1. Historical Background
“Veda” means “knowledge”
Age of Vedic texts: from 300BC to several millennia BC
Sections on medicine, ethics, metaphysics, psychology, architecture, music, astronomy, grammar and so on. And ‘ganita sutras’
Sri Bharati Krsna Tirthaji (1884 – 1960)
Academy of Vedic Mathematics

2. Historical Background
Vedic system was reconstructed between 1911 and 1918
Bharati Krsna wrote one introductory volume in 1958: “Vedic Mathematics” published in 1965
Shankaracharya of Govardhan Matha ,Puri
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4. Academy of Vedic Mathematics www.vedicmaths.org
Sixteen Sutras
They cover all of mathematics, pure and applied
These relate to natural mental functions
This explains why Vedic Maths is so easy
It works the way the mind naturally works
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5. Features of Vedic Mathematics
Natural, powerful
Coherent, unified
Easy to do, easy to understand
Flexible
Creative, fun
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6. “By One More than the One Before”
752 =
(4½)2 =
43 x 47 =
88 x 46 =
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7. Digit Sums
A Digit is a single-figure number: 0,1,2,3,4,5,6,7,8,9.
Sum means add.
So ‘Digit Sum’ means adding the digits in a number. 17 19
123
5030 38
7531
The Digit Sum is found by adding the digits in a number,
and adding again if necessary
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The Nine Point Circle
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
1, 2, 3, 4, 5, 6, 7, 8, 9, 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 9 8 7 6
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Casting out Nines
59190
Adding or subtracting 9s to or from a number does not affect the digit sum
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10. Any group of figures in a number that add up to 9 can be "cast out“
Groups adding to 9
Any group of figures in a number that add up to 9 can be "cast out“
7312
42134
61395
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Left to Right
123
We read, pronounce and write numbers from left to right
The most important/significant figures in a number are at the left
In the Vedic system we can add, subtract, multiply and divide from left to right
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Left to Right
So:-
Mental calculations are easier
We can get at the most significant figures in a calculation first, not last
We can combine the operations
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Addition
8 8
7 6
Practice Add from left to right:
1) ) ) )
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14. The Digit Sum Check for Addition
Find and check the answer using digit sums.
3 2
3 9 +
7 7
2 7 9
4 9 0
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15. Mental Multiplication
7 8
3 x
6 7
7 x
8 8
6 x
6 3
8 x
Practice Multiply from left to right:
1) ) ) 8 6
6 x x x
4) ) 4 9
6 x x
4 3
9 x
6 3
6 x
4 7
6 x
5 3
7 x
6 8
7 x
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16. Longer Multiplications
3 x
6 2 7
4 x
7 x
Practice Multiply from left to right:
1) ) ) )
6 x x x x
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17. The Digit Sum Check for Multiplication
3 8
3 x
6 2 7
4 x
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18. Subtraction from left to right
6 3 –
8 5 – – – – – –
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19. Checking Subtractions
Method 1: Add the second and third rows to get the first row. 8
3 – 5 –
Practice Which is/are wrong? 1) – 2) – 3) –
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20. Checking Subtractions1 2 3 4 5 9 8 7 6
Method 2: Use the digit sums. – 7 – 2
8 3 –
3 6 2 6 5 2 1 6 9
Practice Use digit sums to check these: 1) – 2) – 3) –
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21. “All from 9 and the Last from 10”
8 7 6
5 4 9 1 2 4 4 6 1
The formula All From 9 and the Last From 10
subtracts numbers from the next highest base number.
