The document discusses Vedic mathematics, which originated from the Vedas. It describes 16 sutras or formulae for mental calculation. Examples are provided for using the sutras to multiply and divide numbers mentally. The method of finding square roots through the duplex process is also explained with examples. Contact information is provided at the end for the author.
1. Introduction
Multiplication
Vedic Mathematics
Division
Square Roots . . .
Teaching an Old Dog New Tricks
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Uwe Wystup
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uwe.wystup@mathfinance.com
March 2010
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2. Introduction
Multiplication
Division
1. Introduction
Square Roots . . .
Contact Information veda (Sanskrit) means: knowledge
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Veda Upaveda
Title Page Rigveda Ayurveda
Samaveda Gandharvaveda
Yajurveda Dhanurveda
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Atharvaveda Sthapatyaveda
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Table 1: Vedas and Upavedas (supplementary vedas)
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3. 1.1. The 16 Sutras
are part of a Parisista (Appendix) of the Atharvaveda
1. By one more than the one before
2. All from 9 and the last from 10
Introduction
Multiplication 3. Vertically and crosswise
Division
Square Roots . . .
4. Transpose and apply
Contact Information 5. If the Samuccaya is the same it is zero
6. If one is in ratio the other is zero
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7. By addition and by subtraction
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8. By the completion or non-completion
9. Differential calculus
10. By the deficiency
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11. Specific and general
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12. The remainders by the last digit
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13. The ultimate and twice the penultimate
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14. By one less than the one before
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15. The product of the sum
16. All the multipliers
4. 1.2. Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja
Introduction Explained the sutras in his books (e.g. [2]).
Multiplication
Division Jagadguru Swami Sri Bharati
Square Roots . . . Krsna Tirthaji Maharaja
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(March, 1884 - February 2,
1960) was the Jagadguru
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world; assigned to heads
of Hindu mathas) of the
Govardhana matha of Puri
during 1925-1960. He was
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during the 20th century. He
is particularly known for his
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work on Vedic mathematics.
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6. Introduction 2.1. Example with working base 10
Multiplication
Division
Square Roots . . . 9 - 1 7 - 3 13 + 3 12 + 2
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× × × ×
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6 / 3 3 /1 2 15 / 6 10 / ¯
4
= 63 = 42 = 156 = 96
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Reason: (x + a)(x + b) = x(x + a + b) + ab
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7. Introduction
Multiplication
Division
2.2. Example with working base 100
Square Roots . . .
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91 - 9 111 + 11 108 + 8
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× × ×
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96 - 4 109 + 9 97 - 3
87 / 36 120 / 99 105 / ¯
24
Page 7 of 23 = 8736 = 12099 = 10476
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8. Introduction
Multiplication
2.3. Other working bases (division case)
Division
Square Roots . . .
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100/2=50 100/2=50
49 - 1 54 + 4
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× ×
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49 - 1 46 - 4
2)48 / 01 2)50 / ¯¯
16
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24 / 01 25 / ¯¯
16
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= 2401 = 2484
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9. Introduction
Multiplication
2.4. Other working bases (multiplication case)
Division
Square Roots . . .
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10× 2=20 10× 6=60
19 - 1 62 + 2
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× ×
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19 - 1 48 - 12
× 2)18 / 1 × 6)50 /¯
2
¯
4
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36 / 1 300 /¯
2
¯
4
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= 361 = 2976
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10. Introduction
Multiplication
Division
Square Roots . . .
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2.5. Exercise: Multiply the following mentally
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b 78989 × 99997
c 1222 × 1003
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11. Introduction
Multiplication
Division
3. Division
Square Roots . . .
1
Contact Information • Find the exact decimal representation of 19
.
• Using the “Ekadhika Purva” Sutra it is easy:
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1 1 1 1 1 1
/ 9 4 7 3 6 8 4 2 1
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1 1 1
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12. Introduction
Multiplication
Division
• Start with 1 and then work from right to left multi-
Square Roots . . . plying by 2.
