vedic mathematics : various numbers

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)?


Fractions and Rational Numbers

• 1/1, ½, ¾, 1/60, 1/365, …

• - 1/3, - 2/6, - 6/18, …

– Parts of a whole– Ratios– Percentages


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


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


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


• 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


• 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.


• 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


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


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 = √

History


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

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


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.


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 =


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.

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!


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


vedic mathematics : various numbers

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