prasadvarious numbers1 vedic mathematics : various numbers t. k. prasad tkprasad
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Prasad Various Numbers 1
VEDIC MATHEMATICS : Various Numbers
T. K. Prasadhttp://www.cs.wright.edu/~tkprasad
Numbers
• Whole Numbers 1, 2, 3, …– Counting
• Natural Numbers0, 1, 2, 3, …– Positional number system motivated the
introduction of 0Prasad Various Numbers 2
• Integers
…, -3, -2, -1, 0, 1, 2, 3, …
• Negative numbers were motivated by solutions to linear equations.
• What is x if (2 * x + 7 = 3)?
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Fractions and Rational Numbers
• 1/1, ½, ¾, 1/60, 1/365, …
• - 1/3, - 2/6, - 6/18, …
– Parts of a whole– Ratios– Percentages
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Rational Number
• A rational number is a number that can be expressed as a ratio of two integers (p / q) such that (q =/= 0) and (p and q do not have any common factors other than 1 or -1).
– Decimal representation expresses a fraction as sum of parts of a sequence of powers of 10.
0.125 = 1/10 + 2/100 + 5/1000
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Rationals in decimal system
• - ½ = - 0.5
• 22/7 = 3.142
• 1 / 400 = 0.0025
Terminating decimal
• 1/3 = 0. 3333
- recurs
• 1/7 = 0.142857
--------- recurs
Recurring decimal
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Computing Specific Reciprocals :
The Vedic Way• 1/39
• The decimal representation is recurring.– Start from the rightmost
digit with 1 (9*1=9) and keep multiplying by (3+1), propagating carry.
– Terminate when 0 (with carry 1) is generated.
• The reciprocal of 39 is
0.025641
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• 1
• 41
• 1641
• 25641
• 225641
• 1025641
Computing Reciprocal of a Prime :
The Vedic Way• 1/19
• The decimal representation is recurring.– Start from the rightmost
digit with 1 (9*1=9) and keep multiplying by (1+1), propagating carry.
– Terminate when 0 is generated.
• The reciprocal of 19 is
0.052631578947368421
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• 1
• 168421
• 914713168421
• …
• 05126311151718 914713168421
Computing Recurring Decimals
• The Vedic way of computing reciprocals is very compact but I have not found a general rule with universal applicability simpler than long division.
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• Note how the digits cycle below !
• 1/7 = 0.142857
• 2/7 = 0.285714
• 3/7 = 0.428571
• 4/7 = 0.571428
• 5/7 = 0.714285
• 6/7 = 0.857142
• Rationals are dense.– Between any pair of rationals, there exists
another rational.
• Proof: If r1 and r2 are rationals, then so is their “midpoint”/ “average” .
(r1 + r2) / 2
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Irrational Numbers• Numbers such as √2, √√etc are not
rational.
• Proof: Assume that √2 is rational.
• Then, √2 = p/q, where p and q do not have any common factors (other than 1).
• 2 = p2 / q2 => 2 * q2 = p2
• 2 divides p => 2 * q2 = (2 * r)2
• 2 divides q => ContradictionPrasad Various Numbers 11
Pythagoras’ Theorem
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The Pythagorean Theorem states that, in a right angled triangle, the sum of the squares on the two smaller sides (a,b) is equal to the square on the hypotenuse (c): a2 + b2 = c2
a = 1b = 2c = √
A Proof of Pythagoras’ Theorem
• c2 = a2 + b2
• Construct the “green” square of side (a + b), and form the “yellow” quadrilateral.
• All the four triangles are congruent by side-angle-side property. And the “yellow” figure is a square because the inner angles are 900.
• c2 + 4(ab/2) = (a + b)2
• c2 = a2 + b2
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a b
c
b
a
b
b
a
a
Bhaskara’s Proof of Pythagoras’ Theorem (12th century AD)
• c2 = a2 + b2
• Construct the “pink” square of side c, using the four congruent right triangles. (Check that the last triangle fits snugly in.)
• The “yellow” quadrilateral is a square of side (a-b).
• c2 = 4(ab/2) + (a - b)2
• c2 = a2 + b2
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cb
a-b
a
Algebraic Numbers• Numbers such as √2, √√etc are
algebraic because they can arise as a solution to an algebraic equation.
x * x = 2
x * x = 3
• Observe that even though rational numbers are dense, there are “irrational” gaps on the number line.
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Irrational Numbers• Algebraic Numbers
√2 (=1.4142…), √√Golden ratio
( [[1+ √=1.61803399), etc
• Transcendental Numbers (=3.1415926 …) [pi],
e (=2.71327178 …) [Natural Base], etc = Ratio of circumference of a circle to its diameter
e =
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History
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Baudhayana (800 B.C.) gave an approximation to the value of √2 as:
and an approximate approach to finding a circle whose area is the same as that of a square.
Manava (700 B.C.) gave an approximation to the value of as 3.125.
Non-constructive Proof• Show that there are two irrational numbers a and b
such that ab is rational.• Proof: Take a = b = √2. • Case 1: If √2√2 is rational, then done.• Case 2: Otherwise, take a to be the irrational
number √2√2 and b = √2. • Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is
rational.• Note that, in this proof, we still do not yet know
which number (√2√2) or (√2√2)√2 is rational!
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Complex Numbers
• Real numbers• Rational numbers
• Irrational numbers
• Imaginary numbers• Numbers such as √-1, etc are not real because there does not
exist a real number which when squared yields (-1).
x * x = -1
• Numbers such as √-1 are called imaginary numbers.
• Notation: 5 + 4 √-1 = 5 + 4 i
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