corollaries from the Vedas

To check the factorization of polynomial expressions: "The product of the sum of the coefficients (SC) in the factors is equal to the sum of the coefficients in the product." The SC of the product (the expression) = Product of the SC (in the factors). Additionally, this sub-sūtra helps to fill in the gaps when some factors are known.

### Background information on the Vedas

The word “veda” has two basic meanings. The first, a literal translation of the Sanskrit word, is “knowledge” (Veda). The second, and most common meaning of the word, refers to the sacred ancient literature of Hinduism, the Vedas, a collection of hymns, poetry and Hindu ceremonial formulas (Veda). Believed to be one of the oldest human written records, the Vedas date back over 4000 years (Gaskell, 2000). Traditionally, they were passed down orally and adapted from generation to generation by sacred sages called rishis, before eventually emerging written in Vedic, an ancient form of Sanskrit.

The Vedas are divided into four main sections: the Rig-veda, Sama-veda, Yajur-veda and the Atharva-veda, known collectively as the Samhitas (Veda). The first three, the Rig-veda, Sama-veda, and Yajur-veda are basically ritual handbooks that were used by priests during the Vedic period (1500–500 BCE) (Veda). Vedic mathematics is apparently part of the fourth Veda, Atharva-veda, which is distinct from the others in several ways. First, unlike the religious focus of the other Vedas, the Atharva-veda contains hymns, spells and magical incantations for personal and domestic use (Veda). Also, the Atharva-veda, which was written later than the other Vedas, was not always considered authoritative, but only became so after being accepted by the Brahmans, the highest order of Hindu priests (Veda). Collectively, the Vedas do include information about a huge range of subjects, spanning religion, medicine, architecture, astronomy, etc. (Gaskell, 2000).

Although there is controversy about whether the Vedas themselves actually include references to mathematics, the roots of sophisticated mathematics have actually been traced back to the Vedic era. Ancient Indian Vedic civilizations are known for being skilled in geometry, algebra and computational mathematics complex enough to incorporate things like irrational numbers (Dutta, 2002). Furthermore, all ancient Indian mathematics literature is composed completely in verse; there was a tradition of composing terse sūtras, like those of Vedic mathematics, to ensure that information would be preserved even if written records were damaged or lost (Dutta, 2002).

## Applying the sūtras: some of the actual mathematics

It is not difficult to understand and apply the Vedic mathematical strategies, as long as one does not rely on the sūtras alone for mathematical insight. Those studying Vedic mathematics tend to strongly rely on the examples and explanations Tirthaji provides in his book.

### All from nine and the last from ten

When subtracting from a large power of ten with many columns of zeros, it is not necessary write the notation for "borrowing" from the column on the left. One can instead subtract the last (rightmost) digit from 10 and each other digit from 9. For example, when one is subtracting ten thousand minus 4,679, the leftmost three digits of 4,679--4, 6 and 7--are subtracted from 9, and the rightmost nonzero digit--that is, 9--is subtracted from 10, yielding the solution: 5,321. This method is also used when finding the deficit from the next larger power of ten when setting up a multiplication problem using the "cross-subtraction" method.

#### First corollary, when squaring numbers

"Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)

For instance, in computing the square of 9 we go through the following steps:

1. The nearest power of 10 to 9 is 10. Therefore, let us take 10 as our base.
2. Since 9 is 1 less than 10, decrease it still further to 8. This is the left side of our answer.
3. On the right hand side put the square of the deficiency, which is 1². Hence, the square of nine is 81.

Similarly, 8² = 64, 7² = 49.

For numbers above 10, instead of looking at the deficit we look at the surplus. For example:

$11^2 = \left(11+1\right)cdot 10+1^2 = 121.,$
$12^2 = \left(12+2\right)cdot 10+2^2 = 144.,$
$14^2 = \left(14+4\right)cdot 10+4^2 = 18cdot10+16 = 196.,$
$25^2 = \left[\left(25+5\right)cdot 2\right]cdot 10+5^2 = 625.,$
$35^2 = \left[\left(35+5\right)cdot 3\right]cdot 10+5^2 = 40cdot3cdot10+25 = 1225.,$
and so on.
This method of squaring is based on the fact that $a^2 = \left(a+b\right)\left(a-b\right) + b^2$ where a is the number whose square is to be found and b is the deficit (or surplus) from nearest power of 10.

