Transcript VEDIC MATHEMATICS : Various Numbers

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
• 1/3 = 0. 3333
- recurs
• 22/7 = 3.142
• 1 / 400 = 0.0025
• 1/7 = 0.142857
--------- recurs
Terminating decimal
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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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 (9*1=9) and
keep multiplying by (1+1),
propagating carry.
– Terminate when 0 is
generated.
The reciprocal of 19 is
•
•
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•
•
0.052631578947368421
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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.
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• Note how the digits
cycle below !
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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 √2, √3, √5, 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
=> 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):
a 2 + b 2 = c2
a=1
b=2
c = √5
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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
a
b
b
a
a
b
b
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a
• 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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Bhaskara’s Proof of
Pythagoras’ Theorem (12th century AD)
• c2 = a2 + b2
a-b
a
b
• 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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Algebraic Numbers
• Numbers such as √2, √3, √5, 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…), √3 (=1.732...), √10(= 3.162 ...),
Golden ratio ( [[1+ √5]/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 √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.
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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 = √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:
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5 + 4 √-1 = 5 + 4 i
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