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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
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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
Academy of Vedic Mathematics
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Academy of Vedic Mathematics
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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
Academy of Vedic Mathematics
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Features of Vedic Mathematics
• Natural, powerful
• Coherent, unified
• Easy to do, easy to understand
• Flexible
• Creative, fun
Academy of Vedic Mathematics
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“By One More than the One Before”
0
1
2
3
4
5
6
7
752 =
(4½)2 =
43 x 47 =
88 x 46 =
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8
9 . . . .
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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
38
5030
7531
The Digit Sum is found by adding the digits in a number,
and adding again if necessary
Academy of Vedic Mathematics
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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, 21 . . . .
1, 2, 3, 4, 5, 6, 7, 8, 9,
10
9
9
1
8
8
2
7
3
6
5
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4
1
7
2
3
6
5
4
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Casting out Nines
59190
9899329
Adding or subtracting 9s to or from a number
does not affect the digit sum
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Groups adding to 9
Any group of figures in a number that add up to 9
can be "cast out“
7312
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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:1. Mental calculations are easier
2. We can get at the most significant figures in
a calculation first, not last
3. We can combine the operations
2
2
314 + 432 = 534.06....
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Addition
8 8
4 6 +
7 6
5 5 +
1 8 7
4 4 6 +
Practice Add from left to right:
1) 7 7
66
2) 4 3 7
871
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3) 6 5 4
727
4) 2 4 6 8
3865
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The Digit Sum Check for Addition
Find 32 + 39 and check the answer using digit sums.
32
39+
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77
124 +
279
121 +
490
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Mental Multiplication
7 8
3x
67
7x
88
6x
63
8x
Practice Multiply from left to right:
1) 6 6
6x
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2) 5 3
5x
3) 8 6
7x
4) 6 4
6x
5) 4 9
4x
5 3
4
6
8
7
7x
9
6
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Longer Multiplications
3 8 5
3x
627
4x
5347
7x
Practice Multiply from left to right:
1) 7 2 7
6x
2) 4 3 2
7x
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3) 6 5 4
3x
4) 2 4 6 8
5x
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The Digit Sum Check for Multiplication
3 8
3x
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627
4x
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Subtraction from left to right
6
2
5
1
6
7
3
7 –
5
7 –
5
1
6
7
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6
2
5
5
6
1 –
8
3
3
5
5
7 –
4
2 –
7
3
7
2
3
3
5
2
3
8
6
7
3
8 –
4
1 –
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Checking Subtractions
Method 1: Add the second and third rows to get the first
row.
8
5 6 5
3 –
1 7 7 –
5
3 8 8
Practice Which is/are wrong?
1) 7
2
4
6
7
8
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3
8 –
5
2) 8 0 5
4 5 8 –
3 5 7
3) 7 8 3
1 4 7 –
6 3 6
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9
1
8
Checking Subtractions
2
7
3
6
5
Method 2: Use the digit sums.
5
1
3
6
7
8
7
3
3
5
7
7 – 6
1
8
1
4
6
3
2
7 – 5
6
6
8
4
3
4
3
2
7 – 2
9
6
Practice Use digit sums to check these:
1) 4
2
1
5
7
7
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6
8 –
8
2) 7
3
3
0
6
3
7
8 –
9
3) 9 3 8
1 8 7 –
7 4 1
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“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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“All from 9 and the Last from 10”
Practice
1) 1000 – 777 =
2) 10,000 – 4621 =
3) 100 – 58 =
4) 100,000 – 15032 =
5) 1000,000 – 123456 =
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6) 1000 – 730 =
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“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 – 0.763 =
Rs.100 – Rs. 53 =
Rs.500 – Rs. 77 =
Rs.1000 – Rs.835 =
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Subtraction
7312
3765–
905
447–
64332
28655–
2) 8 1 1
3) 7 6 4 2 3
Practice
1) 5 1 3 2 4
18745–
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345–
23567–
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Subtraction
908
456–
8393
2865–
64935
38657–
Split the number where a bar digit is
followed by a positive digit
Practice
1) 6 1 3 8 1
48747–
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2) 6 1 9
