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Abstract,

Keywords,

Introduction,

Objectives of the study,

Algorithm to carry out Vedic Division by Nikhilam sutra,

Algorithm to carry out Vedic division by Dhwajank (flag digit) method,

Conclusion

Abstract,

Keywords,

Introduction,

Objectives of the study,

Algorithm to carry out Vedic Division by Nikhilam sutra,

Algorithm to carry out Vedic division by Dhwajank (flag digit) method,

Conclusion

1.
International Journal of Mathematics Trends and Technology Volume 67 Issue 2, 166-170, Februray, 2021

ISSN: 2231 – 5373 /doi:10.14445/22315373/IJMTT-V67I2P523 © 2021 Seventh Sense Research Group®

Beauty of Vedic Speed Mathematics in Division

Ms. Neha Dattatrey Yeola

Asst. Prof. Computer Science, Ashoka Center for Business & Computer Studies, Nashik, India

Vedic Mathematics is the former system of Mathematics which was formulated and encapsulated in the modern form by

Jagadguru Swami Bharati Krishna Tirtha Ji. During the period of 1919 – 1927 he formulated a concept of Vedic Maths by

deep meditation and intuition with the help of Vedas and Scriptures. He postulated sixteen sutras (formulae) and 13 sub

sutras (corollaries) of Vedic Mathematics. These sutras are easy to understand, apply and remember. With the help of

these sutras one can calculate faster than conventional maths and hence these are very much helpful in competitive exams

like MPSC, UPSC, GET, JEE, IBPS and many other.

Keywords: Base, Division, Sutras, Vedas, Vedic Mathematics

INTRODUCTION

It is observed that many students as well as adults fear Mathematics and try to avoid it due to inefficiency in carrying out

long multiplication & division, finding square & square roots and cube & cube roots. Swami Bharati Krishna Tirtha Ji

Maharaj, 143rd Shankaracharya of Govardhan Peeth, Puri, thought deeply on this and tried to simplify these processes and

constructed 16 Sutras and 13 Sub sutras by studying ancient Indian scriptures. Swamiji wrote a book “Vedic

Mathematics”, the book and its magic effect of speedy calculations are very much pleasing. It increases speed of

calculations as well as develop interest of students in Mathematics.

OBJECTIVES OF THE STUDY:

1. To enjoy learning Mathematics.

2. To reduce difficult problems to one-line answers.

3. To overcome Maths phobia.

4. To become strong analytical thinker.

5. To improve mental ability, sharpness, creativity and self-confidence.

6. To achieve academic excellence and success in Mathematics.

The present study focused on the magical techniques in Vedic Mathematics for arithmetic division using following

1. Nikhilam Navatahscarmam Dasatah (All from nine, last from 10)

2. Paravartya Yojayet (Transpose and apply)

3. Dhwajank (Flag digit) (Using Urdhvatiryakbhyam sutra)

The conventional form of division has four terms:

(1) 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 (𝐸) (2) 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 (𝑄) (3) 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 (𝐷) (4) 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 (𝑅)

The relation between these four terms is

𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 (𝐸) = 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 (𝑄) × 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 (𝐷) + 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 (𝑅)

1. Nikhilam Navatascaramam Dasatah sutra is applied when the divisor is near to the base and less than base.

Structure of division:

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

ISSN: 2231 – 5373 /doi:10.14445/22315373/IJMTT-V67I2P523 © 2021 Seventh Sense Research Group®

Beauty of Vedic Speed Mathematics in Division

Ms. Neha Dattatrey Yeola

Asst. Prof. Computer Science, Ashoka Center for Business & Computer Studies, Nashik, India

Vedic Mathematics is the former system of Mathematics which was formulated and encapsulated in the modern form by

Jagadguru Swami Bharati Krishna Tirtha Ji. During the period of 1919 – 1927 he formulated a concept of Vedic Maths by

deep meditation and intuition with the help of Vedas and Scriptures. He postulated sixteen sutras (formulae) and 13 sub

sutras (corollaries) of Vedic Mathematics. These sutras are easy to understand, apply and remember. With the help of

these sutras one can calculate faster than conventional maths and hence these are very much helpful in competitive exams

like MPSC, UPSC, GET, JEE, IBPS and many other.

