VEDIC MATHEMATICS : Various Numbers

1. VEDIC MATHEMATICS : Various Numbers T. K. Prasad http://www.cs.wright.edu/~tkprasad Various Numbers

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

3. 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)? Various Numbers

4. Fractions and Rational Numbers • 1/1, ½, ¾, 1/60, 1/365, … • - 1/3, - 2/6, - 6/18, … • Parts of a whole • Ratios • Percentages Various Numbers

5. 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 Various Numbers

6. 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 Various Numbers

7. 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 Various Numbers

8. 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 Various Numbers

9. Computing Recurring Decimals • 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 • The Vedic way of computing reciprocals is very compact but I have not found a general rule with universal applicability simpler than long division. Various Numbers

10. 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 Various Numbers

11. 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 Various Numbers

12. 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 = √5 Various Numbers

13. 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 Various Numbers

14. 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 a c a b b a Various Numbers

15. 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 a-b a c b Various Numbers

16. 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. Various Numbers

17. 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 = Various Numbers

18. 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. Various Numbers

19. Non-constructive Proof • Show that there are two irrational numbers a and b such that abis 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! Various Numbers

20. 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 Various Numbers