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VEDIC MATHEMATICS : Various Numbers T. K. Prasad http://www.cs.wright.edu/~tkprasad Prasad Various Numbers 1 Numbers • Whole Numbers 1, 2, 3, … – Counting • Natural Numbers 0, 1, 2, 3, … – Positional number system motivated the introduction of 0 Prasad 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)? Prasad Various Numbers 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 Prasad Various Numbers 4 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 Prasad Various Numbers 5 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 Prasad Various Numbers 6 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 Prasad • • • • • • Various Numbers 1 41 1641 25641 225641 1025641 7 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 Prasad Various Numbers 1 168421 914713168421 … 05126311151718 914713168421 8 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. Prasad • Note how the digits cycle below ! Various Numbers • • • • • • 1/7 = 0.142857 2/7 = 0.285714 3/7 = 0.428571 4/7 = 0.571428 5/7 = 0.714285 6/7 = 0.857142 9 • 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 Prasad Various Numbers 10 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 Prasad 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): a 2 + b 2 = c2 a=1 b=2 c = √5 Prasad Various Numbers 12 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 Prasad Various Numbers 13 A Proof of Pythagoras’ Theorem • c2 = a2 + b2 a b b a a b b Prasad 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 Various Numbers 14 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 Prasad Various Numbers 15 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. Prasad Various Numbers 16 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= Prasad Various Numbers 17 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. Prasad Various Numbers 18 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! Prasad Various Numbers 19 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: Prasad 5 + 4 √-1 = 5 + 4 i Various Numbers 20