Keywords: Base, Division, Sutras, Vedas, Vedic Mathematics

INTRODUCTION

It is observed that many students as well as adults fear Mathematics and try to avoid it due to inefficiency in carrying out

long multiplication & division, finding square & square roots and cube & cube roots. Swami Bharati Krishna Tirtha Ji

Maharaj, 143rd Shankaracharya of Govardhan Peeth, Puri, thought deeply on this and tried to simplify these processes and

constructed 16 Sutras and 13 Sub sutras by studying ancient Indian scriptures. Swamiji wrote a book “Vedic

Mathematics”, the book and its magic effect of speedy calculations are very much pleasing. It increases speed of

calculations as well as develop interest of students in Mathematics.

OBJECTIVES OF THE STUDY:

1. To enjoy learning Mathematics.

2. To reduce difficult problems to one-line answers.

3. To overcome Maths phobia.

4. To become strong analytical thinker.

5. To improve mental ability, sharpness, creativity and self-confidence.

6. To achieve academic excellence and success in Mathematics.

The present study focused on the magical techniques in Vedic Mathematics for arithmetic division using following

1. Nikhilam Navatahscarmam Dasatah (All from nine, last from 10)

2. Paravartya Yojayet (Transpose and apply)

3. Dhwajank (Flag digit) (Using Urdhvatiryakbhyam sutra)

The conventional form of division has four terms:

(1) 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 (𝐸) (2) 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 (𝑄) (3) 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 (𝐷) (4) 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 (𝑅)

The relation between these four terms is

𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 (𝐸) = 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 (𝑄) × 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 (𝐷) + 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 (𝑅)

1. Nikhilam Navatascaramam Dasatah sutra is applied when the divisor is near to the base and less than base.

Structure of division:

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

2.
Ms. Neha Dattatrey Yeola / IJMTT, 67(2), 166-170, 2021

Nikhilam part of divisor (modified divisor) is placed in leftmost i.e. divisor part. Then the number of digits from the right

side of the dividend equal to the number of zeros in the base are placed in the part of remainder and then remaining portion

of the dividend is placed in the middle part i.e. quotient part.

Here, division will be carried out by using modified divisor and instead of subtraction, addition will be used in Vedic

division which is found to be easier than conventional mathematics.

Algorithm to carry out Vedic Division by Nikhilam sutra:

1. Far left digit in the quotient part is considered as the first digit of quotient. Its product with the modified divisor is

added to the successive digit.

2. The result of this addition is multiplied by the modified divisor and added to next digit.

3. This process is continued up to last digit of the quotient part. The results of the addition in this part will give the

quotient. In this part at each place right most digit (the unit digit) will be placed as it is and extra digit (if exists)

will carry to the immediate left part.

4. If addition in the remainder part is greater than the divisor, then it will be again divided by the divisor and the

quotient obtained here will be added to the original quotient (obtained in quotient part) to get the final quotient

and the last remainder will be considered as the final remainder of the process.

For example: 𝟏𝟓𝟏𝟒𝟐𝟏 ÷ 𝟗𝟗

Here 99 is near to 100, so base is 100.

Deviation is calculated by applying the sutra “Nikhilam Navatascaramam Dasatah”.

Deviation: (9-9) (10-9)

Modified Divisor (M.D.): 01

Step 1: The leftmost digit 1 will be the first digit of the quotient.

Step 2: Multiply 1st quotient digit ‘1’ with M.D. (01) digit wise and resultant product digits are written in the next row

under 2nd & 3rd column of quotient part.

Addition of 2nd column digits of quotient part gives 2nd digit of quotient.