### By one more than the one before

"Ekādhikena Pūrveṇa" is the Sanskrit term for "[by] One more than the previous one". It provides a simple way of calculating values like 1/x9 (e.g: 1/19, 1/29, etc). The sūtra can be used for multiplying as well as dividing algorithms.

Example: let's calculate 1/19. In this case, x = 1 . For the multiplication algorithm (working from right to left), the method is to start by denoting the dividend, 1, as the first (rightmost) digit of the result. Then multiply that digit by 2 (i.e.: x + 1 ), and denote that next digit to its left. If the result of this multiplication was greater than 10, denote (value – 10) and keep the "1" as "carry over" which you'll add to the next digit directly after multiplying.

The preposition "by" means the operations this formula concerns are either multiplication or division. [In case of addition/subtraction preposition "to" or "from" is used.] Thus this formula is used for either multiplication or division. It turns out that it is applicable in both operations.

Note: This sūtra can also be applied to multiplication of numbers with the same first digit and the sum of their last unit digits is 10.

An interesting sub-application of this formula is in computing squares of numbers ending in five. Examples:

35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25
or by the sūtra, multiply "by one more than the previous one."
35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25

The latter portion is multiplied by itself (5 by 5) and the previous portion is square of first digit or first two digit (3×3) or (12×12) and adding the same digit in that figure (3or12) resulting in the answer 1225.

(Proof) This is a simple application of $\left(a+b\right)^2=a^2+2ab+b^2$ when $a=10c$ and $b=5$, i.e.

$\left(10c+5\right)^2=100c^2+100c+25=100c\left(c+1\right)+25.,$

It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Examples:

37 × 33 = (3 × 4),7 × 3 = 12,21
29 × 21 = (2 × 3),9 × 1 = 6,09

This uses $\left(a+b\right)\left(a-b\right)=a^2-b^2$ twice combined with the previous result to produce:

$\left(10c+5+d\right)\left(10c+5-d\right)=\left(10c+5\right)^2-d^2=100c\left(c+1\right)+25-d^2=100c\left(c+1\right)+\left(5+d\right)\left(5-d\right)$.

We illustrate this sūtra by its application to conversion of fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.

#### Method 1: example: using multiplication to calculate 1/19

For 1/19, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating decimal. (If the denominator contains only factors 2 and 5, the result is a purely non-circulating decimal, else it is a mixture of the two: a short non-circulating sequence of digits, followed by an endless repetition.) Each factor of 2 or 5 or 10 in the denominator gives one fixed decimal digit.

So we start with the last digit of the result, being the dividend:

1

Multiply this by "one more", that is, 2 (this is the "key" digit from 'Ekādhikena')

21

Multiplying 2 by 2, followed by multiplying 4 by 2

421 → 8421

Now, multiplying 8 by 2, sixteen

68421
1 ← carry

multiplying 6 by 2 is 12 plus 1 carry gives 13

368421
1 ← carry

Continuing

7368421 → 47368421 → 947368421
1

Now we have 9 digits of the answer. There are a total of 18 digits (= denominator − numerator) in the answer computed by complementing the lower half (with its complement from nine):

052631578
947368421

Thus the result is 1/19 = 0.052631578,947368421 repeating.

```1
21
421
8421
68421 (carry 1) – we got 16, so we keep 6 and carry 1
368421 (carry 1) – we get 6*2 + carry 1 = 13, so we keep 3 and carry onedo this to eighteen digits (19–1. If you picked up 1/29,
you'll have to do it till 28 digits). You'll get the following
1/19 = 052631578947368421
10100111101011000Run this on your favorite calculator and check the result!
```

#### Method 2: example: using division to calculate 1/19

The earlier process can also be done using division instead of multiplication. We start again with 1 (dividend of "1/x9"), dividing by 2 (" x + 1 "). We divide 1 by 2, answer is 0 with remainder 1

result .0

Next 10 divided by 2 is five

.05

Next 5 divided by 2 is 2 with remainder 1

.052

next 12 (remainder,2) divided by 2 is 6

.0526

and so on.