1000 – 864 =
1000 – 307 =
10000 – 6523 =
100 – 63 =
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22. “All from 9 and the Last from 10”
Practice
1) 1000 – 777 = 2) 10,000 – 4621 =
3) 100 – 58 = 4) 100,000 – =
5) 1000,000 – = 6) 1000 – 730 =
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23. “All from 9 and the Last from 10”
1000 – 72 =
100,000 – 503 =
700 – 66 =
4000 – 333 =
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Variations
5000 – 47 =
40,000 – 33 =
1 – =
Rs.100 – Rs. 53 =
Rs.500 – Rs. 77 =
Rs.1000 – Rs.835 =
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Subtraction –
9 0 5
4 4 7 – –
Practice 1) –
2) 8 1 1
3 4 5 – 3) –
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26. Split the number where a bar digit is followed by a positive digit
Subtraction
9 0 8
4 5 6 – – –
Split the number where a bar digit is followed by a positive digit
Practice 1) –
2) 6 1 9
1 3 5 – 3) –
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27. Multiplying Numbers Near a Base
100 89 97
– 11 98 97
– 02
– 3
– 3 86
/ 33 95
/ 06 89
– 11
– 11
/ 21 1 78
= 79/21
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Practice: Multiply
8 9
9 6
9 7
9 4
9 1
9 5
9 9
8 8
9 8
8 8
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Proofs
100 89 97
– 11
Arithmetic proof:
89 × 97 = 89×100 – 89×3
= 8900 – 100×3 + 11×3
= 8900 – ×3 =
= 8633
– 3 86
/ 33
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Proofs
100 89 97
– 11
Geometric proof:
– 3 86
/ 33
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Proofs
100 89 97
– 11
(x – a)(x – b) = x(x – a – b) + ab
– 3 86
/ 33
Algebraic proof:
(100 – a)(100 – b) = 100(100 – a – b) + ab
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Mentally 89 97
– 11
– 3 86
/ 33
96 x 93
97 x 94
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Numbers Over 100
103
104
+ 03
123
103
+ 23
+ 4
+ 3
107
/ 12
126
/ 69
102
103
+ 02
+ 3
105
/ 06
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Practice: Multiply
106 x 107
104 x 108
102 x 103
111 x 112
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35. One Number Under and One Number Over 100
105 97
+ 5
104 93
+ 4
– 3
– 7
= 101 / 85
= 96 / 72
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Proportionately
204 x 107
48 x 97
98 x 206
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Proportionately
200
189
197
–11
304
307
+ 4
300
– 3
+ 7 2x
186
/ 33 3x
311
/ 28
= 372 / 33
= 933 / 28
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38. Practice: Multiply Using the Base Method
307
308
2 0 1
2 3 4
1 9 8
1 8 8
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Base Multiplication
– 2 6
– 4 12 13
+ 2 8 7
– 3
– 4
+ 3 5
/ 6 2
/ 6
= 3 / 6 15
/ 6 1
Practice Multiply: 8 12 14 13 14
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Proportionately 20 30 21 22
+ 1 28 27
– 2
+ 2
– 3 2x
23 / 2 3x
25 / 6
= 46 / 2
= 75 / 6
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41. Practice: Multiply Using the Base Method
4 1
4 2
3 3
3 2
5 1
5 6
1 9
1 8
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42. Numbers Near Large Bases
879
x 997
10003 x
69978 x
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43. “Vertically and Crosswise”
2 3 ×
3 3 ×
6 7 ×
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Practice Multiply:
5 3 ×
7 2 ×
4 4 ×
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The Vedic Method
Can multiply numbers of any size in one line
The simple pattern makes it easy to remember
Easy to explain
Can multiply from left to right or from right to left
Reversible
Algebraic products can be done the same way
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Division ?
8 ×
2 4
3 x 8 = 24
24 ÷ 8 = 3
3 1 ×
4 2 ×
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47. Find the Missing Number
5 4 ×
7 3 ×
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48. Comparison with Conventional Methods
Vedic Conventional
6 7 ×
6 7 ×
Multiplication
5 2 ×
52)
Division
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Algebra
2x + 3
3x ×
3x – 2
x ×
+ 19x
+ 15
+ 19x
– 14
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Algebraic Division
3x ×
+ 19x
+ 15
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51. Extending to 3-figure Numbers ×
104 x 53 ×
112 x 113
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52. Answers 2-Figures at a Time
11 3 ×
1 13 ×
113 x 203
113 x 203 ×
1202 x 1103
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Bar Numbers
39 x 32
Bar numbers are extremely useful
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Extending the Pattern
504 x 321
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Practice Multiply: × × ×
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Extending the Pattern
3-figure numbers
4-figure numbers
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4-Figure Numbers × ×
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Practice Multiply: × ×
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59. “Vertically and Crosswise”
Numbers of any size can be multiplied in one line
Numbers of any size can be divided in one line
The multiplication method simplifies for squaring numbers
And this is reversible to do square roots (in one line)
And quadratic and higher order equations
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60. Adding/subtracting Fractions
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61. Adding/subtracting Fractions
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Practice: Find
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Three Fractions
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Greatest or Least
Which fraction is greater?