Step 3: Now multiply 2nd quotient digit ‘2’ with M.D. digit- wise and resultant product digits are written in the next row

under 3rd column in quotient part and 4th column in remainder part.

167

Nikhilam part of divisor (modified divisor) is placed in leftmost i.e. divisor part. Then the number of digits from the right

side of the dividend equal to the number of zeros in the base are placed in the part of remainder and then remaining portion

of the dividend is placed in the middle part i.e. quotient part.

Here, division will be carried out by using modified divisor and instead of subtraction, addition will be used in Vedic

division which is found to be easier than conventional mathematics.

Algorithm to carry out Vedic Division by Nikhilam sutra:

1. Far left digit in the quotient part is considered as the first digit of quotient. Its product with the modified divisor is

added to the successive digit.

2. The result of this addition is multiplied by the modified divisor and added to next digit.

3. This process is continued up to last digit of the quotient part. The results of the addition in this part will give the

quotient. In this part at each place right most digit (the unit digit) will be placed as it is and extra digit (if exists)

will carry to the immediate left part.

4. If addition in the remainder part is greater than the divisor, then it will be again divided by the divisor and the

quotient obtained here will be added to the original quotient (obtained in quotient part) to get the final quotient

and the last remainder will be considered as the final remainder of the process.

For example: 𝟏𝟓𝟏𝟒𝟐𝟏 ÷ 𝟗𝟗

Here 99 is near to 100, so base is 100.

Deviation is calculated by applying the sutra “Nikhilam Navatascaramam Dasatah”.

Deviation: (9-9) (10-9)

Modified Divisor (M.D.): 01

Step 1: The leftmost digit 1 will be the first digit of the quotient.

Step 2: Multiply 1st quotient digit ‘1’ with M.D. (01) digit wise and resultant product digits are written in the next row

under 2nd & 3rd column of quotient part.

Addition of 2nd column digits of quotient part gives 2nd digit of quotient.

Step 3: Now multiply 2nd quotient digit ‘2’ with M.D. digit- wise and resultant product digits are written in the next row

under 3rd column in quotient part and 4th column in remainder part.

167

3.
Ms. Neha Dattatrey Yeola / IJMTT, 67(2), 166-170, 2021

Addition of 3rd column digits of quotient part gives 3rd digit of the quotient.

Step 4: Multiply 3rd quotient digit ‘4’ with M.D. digit-wise and resultant product digits are written in the next row under

4th and 5th column in remainder part.

Addition of 4th and 5th column digits of remainder part gives the remainder.

As we reached up to the unit place digit of dividend, the job of M.D. is over.

Remainder must be non-negative number less than divisor always.

Answer: Quotient = 124, Remainder = 65

2. Paravartya Yojayet sutra is applied when the divisor is near to the base and greater than base.

In this method, obtain difference between divisor and base, then give negative sign to each digit of the difference. The

number thus obtained is called paravartya of the divisor. Now in spite of using Nikhilam use paravartya and follow the

same procedure that we have followed in Nikhilam method.

For Example: 𝟏𝟑𝟔𝟗𝟕𝟐 ÷ 𝟏𝟐𝟏

Remainder must be non-negative number less than divisor always.

Answer: Quotient = 1132, Remainder = 00

3. Dhwajank method (Flag digit): This is the universal method of division. In this method divisor is splitted into two

parts. One part i.e. left part is called as the principal divisor and the remaining part on the right side is called as

“Dhwajank”(flag digit). It is also called as flag number. Both principal divisor and dhwajank are placed in divisor part

but the division is carried out by only the principal divisor.

168

Addition of 3rd column digits of quotient part gives 3rd digit of the quotient.

Step 4: Multiply 3rd quotient digit ‘4’ with M.D. digit-wise and resultant product digits are written in the next row under

4th and 5th column in remainder part.

Addition of 4th and 5th column digits of remainder part gives the remainder.

