PrasadVarious Numbers1 VEDIC MATHEMATICS : Various Numbers T. K. Prasad

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

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

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

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 = 1/10 + 2/ /1000 PrasadVarious Numbers5

Rationals in decimal system - ½ = /7 = / 400 = Terminating decimal 1/3 = recurs 1/7 = recurs Recurring decimal PrasadVarious Numbers6

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 PrasadVarious Numbers

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 PrasadVarious Numbers …

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 = /7 = /7 = /7 = /7 = /7 =

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

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

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 = √ 

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

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 c 2 + 4(ab/2) = (a + b) 2 c 2 = a 2 + b 2 PrasadVarious Numbers14 ab c b a b b a a

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

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

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

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

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

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: √-1 = i PrasadVarious Numbers20