Other fractions can sometimes be converted into the format of "d/x9"; as another example, consider 1/7, this is the same as 7/49 which has 9 as the last digit of the denominator. The previous digit is 4, by one more is 5. So we multiply (or divide) by 5, that is:

…7 → 57 → 857 → 2857 → 42857 → 142857 → .142,857 (stop after 7 − 1 digits) 3 2 4 1 2

### Multiplying by 11

11×35= 385

(1) The five in the ones place of the answer is taken from the five in 35.
(2) The eight in the answer is the sum of 35 (3+5=8).
(3) The three in the hundreds place of the answer is taken from the three in 35.

However, if in step #2 the sum is greater than 9, the sum's left digit is added the first digit of the number multiplied by 11. For example:

11×59= 649

(1) The nine in the ones place of the answer is taken from the nine in 59.
(2) The four in the answer is the right digit in the sum of 59 (5+9=14)
(3) The six in the hundreds place of the answer is taken from the sum of the five in 59 and the digit in the tens place from the sum of 59 (5+9=14) --> (5+1=6)

The steps for multiplying a three-digit number by 11 are as follows:

11×768= 8448

(1) The 8 in the ones place of the answer is taken from the eight in 768.
(2) The 4 in the tens place of the answer is taken from the sum of 8, in the ones place of 768, and 6, in the tens place of 768 (8+6=14). As 14 is greater than 9, the 1 is carried over to step 3.
(3) The 4 in the hundreds place of the answer is taken from the sum of 6, in the tens place of 768, and 7, in the hundreds place of 768, plus the carried 1 from step 2 (6+7+1=14). As 14 is greater than 9, the 1 is carried over to step 4.
(4) The 8 in the thousandths place of the answer is taken from the sum of 7, in the hundreds place of 768, plus the carried 1 from step 3 (7+1=8).

### Vertically and crosswise

This formula applies to all cases of multiplication and is very useful in division of one large number by another large number.

For example, to multiply 23 by 12:

`      2           3`
`      |     ×     |`
`      1           2`
`     2×1 2×2+3×1 3×2`
`      2     7     6`
So 23×12=276.

When any of these calculations exceeds 9 then a carry is required.

This is the equivalent of (10a+b)(10c+d)=100ac+10(ad+bc)+bd.

### Transpose and apply

This formula complements "all from nine and the last from ten", which is useful in divisions by large numbers. This formula is useful in cases where the divisor consists of small digits. This formula can be used to derive the Horner's process of Synthetic Division.

### When the samuccaya is the same, that samuccaya is zero

This formula is useful in solution of several special types of equations that can be solved visually. The word samuccaya has various meanings in different applications. For instance, it may mean a term which occurs as a common factor in all the terms concerned. A simple example is equation "12x + 3x = 4x + 5x". Since "x" occurs as a common factor in all the terms, therefore, x = 0 is a solution. Another meaning may be that samuccaya is a product of independent terms. For instance, in (x + 7) (x + 9) = (x + 3) (x + 21), the samuccaya is 7 × 9 = 3 × 21, therefore, x = 0 is a solution. Another meaning is the sum of the denominators of two fractions having the same numerical numerator, for example: 1/ (2x − 1) + 1/ (3x − 1) = 0 means we may set the denominators equal to zero, 5x – 2 = 0.

Yet another meaning is "combination" or total. This is commonly used. For instance, if the sum of the numerators and the sum of denominators are the same then that sum is zero. Therefore,

$\left\{2x+9 over 2x+7\right\}=\left\{2x+7 over 2x+9\right\}.$

Therefore, 4x + 16 = 0 or x = −4.

This meaning ("total") can also be applied in solving quadratic equations. The total meaning can not only imply sum but also subtraction. For instance when given N1/D1 = N2/D2, if N1 + N2 = D1 + D2 (as shown earlier) then this sum is zero. Mental cross multiplication reveals that the resulting equation is quadratic (the coefficients of x² are different on the two sides). So, if N1D1 = N2D2 then that samuccaya is also zero. This yields the other root of a quadratic equation.

