VEDIC MATHEMATICS : Various Numbers

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VEDIC MATHEMATICS Various Numbers

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

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Numbers

• Whole Numbers
• 1, 2, 3,
• Counting
• Natural Numbers
• 0, 1, 2, 3,
• Positional number system motivated the
  introduction of 0

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• 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 (919) and
  keep multiplying by (31), propagating carry.
• Terminate when 0 (with carry 1) is generated.
• The reciprocal of 39 is 0.025641

• 1
• 41
• 1641
• 25641
• 225641
• 1025641

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Computing Reciprocal of a Prime The Vedic Way

• 1/19
• The decimal representation is recurring.
• Start from the rightmost digit with 1 (919) and
  keep multiplying by (11), propagating carry.
• Terminate when 0 is generated.
• The reciprocal of 19 is 0.052631578947368421

• 1
• 168421
• 914713168421
• 05126311151718 914713168421

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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.

• 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

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• 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 v2, v3, v5, etc are not rational.
• Proof Assume that v2 is rational.
• Then, v2 p/q, where p and q do not have any
  common factors (other than 1).
• 2 p2 / q2 gt 2 q2 p2
• 2 divides p gt 2 q2 (2 r)2
• 2 divides q gt Contradiction

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Pythagoras Theorem
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 1 b 2 c v5
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History

• Pythagoras (500 B.C.)
• Euclid (300 B.C)
• Proof in Elements
• Book 1 Proposition 47
• Baudhayana (800 B.C.)
• Used in Sulabh Sutras
• (appendix to Vedas).
• Bhaskara (12th Century AD)
• Proof given later

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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

a
b
b
c
a
b
a
b
a
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Bhaskaras 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

a-b
a
c
b
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Algebraic Numbers

• Numbers such as v2, v3, v5, 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
• v2 (1.4142), v3 (1.732...), v10( 3.162 ...),
  Golden ratio ( 1 v5/2 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
Baudhayana (800 B.C.) gave an approximation to
the value of v2 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.
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Non-constructive Proof

• Show that there are two irrational numbers a and
  b such that ab is rational.
• Proof Take a b v2.
• Case 1 If v2v2 is rational, then done.
• Case 2 Otherwise, take a to be the irrational
  number v2v2 and b v2.
• Then ab (v2v2)v2 v2v2v2 v22 2 which is
  rational.
• Note that, in this proof, we still do not yet
  know which number (v2v2) or (v2v2)v2 is
  rational!

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Complex Numbers

• Real numbers
• Rational numbers
• Irrational numbers
• Imaginary numbers
• Numbers such as v-1, etc are not real because
  there does not exist a real number which when
  squared yields (-1).
• x x -1
• Numbers such as v-1 are called imaginary numbers.
• Notation 5 4 v-1 5 4 i