Download Vedic Mathematics

10/19/11
Vedic
Mathematics
Contributions of Hindu
Mathematicians of
Ancient Times!
What are Vedas?
•  Veda means wisdom, knowledge or vision
•  The laws of the four Vedas regulate the social,
legal, domestic and religious customs of the
Hindus to present day
•  As the ancient Hindus seldom kept any historical
record of their religious, literary and political
realization, it is difficult to determine the period of
the Vedas with precision.
•  Compositions of the Vedas were handed down
through generations by the word of mouth.
•  The Vedas were mainly compiled around (c. 1500
BC)
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What is Vedic Mathematics?
•  Ancient system of Mathematics which was rediscovered from
the Vedas.
•  It is a unique technique of calculations based on simple
principles and rules, with which any mathematical problem
involving algebra, arithmetic, geometry and trigonometry can be
solved mentally.
•  It is based on 16 simple mathematical formulae or Sutras (and
13 sub-formulae or sub sutras) from the Uup-Veda Sthapatya
Veda of Atharva Vedas as discovered by Sri Bharati Krishna
Tirthaji. (1884 - 1960)
What does Mathematics have to do with Hinduism?
Just as the feathers of a peacock and the jewel-stone of a snake are placed at
the highest point of the body, so is the position of mathematics the highest
amongst all branches of the Vedas and the Shastras.
The mantra at the end of the annahoma invokes powers of
ten from a hundred to a trillion.
Hail to śata (102), sahasra (103), ayuta (104), niyuta (105),
prayuta (106), arbuda (107), nyarbuda (108), samudra (109),
madhya (1010), anta (1011), parārdha (1012),
the dawn (uśas), the twilight (vyuṣṭi), the one
which is going to rise (udeṣyat), the one
which is rising (udyat), the one which has just
risen (udita), the heaven (svarga), the world
(loka) and all.
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The sutras list the rules of the sacrificial fire altars:
Most mathematical problems considered spring from
a single theological requirement:
a) When constructing fire altars of different
shapes, the altars should occupy the same area.
b) The altars were required to be constructed of five layers of burnt
brick, with the further condition that each layer consist of 200
bricks and that no two adjacent layers have congruent
arrangements of bricks.
“The diagonal rope of an oblong rectangle produces
both which the flank and the horizontal ropes
produce separately”
16 Sutras (formulae)
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1. Ekādhikena Pūrvena means:“By one more than the
previous one”.
It is used in 'Squaring of numbers ending in 5'
Vedic Method
65 X 65 = 4225
(multiply the previous digit
6 by one more than itself 7.
Then write 25 )
Conventional Method:
65
X65
325
+3 9 0 0
4225
2. Nikhilam Navatas' Chandmam Dasatah
“all from 9 and the last from 10”
This formula can be very effectively applied in multiplication of
numbers, which are nearer to bases like 10, 100, 1000 i.e., to the
powers of 10
Conventional Method
97
103
X94
X105
388
515
+8 7 3 0
0000
9, 1 1 8
+1 0 3 0 0
1 0, 8 1
5
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3. Urdhva Tiryagbhyam
“Vertically and cross wise”
75
X 32
Step 1: 5×2=10, write down
0 and carry 1
Step 2: 7×2 + 5×3 =
14+15=29, add to it
previous carry over value
1, so we have 30, now
write down 0 and carry 3
Step 3: 7×3=21, add
previous carry over value
of 3 to get 24, write it
down.
So we have 2400 as the
answer.
Multiplication of Numbers with 3 digits
103 X 105
Step 1: 3 x 5 = 15
Retain 5 carry over 1
Step 2: 5 x 0 + 0 x 3 = 0
Add carried over 1 and retain
Step 3: 5 x 1 + 0 x 0 + 1 x 3 = 8
Retain 8
Step 4: 0 x 1 + 1 x 0 = 0
Retain 0
Step 5: 1 x 1 = 1
Combining the numbers we get:
10815
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Example:
45x – 23y = 113
23x – 45y = 91
Gunita Samuccayah – Samuccayah Gunitah
'the product of the sum'
This is used to factorize homogeneous equations of second
degree in three variables x, y and z.
Example :
3x 2 + 7xy + 2y 2+ 11xz + 7yz + 6z 2
Eliminate z and retain x, y ;
factorize 3x 2 + 7xy + 2y 2 =
(3x + y) (x + 2y)
Eliminate y and retain x, z;
factorize 3x 2 + 11xz + 6z 2 =
(3x + 2z) (x + 3z)
Explanation
Eliminate z by putting z = 0 and retain
x
and y and factorize the obtained
quadratic in x and y
Similarly eliminate y and retain x and
z
and factorize the quadratic in x and z.
Fill the gaps thus
With these two sets of factors, fill in
the factors are:
the
(3x + y + 2z) (x + 2y + 3z) gaps caused by the elimination
process of
z and y respectively. This gives
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Some Famous Indian Mathematicians and their
contributions.
Aryabhatta (475 A.D. -550 A.D.)
- laid the foundations of algebra
- gave the value of pi as 3.1416, claiming, for the first
time, that it was an approximation. He gave it in the
form that the approximate circumference of a circle of
diameter 20000 is 62832.
- gave methods for extracting square roots, summing
arithmetic series, solving indeterminate equations of
the type ax - by = c, and also gave what later
came to be known as the table of Sines.
Brahmagupta (598 A.D. -665 A.D.)
- introduced negative numbers and operations of zero in
arithmetic
- formulated the rule of three and proposed rules for the solution
of quadratic and simultaneous equations.
- the formula for the area of a cyclic quadrilateral as where s is
the semi perimeter.
- was the first mathematician to treat algebra and arithmetic as
two different branches of mathematics.
- gave the solution of the indeterminate equation
Nx² + 1 = y².
- He is also the founder of Numerical Analysis.
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Bhaskara (1114 A.D. -1185 A.D.)/ Bhaskaracharaya
- was the first to declare that any number divided by zero is infinity
and that the sum of any number and infinity is also infinity
- wrote Siddhanta Siromani (1150 A.D.). It is divided into four
sections -Lilavati (a book on arithmetic), Bijaganita (algebra),
Goladhayaya (chapter on sphere -celestial globe), and Grahaganita
(mathematics of the planets)
- introduced the cyclic method to solve algebraic equations. 6
centuries later, European mathematicians like Galois, Euler and
Lagrange rediscovered this method and called it "inverse cyclic".
- Bhaskara's works developed into differential calculus. He
gave an example of what is now called "differential
coefficient" and the basic idea of what is now called
"Rolle's theorem". Unfortunately, later Indian
mathematicians did not take any notice of this. 5
centuries later, Newton and Leibniz developed this
subject
Srinivasa Aaiyangar Ramanujan (1887-1920)
- best known for his work on hypergeometric series and
continued fractions.
- In January 1913 Ramanujan sent some of his work to G. H.
Hardy, Cayley lecturer in mathematics at Cambridge.
Hardy noticed that whereas Ramanujan had rediscovered,
and gone far beyond, some of the latest conclusions of
Western mathematicians, he was completely ignorant of
some of the most fundamental areas.
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References
Kandasamy, W. B. V., & Smarandache, F. (2006). Vedic
mathematics - ‘vedic’ or ‘mathematics’: a fuzzy & neutrosophic
analysis. Ann Arbor, MI: Automaton. Retrieved from
http://fs.gallup.unm.edu//vedicmath.pdf
Das, D. R. N. (2008, January 17). Ancient Hindu Civilisation and
Mathematics. Retrieved from http://www.scribd.com/doc/
1032298/Ancient-Hindu-Civilisation-and-Mathematics
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