Yet interpretation of "total" is applied in multi-term RHS and LHS. For instance, consider

$\left\{1 over x-7\right\}+\left\{1 over x-9\right\}=\left\{1 over x-6\right\}+\left\{1 over x-10\right\}.$

Here D1 + D2 = D3 + D4 = 2x − 16. Thus x = 8.

There are several other cases where samuccaya can be applied with great versatility. For instance "apparently cubic" or "biquadratic" equations can be easily solved as shown below:

$\left(x-3\right)^3+\left(x-9\right)^3=2\left(x-6\right)^3.$

Note that x − 3 + x − 9 = 2 (x − 6). Therefore (x − 6) = 0 or x = 6.
This would not work for the apparently quadratic $\left(x-3\right)^2+\left(x-9\right)^2=2\left(x-6\right)^2$, which has no real or complex solutions.

Consider

$\left\{\left(x+3\right)^3 over \left(x+5\right)^3\right\}=\left\{x+1 over x+7\right\}.$

Observe: N1 + D1 = N2 + D2 = 2x + 8. Therefore, x = −4.

This formula has been extended further.

### If one is in ratio, the other one is zero

This formula is often used to solve simultaneous linear equations which may involve big numbers. But these equations in special cases can be visually solved because of a certain ratio between the coefficients. Consider the following example:

6x + 7y = 8
19x + 14y = 16

Here the ratio of coefficients of y is same as that of the constant terms. Therefore, the "other" variable is zero, i.e., x = 0. Hence, mentally, the solution of the equations is x = 0 and y = 8/7.

(alternatively:

19x + 14y = 16 is equivalent to:
(19/2)x +7y = 8.

Thus it is obvious that x has to be zero, no ratio needed, just divide by 2!)

Note that it would not work if both had been "in ratio". For then we have the case of coinciding lines with an infinite number of solutions.:

6x + 7y = 8
12x + 14y = 16

This formula is easily applicable to more general cases with any number of variables. For instance

ax + by + cz = a
bx + cy + az = b
cx + ay + bz = c

which yields x = 1, y = 0, z = 0.

A corollary says solving "by addition and by subtraction." It is applicable in case of simultaneous linear equations where the x- and y-coefficients are interchanged. For instance:

45x − 23y = 113
23x − 45y = 91

By addition: 68x − 68 y = 204 → 68 (xy) = 204 → xy = 3.

By subtraction: 22x + 22y = 22 → 22 (x + y) = 22 → x + y = 1.

Again, by addition, we eliminate the y-terms: 2x = 4, so x = 2.

Or, by subtraction, we eliminate the x-terms: – 2y = 2, and so y = – 1.

The solution set is {2,-1}.

## Applications

The most notable application of Vedic mathematics is in education. Vedic mathematical strategies may prove to be a useful resource for teachers and students, who may find elements of it easier and more accessible to teach and learn than conventional mathematics. In particular, these strategies may be an invaluable resource to students that already struggle with mathematics, and could benefit from alternative approaches.

One attempt at incorporating Vedic mathematics into education was made by Mark Gaskell, the head of mathematics at the Maharishi School Lancashire, England (Gaskell, 2000). The school has developed a Vedic mathematics curriculum equivalent to the national one with impressive results. According to Gaskell, the alternative curriculum has resulted in livelier classes, greater student enjoyment and understanding, and improved academic performance (2000). In fact, the first set of students to complete the course were each able to not only pass, but achieve over 80%, on the General Certificate of Secondary Education, a proficiency test taken by all secondary school British students, a year earlier than their peers in the regular curriculum (2000). If harnessed appropriately, there seems to be great potential for how Vedic mathematics can be used to teach, learn and understand mathematics. Perhaps the most important aspect of including Vedic mathematics in an education system will be taking the step towards becoming open to conceptually different mathematical approaches — approaches that could one day free and transform mathematics education.

