VEDIC MATHEMATICS : Arithmetic Operations

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VEDIC MATHEMATICS Arithmetic Operations

• T. K. Prasad
• http//www.cs.wright.edu/tkprasad

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Positional Number System
TEN-THOUSANDS THOUSANDS HUNDREDS TENS UNITS
43210 4 10,000 3 1,000 2 100 1
10 0
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Two Digit Multiplication (above the base) using
Vedic Approach

1. Method Vertically and Crosswise Sutra
2. Correctness and Applicability

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Method Multiply 13 12

• Write the first number to be multiplied and
  excess over 10 in the first row, and the second
  number to be multiplied and excess over 10 in the
  second row.
• 13 3
• 12 2

5

• 13 3
• 12 2
• To determine the 3-digit product
• add crosswise to obtain the left digits
• (13 2) (12 3) 15
• and
• multiply the excess vertically to obtain the
  right digit.
• (3 2) 6
• 13 12 156

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Another Example

• 12 14
• 12 2
• 14 4
• 16 8
• 12 14 168

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Questions

• Why do both crosswise additions yield the same
  result?
• Why does this method yield the correct answer for
  this example?
• Does this method always work for any pair of
  numbers?

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Proof Sketch

• (12 4) (14 2) 16
• Why are they same?
• That is, the sum of first number and excess over
  10 of the second number, and .
• (12 (14 10)) (1214 10) (26 10) 16
• (14 (12 10)) (1412 10) (26 10) 16

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Correctness ArgumentTwo possibilities

• 12 (10 2)
• 14 (10 4)
• 12 14
• (10 2) 14
• 10 14 2 14
• 10 14 2 (10 4)
• 10 14 2 10 (2 4)
• 10 (142) 8
• 10 16 8
• 168

• 12 (10 2)
• 14 (10 4)
• 12 14
• 12 (10 4)
• 12 10 12 4
• 12 10 (10 2) 4
• 12 10 10 4 (2 4)
• 10 (12 4) 8
• 10 16 8
• 168

Right digit Vertical Product
Right digit Vertical Product
Left digits Crosswise Addition
Left digits Crosswise Addition
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Another Example

• 15 12
• 15 5
• 12 2
• 17 10
• 18 0

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Yet Another Example

• 17 15
• 17 7
• 15 5
• 22 35
• 223 5
• 25 5
• Need proof to feel comfortable!

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Method Multiply 113 106

• Write the first number to be multiplied and
  excess over 100 in the first row, and the second
  number to be multiplied and excess over 100 in
  the second row.
• 113 13
• 106 6

13

• 113 13
• 106 6
• To determine the 5-digit product
• add crosswise to obtain the left digits
• (113 6) (106 13) 119
• and
• multiply the excess vertically to obtain the
  right digits.
• (13 6) 78
• 113 106 11978

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Questions

• Why do both crosswise additions yield the same
  result?
• Why does this method yield the correct answer for
  this example?
• Does this method always work for any pair of 3
  digit numbers?

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Proof Sketch

• (113 6) (106 13) 119
• Why are they same?
• (113 (106 100)) (113 106 100) 119
• (106 (113 100)) (106 113 100) 119

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Correctness of Product Two possibilities

• 113 (100 13)
• 106 (100 6)
• 113 106
• 113 (100 6)
• 113 100 (100 13) 6
• 113 100 100 6 (13 6)
• 100 (113 6) 78
• 100 119 78
• 11978

• 113 (100 13)
• 106 (100 6)
• 113 106
• (100 13) 106
• 100 106 13 (100 6)
• 100 106 13 100 (13 6)
• 100 (106 13) 78
• 100 119 78
• 11978

Right digits Vertical Product
Right digits Vertical Product
Left digits Crosswise Addition
Left digits Crosswise Addition
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Another Example

• 160 180
• 160 60
• 180 80
• 240 4800
• 288 00
• Note that, the product of the excess over 100 has
  more than two digits. However, the weight
  associated with 240 and 48 are both 100, and
  hence they can be combined.

Breakdown?!
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Yet Another Example

• 190 199
• 190 90
• 199 99
• 289 8910
• 28989 10
• 378 10
• This approach is valid with suggested
  modifications!

Breakdown?!
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More Shortcuts
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Quick squaring of numbers that end in 5

• 15 15
• 225
• (12) (55)
• 75 75
• 5625
• (78) (55)
• 95 95
• 9025
• (910) (55)

• Proof Let the two digit number be written as D5.
• D5 D5
• (D10 5) (D10 5)
• (DD100) (D250) 55
• (D(D1))100 25

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Quick Multiplication Special Case

• Proof Let two digit numbers be AB and AC.
• AB AC
• (A10 B) (A10 C)
• (AA100) (A10(BC)) BC
• (AA)100 (A)(BC)10 (BC)
• For BC10, this reduces to
• A(A1)100 BC
• For A12, B8 and C2, this reduces to
• (12)(13)100 16
  15616

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Quicking squaring of numbers that begin with 5

• 51 51
• (551)100 (11)
• 2601
• 57 57
• (557) 100 (77)
• 3249
• 59 59
• (559) 100 (99)
• 3481

• Proof Let the two digit number be written as 5D.
• 5D 5D
• (50 D) (50 D)
• (25 D)100 (DD)

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Quick squaring of two digit numbers

• Proof Let two digit numbers be AB.
• AB AB
• (A10 B) (A10 B)
• (AA)100 2(A10)B BB
• (AA)100 20(AB) (BB)
• For AB79, this reduces to 4900206381
• 
  49811260 6241
• For AB116, this reduces to 12100206636
• 
  121361320 13456

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Generalized Multplication Using Working Base
25

• 23 3
• 24 4
• To determine the product, choose working base as
  20
• add crosswise to obtain the left digits with
  weight 20
• (23 4) (24 3) 27
• multiply the excess vertically to obtain the
  right digits.
• (3 4) 12
• 23 24 27 20 12
• 540 12
• 23 24 552

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• 723 23
• 724 24
• To determine the product, choose working base as
  700
• add crosswise to obtain the left digits with
  weight 700
• (723 24) (724 23) 747
• multiply the excess vertically to obtain the
  right digits.
• (23 24) 552
• 723 724 747 700 552
• 522900 552
• 723 724 523452

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• 783 -17
• 775 -25
• To determine the product, choose working base as
  800
• add crosswise to obtain the left digits with
  weight 800
• (783 - 25) (775 - 17) 758
• multiply the excess vertically to obtain the
  right digits.
• (17 25) 425
• 783 775 758 800 425
• 606400 425
• 783 775 606825

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• 532 32
• 472 -28
• To determine the product, choose working base as
  1000/2
• add crosswise to obtain the left digits with wt.
  1000/2
• (532 - 28) (472 32) 504
• multiply the excess vertically to obtain the
  right digits.
• (32) (-28) 896
• 532 472 (504 / 2)1000 (104 -1000)
• 252000 104 - 1000
• 532 472 251104