135–
3) 8 6 8 2 9
38564–
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Multiplying Numbers Near a Base
100
98 – 02
97 – 3
95 / 06
89 – 11
97 – 3
86 / 33
89 – 11
89 – 11
78 / 21 = 79/21
1
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Practice: Multiply
89
96
97
94
91
95
99
88
98
98
88
88
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Proofs
100
89 – 11
97 – 3
86 / 33
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Arithmetic proof:
89 × 97 = 89×100 – 89×3
= 8900 – 100×3 + 11×3
= 8900 – 300 + 11×3
= 8600 + 33
= 8633
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Proofs
100
89 – 11
97 – 3
86 / 33
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Geometric proof:
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Proofs
100
89 – 11
97 – 3
86 / 33
(x – a)(x – b) = x(x – a – b) + ab
Algebraic proof:
(100 – a)(100 – b) = 100(100 – a – b) + ab
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Mentally
89 – 11
97 – 3
86 / 33
96 x 93
97 x 94
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Numbers Over 100
103 + 03
104 + 4
107 / 12
123 + 23
103 + 3
126 / 69
102 + 02
103 + 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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One Number Under and One Number Over 100
105 + 5
97 – 3
102 / 15
= 101 / 85
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104 + 4
93 – 7
97 / 28
= 96 / 72
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Proportionately
204 x 107
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48 x 97
98 x 206
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Proportionately
200
189 –11
197 – 3
2x 186 / 33
= 372 / 33
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300
304
307
3x 311
= 933
+4
+7
/ 28
/ 28
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Practice: Multiply Using the Base Method
201
234
307
308
198
188
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Base Multiplication
8–2
7–3
5/ 6
6–4
6–4
2/ 6 =3/6
12 + 2
13 + 3
15 / 6
1
Practice Multiply:
8
8
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12
14
13
14
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Proportionately
20
21 + 1
22 + 2
2x 23 / 2
= 46 / 2
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30
28 – 2
27 – 3
3x 25 / 6
= 75 / 6
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Practice: Multiply Using the Base Method
41
42
33
32
51
56
19
18
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Numbers Near Large Bases
879
x 997
10003
x 12331
69978
x 99997
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“Vertically and Crosswise”
2
4
3
1 ×
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3
4
3
4 ×
6
5
7
2 ×
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Practice Multiply:
5
3
3
2 ×
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7
7
2
3 ×
4
4
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
3 x 8 = 24
3 1 ×
9 9 2
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24 ÷ 8 = 3
?
8 ×
2 4
4 2 ×
2 3 5 2
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Find the Missing Number
5 4 ×
3 4 5 6
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7 3 ×
2 3 3 7
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Comparison with Conventional Methods
Multiplication
Division
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Vedic
6 7
5 2 ×
52 ×
3484
Conventional
6 7
5 2 ×
52) 3 4 8 4
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Algebra
2x + 3
3x + 5 ×
2
6x + 19x + 15
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3x – 2
x + 7 ×
2
3x + 19x – 14
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Algebraic Division
3x + 5 ×
2
6x + 19x + 15
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Extending to 3-figure Numbers
104 x 53
10
5
112 x 113
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4
3 ×
11
2
11
3 ×
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Answers 2-Figures at a Time
113 x 203 11 3
20 3 ×
1202 x 1103
113 x 203 1 13
2 03 ×
12 02
11 03 ×
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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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5 0 4
4 1 3
3 2 1
5 2 3
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Practice Multiply:
4 3 2
5 1 3 ×
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3 4 5
2 0 7 ×
4 4 4
4 4 4 ×
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Extending the Pattern
3-figure numbers
4-figure numbers
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4-Figure Numbers
2 4 3 2
3 5 1 3 ×
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3 4 5 2
2 0 3 4 ×
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Practice Multiply:
4 1 3 2
5 4 1 3 ×
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4 2 4 4
4 3 4 4 ×
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“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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Adding/subtracting Fractions
2 1
 