## Controversy and criticism

There has been much controversy amongst Indian scholars about Tirthaji’s claims that the mathematics is Vedic and that it encompasses all aspects of mathematics (Kansara, 2000). First, Tirthaji’s description of the mathematics as Vedic is most commonly criticised on the basis that, thus far, none of the sūtras can be found in any extant Vedic literature (Williams, 2000). However, trying to locate Tirthaji’s references in the Vedic literature would be extremely difficult as it is possible that Tirthaji rediscovered and reconstructed the sūtras from stray references scattered throughout the Atharva-veda, making it difficult to trace them (Trivedi, 1965). In response to criticisms that the sūtras cannot be located within the texts, several people have explained how textual references should not be the basis for evaluating the Vedicity of the mathematics (Agrawala, 1992). Some propose that Vedic mathematics is different than other scientific work because it is not pragmatically worked out, but is based on a direct revelation, or an “intuitional visualisation” of fundamental mathematical truths (Agrawala, 1992; Pratyagatmananda, 1965). Tirthaji has been described as having the same “reverential approach” towards the Vedas as the ancient rishis that formed them. Thus, it seems as though some believe that Tirthaji may not have found the sūtras within the Vedas, but that he received them spiritually as the rishis did, which should validate them as Vedic.

The controversy about the Vedicity of the mathematics is further confused by the double meaning of veda. Since veda can be translated to mean ‘knowledge’, it is also possible that Vedic mathematics simply refers to the fact that the sūtras are supposed to present all knowledge of mathematics. Tirthaji’s definition of veda does not clearly clarify whether he uses it to represent ‘all knowledge’ or the Vedic texts; rather, it seems that he uses it to refer to both:

Considering the lack of references to the sūtras, coupled with the fact that the language style does not seem Vedic, some propose that the sūtras were simply composed by Tirthaji himself (Agrawala, 1992). In that case, one must consider what motivated Tirthaji to attribute the mathematical sūtras to the ancient texts. Was it because they are from the Vedas, or does claiming so give them more credibility? Other areas of controversy regarding Vedic mathematics focus on the actual mathematics itself. Tirthaji’s assertion that the 16 sutras of Vedic mathematics encompass all branches of mathematics is an extreme one even if true, and so it is not surprising that many mathematicians challenge it (Kansara, 2000). They point to the inconsistency between the topics addressed by the system (such as decimal fractions) and the known mathematics of early India, the substantial extrapolations from a few words of a sūtra to complex arithmetic strategies, and the restriction of applications to convenient, special cases. They further say that such arithmetic as is sped up by application of the sūtras can be performed on a computer or calculator anyway, making their knowledge rather irrelevant in the modern world.

They are also worried that it deflects attention from genuine achievements of ancient and modern Indian mathematics and mathematicians.

## References

Agrawala, V. S. (1992). General editor's note. Vedic mathematics (pp. v-viii)Motilal Banarsidass Publishers Private Limited.

Dutta, . (2002). Mathematics in Ancient India. Seattle, Wash.: Resonance Media.

Gaskell, M. (2000). Try a sūtra. The Times Educational Supplement, , M10.

Glover, J. (2002, Vedic Mathematics Today (Only a Matter of 16 Sutras). Education Times.

Kansara, N. M. (2000). Vedic sources of the Vedic mathematics. Sambodhi Vol. XXIII.

Pratyagatmananda, S. (1965). Forward. Vedic mathematics (pp. ix-xii). Delhi: Motilal Banarsidass Publishers Private Limited.

Tirthaji, Bharati Krsna Maharaja. (1992). In Agrawala V. S. (Ed.), Vedic mathematics. Delhi: Motilal Banarsidass Publishers Private Limited.

Trivedi, S. M. (1965). My beloved gurudeva. Vedic mathematics (pp. xxiii). Delhi: Motilal Banarsidass Publishers Private Limited.

Veda. In L. L. Bram, & N. H. Dickey (Eds.), Funk and Wagnalls New Encyclopedia (pp. 417–418). Funk and Wagnalls L.P.

Williams, K. (2000). The sūtras of Vedic mathematics. Baroda: Oriental Institute.