As we reached up to the unit place digit of dividend, the job of M.D. is over.

Remainder must be non-negative number less than divisor always.

Answer: Quotient = 124, Remainder = 65

2. Paravartya Yojayet sutra is applied when the divisor is near to the base and greater than base.

In this method, obtain difference between divisor and base, then give negative sign to each digit of the difference. The

number thus obtained is called paravartya of the divisor. Now in spite of using Nikhilam use paravartya and follow the

same procedure that we have followed in Nikhilam method.

For Example: 𝟏𝟑𝟔𝟗𝟕𝟐 ÷ 𝟏𝟐𝟏

Remainder must be non-negative number less than divisor always.

Answer: Quotient = 1132, Remainder = 00

3. Dhwajank method (Flag digit): This is the universal method of division. In this method divisor is splitted into two

parts. One part i.e. left part is called as the principal divisor and the remaining part on the right side is called as

“Dhwajank”(flag digit). It is also called as flag number. Both principal divisor and dhwajank are placed in divisor part

but the division is carried out by only the principal divisor.

168

4.
Ms. Neha Dattatrey Yeola / IJMTT, 67(2), 166-170, 2021

The arrangement is like a flag. Write principal divisor in the base part and flag digit in the above part as shown in the

following figure.

Algorithm to carry out Vedic division by Dhwajank (flag digit) method:

1. The number of digits from right side of the dividend, equal to the number of the digits of Dhwajank are placed in

the remainder part and the remaining part of the dividend is placed into the quotient part.

2. At every stage of the division, principal divisor divides the true dividend and remainder is written before the

successive digit of the dividend, which forms the gross dividend (G.D.)

3. At every stage of division, applying Urdhvatiryagbhyam sutra (vertically and crosswise) product is subtracted

from gross dividend and the result of subtraction is considered as the true dividend. Now divide the true dividend

by the principal divisor, we get the successive digit of the quotient as well as the remainder at that stage and the

process continues by placing this remainder before the next digit.

4. At any stage if Urdhvatiryag product is greater than the gross dividend, division process can’t step forward. In this

case decrease previous digit of the quotient by 1 to get the proper gross dividend for subtracting the Urdhvatiryag

product.

For example: 𝟐𝟑𝟕𝟓𝟒 ÷ 𝟕𝟒

Here we split the divisor 74 in two parts, 7 as the principal divisor and 4 as the flag digit. As there is 1 flag digit,

remainder part contains 1 digit only.

Step 1: Divide 2 by principal divisor 7, we get the quotient 0 and remainder 2.

Step 2: Place remainder 2 before next dividend digit 3, we get 23 as gross dividend. Now subtract Urdhva (vertical)

product of flag digit 4 and the first quotient digit 0 (i.e. 4 × 0 = 0), we get true dividend 23 − 0 = 23. Now divide it

by principal divisor 7, we get second digit of quotient as 3 and remainder 2.

Step 3: Place the remainder 2 before next dividend digit 7, we get 27 as gross dividend. Now subtract Urdhva product

of flag digit 4 and the second quotient digit 3, we get true dividend 27 − 12 = 15. Now dividing it by principal

divisor 7, we obtain third digit of quotient as 2 and remainder 1.

169

The arrangement is like a flag. Write principal divisor in the base part and flag digit in the above part as shown in the

following figure.

Algorithm to carry out Vedic division by Dhwajank (flag digit) method:

1. The number of digits from right side of the dividend, equal to the number of the digits of Dhwajank are placed in

the remainder part and the remaining part of the dividend is placed into the quotient part.

2. At every stage of the division, principal divisor divides the true dividend and remainder is written before the

successive digit of the dividend, which forms the gross dividend (G.D.)

3. At every stage of division, applying Urdhvatiryagbhyam sutra (vertically and crosswise) product is subtracted

from gross dividend and the result of subtraction is considered as the true dividend. Now divide the true dividend

by the principal divisor, we get the successive digit of the quotient as well as the remainder at that stage and the

process continues by placing this remainder before the next digit.