Which fraction is the greatest?
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65. Unifying the Four Operations
Addition Subtraction Multiplication Division + – × ÷
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Algebraic Fractions
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67. Equation of a Line through Two Given Points
Find the equation of the line through the points (7,4) and (5,1). O
(5,1)
(7,4) l
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68. Equation of a Line through Two given Points
Find the equation of the line through the points (0,-4) and (-2,3).
Practice 1 Find the equation of the line through the points:-
1) (4,9), (1,2) 2) (8,5), (-3,1) 3) (5,0), (-2,-5)
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69. Equation of a Line through a given Point and Parallel to a given Line
Find the equation of the line through the point (5,7)
and parallel to the line 2x + 3y = 5. O
(5,7) l
2x + 3y = 5
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70. Equation of a Line through a given Point and Parallel to a given Line
Find the equation of the line through the point (-3,2)
and parallel to the line x - 5y = 1.
Practice 2 Find the equation of the line through the given point
and parallel to the given line:-
1) (4,1), 2x + y = ) (-2,5), 3x – 5y = 2
3) (0,-3), 5y + 2x = 2
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Equation of a Line through a given Point and Perpendicular to a given Line
Find the equation of the line through the point (3,1)
and perpendicular to the line 2x + 3y = 5. O
(3,1) l
2x + 3y = 5
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Equation of a Line through a given Point and Perpendicular to a given Line
Find the equation of the line through the point (2,5)
and perpendicular to the line 3x - y = 2.
Practice 3 Find the equation of the line through the given point
and perpendicular to the given line:-
1) (7,2), 2x + y = ) (-3,4), 5x – 2y = 3
3) (0,-1), 4y + x = 2
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Dividing by 19
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74. Right to Left / 9-Point Circle1
9,0 2 3 6 8 7 5 4
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Divisor Ending in 9
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A Short Cut
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Proportionately
(to 5 decimal places)
(to 5 decimal places)
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COMPOUND ANGLES
If sinA = (A is obtuse) and cosB = (B is acute) find tan(A-B).
A B A-B Tan(A-B) =
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79. TRIGONOMETRIC EQUATIONS
x c s 1 a
x-a x
x 0 5 5 1 2 R a
Vedic method
from “Advanced Mathematics 1” by Celia, Nice & Elliott, Page 119.
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80. APPLICATIONS OF TRIPLES
Trigonometry Coordinate geometry (2 & 3 dimensions) Transformations (2 & 3 dimensions) Simple Harmonic Motion Projectiles Complex numbers Hyperbolic functions Conics Astronomy
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81. SOLUTION OF A QUADRATIC EQUATION
Find, correct to 2 decimal places, the roots of the equation
3x2 – 5x – 7 = 0.
Comparing 3x2 – 5x – 7 = 0 with ax2 + bx + c = 0;
a = 3, b = -5, c = -7.
from “General Mathematics” Book 3, by Channon & Smith, Page 195.
a = 3,
D1 = 6x – 5 = 13
Vedic method
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82. INTEGRATION BY ‘PARTS’
Let
let
from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 313. {
Vedic method
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83. SQUARE ROOT OF A COMPLEX NUMBER
Equating real and imaginary parts gives:
a2 – b2 = 15 (1)
and 2ab = 8 (2)
Vedic method
Thus a2 – 16 = 0 or a2 + 1 = 0
But a is real so a2 + 1 = 0 gives no suitable values
Referring to equation (2) we have a b 4 1 -4 -1
from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 536.
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84. Features of Vedic Mathematics
Works the way the mind works
More unified
Can be a mental system which develops memory and mental agility
Very flexible: choice of methods
left to right or right to left
Encourages creativity and innovation
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