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. 0 5 2 6 3 1 5 7 8
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1 1 1 1 1 1
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/ 9 4 7 3 6 8 4 2 1
1 1 1
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13. • A further shortcut is the insight that
Introduction
Multiplication
Division
. 0 5 2 6 3 1 5 7 8
Square Roots . . .
+ 9 4 7 3 6 8 4 2 1
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= 9 9 9 9 9 9 9 9 9
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1
• The same works for all periodic decimals, e.g. 7
. 1 4 2
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+ 8 5 7
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= 9 9 9
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14. Introduction
Multiplication
Division
Square Roots . . .
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4. Square Roots (Vargamula)
mathfinance.com • Square numbers only have digit sums 1, 4, 7, 9
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• and they only end in 1, 4, 5, 6, 9, 0.
• If the given number has n digits, then the square
root will contain n or n+1 digits.
2 2
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• Systematic computation of an exact square root re-
quires the Dvandvayoga (Duplex) process.
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15. Introduction
Multiplication
Division
Square Roots . . .
4.1. Duplex Process (Dvandvayoga)
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D(4) = 16 (1)
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D(1034) = 8 (4)
D(10345) = 19 (5)
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16. 4.2. Square Root of a Perfect Square
√
Introduction Find 1849.
Multiplication
Division
Group in pairs, taking a single extra digit on the left as
Square Roots . . .
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extra digit.
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8) 2
4
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4 is the largest integer whose square does not exceed 18.
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18/4 is 4 with remainder 2.
The divisor 8 is two times 4.
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17. Introduction
Multiplication
Next we divide 24 by the divisor 8. This gives 3 remainder
Division
0, placed as
Square Roots . . .
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8) 2 0
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4 3
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the last answer figure 3, i.e. 09 − D(3) = 09 − 32 =
09 − 9 = 0. This means that the answer is exactly 43.
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18. 1 3 — 6 9
6) 4
Introduction 3
Multiplication
Division
Square Roots . . .
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3 is the largest integer whose square does not exceed 13.
13/3 is 3 with remainder 4.
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The divisor 6 is two times 3.
Title Page Next we divide 46 by the divisor 6. This gives 7 remainder
4, placed as
1 3 — 6 9
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49 − D(7) = 0, so 37 is the exact square root of 1369.
19. 4.3. Larger Numbers
Introduction
2 9 3 7 6 4
Multiplication
Division 10) 4
Square Roots . . .
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2 9 3 7 6 4
10) 4 3
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5 4 .
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37 − D(4) = 37 − 16 = 21 = 2 × 10 + 1.
21. Introduction
Multiplication
Division
Square Roots . . . 4.4. General Square Roots
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Find the first 5 figures of the square root of 38:
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3 8 . 0 0 0 0 0
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12) 2 8 7 10 8
6 . 1 6 4 4
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22. 5. Contact Information
Introduction
Multiplication
Division
Square Roots . . .
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Uwe Wystup
MathFinance AG
mathfinance.com Mainluststraße 4
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60329 Frankfurt am Main
Germany
+49-700-MATHFINANCE
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http://www.mathfinance.com/wystup/papers.php
http://www.mathfinance.com/seminars/vedic.php
23. References
[1] Datta, B. and Singh, A.N. (1962). History of Hindu Mathematics. Asia
Introduction Publishing House, Calcutta.
Multiplication
[2] Maharaja, Bharati Krsna Tirthaji (1992). Vedic Mathematics, Motilal
Division
Banarsidass Publishers Private Ltd, Delhi.
Square Roots . . .
Contact Information [3] Schonard, A. and Kokot, C. (2006). Der Mathekn¨ller. http://www.
u
matheknueller.de.
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[4] Williams, K.R. (2002). Vedic Mathematics - Teacher’s Manual. Ad-
vanced Level. Motilal Banarsidass Publishers Private Limited, Delhi.
Title Page http://www.mlbd.com
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24. Index
Introduction
Multiplication
division, 11
Division
duplex, 15
dvandvayoga, 15
Square Roots . . .
Contact Information multiplication, 5
square root, 14
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Title Page vedas, 2
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