3 5
1 3
 
4 7
2 1
 
3 5
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Adding/subtracting Fractions
3 5
 
4 6
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7 11


18 30
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Practice: Find
3 2
 
7 5
4 2
 
5 3
5 1
 
6 9
11 7


15 25
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Three Fractions
1 1 2
  
2 3 5
1 2 4
  
3 5 7
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Greatest or Least
Which fraction is greater?
5
7
or
7
10
Which fraction is the greatest?
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3 4 5
, ,
8 9 11
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Unifying the Four Operations
Addition
Subtraction
Multiplication
Division
4 1
+
5
3
4
1
–
5
3
4
1
×
5
3
4
1
÷
5
3
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Algebraic Fractions
2x x
8x + 3x
11x
+ =
=
12
12
3 4
1
2
x + 3 + 2  x + 2
3x + 7
+
=
=
x +2
x +3
 x + 2 x + 3 
 x + 2  x + 3 
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Equation of a Line through Two Given Points
Find the equation of the line through the points (7,4) and (5,1).
l (7,4)
l (5,1)
O
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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)
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2) (8,5), (-3,1)
3) (5,0), (-2,-5)
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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.
l (5,7)
O
2x + 3y = 5
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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 = 3
2) (-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.
l (3,1)
O
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 = 1
2) (-3,4), 5x – 2y = 3
3) (0,-1), 4y + x = 2
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Dividing by 19
1
19
11
19
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Right to Left / 9-Point Circle
.
.
1 = 0. 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
1
1
1 1 1 1
1
1 1
19
1
19
9,0
1
8
7
2
3
6
5
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4
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Divisor Ending in 9
17
29
15
39
29
49
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A Short Cut
.
= 0.1 0 5 12 6 3 11 15 17 18
1
19
11
19
.
=0.1 51 718 91 4 71316 8
.
4 2 1 10 5 12 6 31 1
9 1 4 7 13 16 8 4 2 1
17
29
.
.
=0.2 51 8 6 2202628191615 5 21 71 2
.
4 11 2327 9 3 110131 42 4 8221 7
6
39
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Proportionately
4
7
8
13
4
33
7
11
5
23
3
17
(to 5 decimal places)
(to 5 decimal places)
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COMPOUND ANGLES
4
5
If sinA = (A is obtuse) and cosB = (B is acute)
5
13
find tan(A-B).
A
-3 4 5
B
5 12 13
A-B 33 56 65
56
Tan(A-B) =
33
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TRIGONOMETRIC EQUATIONS
Find the general solution of the equation 2sinx - cosx = 2
2sinx  cosx  Rsinxcos   Rcosxsin   Rsin(x  ).
If 2  Rcos  and 1  Rsin ,i.e.R2  22  12 giving R  5 and
cos  
2
1
, sin  
giving   26 34'.
5
5
R
1

x

x-
x
x
c
-1
2
-4
0
s
2
1
3
5
1
+
5
5
2
 2sinx  cosx  5 sin(x  26 34').
Vedic method
Thus the equation 2sin x  cosx  2
becomes 5 sin(x  26 34')  2
2
 sin(x  26 34') 
 0.8945
5
 x  26 34'  180n  ( 1)n (63 26')
 x  180n  ( 1)n (63 26')  26 34'
from “Advanced Mathematics 1” by Celia, Nice & Elliott, Page 119.
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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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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.
b  b2  4ac
Then x =
.
2a
So x =
5
2
 5 
 4.3.  7 
2.3

5  25  84
6

5  109
6

5  10.44
6

15.44
5.44
or
6
6
a = 3,
D1 = 6x – 5 = 13
.
13 7 . 0 0 0
5
2
11
3. 4 3
x  2.57,  0.91.
Vedic method
 2.57 or  0.91.
from “General Mathematics” Book 3, by Channon & Smith, Page 195.
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INTEGRATION BY ‘PARTS’
Find  x 2sinxdx.
Let
{
v  x2

du
 sin x 
dx
dv
 2x
dx
u   cos x
du
dv
Then  v
dx  uv   u
dx
dx
dx
gives  x2sinxdx = (-cosx)(x2 ) -  (-cosx)(2x)dx
 x sinxdx = -x cosx +2xsinx +2cosx +K
2
2
Vedic method
 x2 cos x  2 x cos xdx
Thus, for  x cos xdx,
let
{
vx

du
 cos x 
dx
dv
1
dx
u  sin x
giving  x cos xdx  (sin x)(x)   (sin x)(1)dx
 xsinx  cosx  K
Hence  x2sinxdx = -x2cosx +2xsinx +2cosx +K
from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 313.
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SQUARE ROOT OF A COMPLEX NUMBER
Find 15 + 8i
Let 15  8i = a + bi, where a and b are real

15  8i  (a  bi)2
 a2  b2  2abi
Equating real and imaginary parts gives:
a2 – b2 = 15
and
2ab = 8
Using b 
(1)
(2)
15  8i    4  i
Vedic method
4
16
in (1) gives a2  2  15.
a
a
 a 4  15a2  16  0
 (a2 - 16)(a2 + 1) = 0
Thus a2 – 16 = 0 or a2 + 1 = 0
But a is real so a2 + 1 = 0 gives no suitable values
so a  4
Referring to equation (2) we have
Hence 15  8i = 4 + i or -4 - i
a
b
4
1
-4
-1
   4  i
from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 536.
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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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