1. PrasadVarious Numbers1 VEDIC MATHEMATICS : Various Numbers T. K. Prasad http://www.cs.wright.edu/~tkprasad

2. Numbers Whole Numbers 1, 2, 3, … –Counting Natural Numbers 0, 1, 2, 3, … –Positional number system motivated the introduction of 0 PrasadVarious Numbers2

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)? PrasadVarious Numbers3

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 PrasadVarious Numbers4

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 PrasadVarious Numbers5

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 PrasadVarious Numbers6

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 PrasadVarious Numbers7 1 41 1 641 2 5641 2 25641 1 025641

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 PrasadVarious Numbers8 1 1 68421 9 1 47 1 3 1 68421 … 05 1 263 1 1 1 5 1 7 1 8 9 1 47 1 3 1 68421

9. 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. PrasadVarious Numbers9 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

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 PrasadVarious Numbers10

11. 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 = p 2 / q 2 => 2 * q 2 = p 2 2 divides p => 2 * q 2 = (2 * r) 2 2 divides q => Contradiction PrasadVarious Numbers11

12. Pythagoras’ Theorem PrasadVarious Numbers12 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 = c 2 a = 1 b = 2 c = √ 

13. History PrasadVarious Numbers13 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 (12 th Century AD) : Proof given later

14. A Proof of Pythagoras’ Theorem c 2 = a 2 + b 2 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 90 0. c 2 + 4(ab/2) = (a + b) 2 c 2 = a 2 + b 2 PrasadVarious Numbers14 ab c b a b b a a

15. Bhaskara’s Proof of Pythagoras’ Theorem (12 th century AD) c 2 = a 2 + b 2 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). c 2 = 4(ab/2) + (a - b) 2 c 2 = a 2 + b 2 PrasadVarious Numbers15 c b a-b a

16. 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. PrasadVarious Numbers16

17. Irrational Numbers Algebraic Numbers √2 (=1.4142…), √  √  Golden ratio ( [[1+ √  =1.61803399), etc Transcendental Numbers P(=3.1415926 …) [pi], e (=2.71327178 …) [Natural Base], etc  = Ratio of circumference of a circle to its diameter e = PrasadVarious Numbers17

18. History PrasadVarious Numbers18 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.

19. Non-constructive Proof Show that there are two irrational numbers a and b such that a b 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 a b = (√2 √2 ) √2 = √2 √2·√2 = √2 2 = 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! PrasadVarious Numbers19

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 PrasadVarious Numbers20