4. At any stage if Urdhvatiryag product is greater than the gross dividend, division process can’t step forward. In this

case decrease previous digit of the quotient by 1 to get the proper gross dividend for subtracting the Urdhvatiryag

product.

For example: 𝟐𝟑𝟕𝟓𝟒 ÷ 𝟕𝟒

Here we split the divisor 74 in two parts, 7 as the principal divisor and 4 as the flag digit. As there is 1 flag digit,

remainder part contains 1 digit only.

Step 1: Divide 2 by principal divisor 7, we get the quotient 0 and remainder 2.

Step 2: Place remainder 2 before next dividend digit 3, we get 23 as gross dividend. Now subtract Urdhva (vertical)

product of flag digit 4 and the first quotient digit 0 (i.e. 4 × 0 = 0), we get true dividend 23 − 0 = 23. Now divide it

by principal divisor 7, we get second digit of quotient as 3 and remainder 2.

Step 3: Place the remainder 2 before next dividend digit 7, we get 27 as gross dividend. Now subtract Urdhva product

of flag digit 4 and the second quotient digit 3, we get true dividend 27 − 12 = 15. Now dividing it by principal

divisor 7, we obtain third digit of quotient as 2 and remainder 1.

169

5.
Ms. Neha Dattatrey Yeola / IJMTT, 67(2), 166-170, 2021

Step 4: By following same procedure as in step 2 and step 3, we get quotient as 1 and remainder 0.

Step 5: Finally subtract vertical multiplication of remainder 0 and flag digit from 4 (remainder part), we get the final

remainder as 0 and final quotient 321.

Answer: Quotient = 321, Remainder = 0

Remainder must be non-negative number less than divisor always.

CONCLUSION

The present paper gives different techniques for arithmetic division using three Vedic sutras, which provides correct and

fast solutions as compare to the method in conventional mathematics. Nikhilam and Paravartya Sutra can be applied to

limited problems i.e. when divisor is near to base whereas Dhwajank method can be applied to any divisor. These methods

make solving problems easier and interesting, that would be beneficial to those who are afraid of Maths.

REFERENCES

[1] Vedic Mathematics by Jagadguru Swami Sri Bharti Krishna Tirtha ji Maharaj, 1965, Motilal Banarasidas, New Delhi.

[2] Enjoy Vedic Mathematics by Dr. Ramesh Kolhuru and Shriram M. Chauthaiwale, 2010, Art of Living publications, Bangalore.

[3] Elements of Vedic Mathematics by Udayan S. Patankar and Sunil M. Patankar, 2018, TTU press.

170

Step 4: By following same procedure as in step 2 and step 3, we get quotient as 1 and remainder 0.

Step 5: Finally subtract vertical multiplication of remainder 0 and flag digit from 4 (remainder part), we get the final

remainder as 0 and final quotient 321.

Answer: Quotient = 321, Remainder = 0

Remainder must be non-negative number less than divisor always.

CONCLUSION

The present paper gives different techniques for arithmetic division using three Vedic sutras, which provides correct and

fast solutions as compare to the method in conventional mathematics. Nikhilam and Paravartya Sutra can be applied to

limited problems i.e. when divisor is near to base whereas Dhwajank method can be applied to any divisor. These methods

make solving problems easier and interesting, that would be beneficial to those who are afraid of Maths.

REFERENCES

[1] Vedic Mathematics by Jagadguru Swami Sri Bharti Krishna Tirtha ji Maharaj, 1965, Motilal Banarasidas, New Delhi.

[2] Enjoy Vedic Mathematics by Dr. Ramesh Kolhuru and Shriram M. Chauthaiwale, 2010, Art of Living publications, Bangalore.

[3] Elements of Vedic Mathematics by Udayan S. Patankar and Sunil M. Patankar, 2018, TTU press.

170