Historical Background “Veda” means “knowledge” Age of Vedic texts: from 300BC to several millennia BC Sections on medicine, ethics, metaphysics, psychology, architecture, music, astronomy, grammar and so on. And ‘ganita sutras’ Sri Bharati Krsna Tirthaji (1884 – 1960) Academy of Vedic Mathematics www.vedicmaths.org

Historical Background Vedic system was reconstructed between 1911 and 1918 Bharati Krsna wrote one introductory volume in 1958: “Vedic Mathematics” published in 1965 Shankaracharya of Govardhan Matha ,Puri Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Sixteen Sutras They cover all of mathematics, pure and applied These relate to natural mental functions This explains why Vedic Maths is so easy It works the way the mind naturally works Academy of Vedic Mathematics www.vedicmaths.org

Features of Vedic Mathematics Natural, powerful Coherent, unified Easy to do, easy to understand Flexible Creative, fun Academy of Vedic Mathematics www.vedicmaths.org

“By One More than the One Before” 0 1 2 3 4 5 6 7 8 9 . . . . 752 = (4½)2 = 43 x 47 = 88 x 46 = Academy of Vedic Mathematics www.vedicmaths.org

Digit Sums A Digit is a single-figure number: 0,1,2,3,4,5,6,7,8,9. Sum means add. So ‘Digit Sum’ means adding the digits in a number. 17 19 123 5030 38 7531 The Digit Sum is found by adding the digits in a number, and adding again if necessary Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org The Nine Point Circle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 . . . . 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 9 8 7 6 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Casting out Nines 59190 9899329 Adding or subtracting 9s to or from a number does not affect the digit sum Academy of Vedic Mathematics www.vedicmaths.org

Any group of figures in a number that add up to 9 can be "cast out“ Groups adding to 9 Any group of figures in a number that add up to 9 can be "cast out“ 7312 42134 61395 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Left to Right 123 We read, pronounce and write numbers from left to right The most important/significant figures in a number are at the left In the Vedic system we can add, subtract, multiply and divide from left to right Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Left to Right So:- Mental calculations are easier We can get at the most significant figures in a calculation first, not last We can combine the operations Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Addition 8 8 4 6 + 7 6 5 5 + 1 8 7 4 4 6 + Practice Add from left to right: 1) 7 7 2) 4 3 7 3) 6 5 4 4) 2 4 6 8 6 6 8 7 1 7 2 7 3 8 6 5 Academy of Vedic Mathematics www.vedicmaths.org

The Digit Sum Check for Addition Find 32 + 39 and check the answer using digit sums. 3 2 3 9 + 7 7 1 2 4 + 2 7 9 1 2 1 + 4 9 0 Academy of Vedic Mathematics www.vedicmaths.org

Mental Multiplication 7 8 3 x 6 7 7 x 8 8 6 x 6 3 8 x Practice Multiply from left to right: 1) 6 6 2) 5 3 3) 8 6 6 x 5 x 7 x 4) 6 4 5) 4 9 6 x 4 x 4 3 9 x 6 3 6 x 4 7 6 x 5 3 7 x 6 8 7 x Academy of Vedic Mathematics www.vedicmaths.org

Longer Multiplications 3 8 5 3 x 6 2 7 4 x 5 3 4 7 7 x Practice Multiply from left to right: 1) 7 2 7 2) 4 3 2 3) 6 5 4 4) 2 4 6 8 6 x 7 x 3 x 5 x Academy of Vedic Mathematics www.vedicmaths.org

The Digit Sum Check for Multiplication 3 8 3 x 6 2 7 4 x Academy of Vedic Mathematics www.vedicmaths.org

Subtraction from left to right 6 3 2 7 – 8 5 3 7 – 5 6 5 1 7 7 – 6 3 4 2 5 2 – 7 5 6 3 3 2 7 8 – 5 6 5 6 1 7 5 1 – 7 3 3 4 2 3 8 1 – Academy of Vedic Mathematics www.vedicmaths.org

Checking Subtractions Method 1: Add the second and third rows to get the first row. 8 3 – 5 5 6 5 1 7 7 – 3 8 8 Practice Which is/are wrong? 1) 7 6 3 2 7 8 – 4 8 5 2) 8 0 5 4 5 8 – 3 5 7 3) 7 8 3 1 4 7 – 6 3 6 Academy of Vedic Mathematics www.vedicmaths.org

Checking Subtractions 1 2 3 4 5 9 8 7 6 Method 2: Use the digit sums. 5 6 5 1 7 7 – 3 8 8 7 7 1 3 3 4 7 – 3 6 6 2 8 3 4 7 – 3 6 2 6 5 2 1 6 9 Practice Use digit sums to check these: 1) 4 5 6 2 7 8 – 1 7 8 2) 7 0 7 3 6 8 – 3 3 9 3) 9 3 8 1 8 7 – 7 4 1 Academy of Vedic Mathematics www.vedicmaths.org

“All from 9 and the Last from 10” 8 7 6 5 4 9 1 2 4 4 6 1 The formula All From 9 and the Last From 10 subtracts numbers from the next highest base number. 1000 – 864 = 1000 – 307 = 10000 – 6523 = 100 – 63 = Academy of Vedic Mathematics www.vedicmaths.org

“All from 9 and the Last from 10” Practice 1) 1000 – 777 = 2) 10,000 – 4621 = 3) 100 – 58 = 4) 100,000 – 15032 = 5) 1000,000 – 123456 = 6) 1000 – 730 = Academy of Vedic Mathematics www.vedicmaths.org

“All from 9 and the Last from 10” 1000 – 72 = 100,000 – 503 = 700 – 66 = 4000 – 333 = Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Variations 5000 – 47 = 40,000 – 33 = 1 – 0.763 = Rs.100 – Rs. 53 = Rs.500 – Rs. 77 = Rs.1000 – Rs.835 = Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Subtraction 7 3 1 2 3 7 6 5 – 9 0 5 4 4 7 – 6 4 3 3 2 2 8 6 5 5 – Practice 1) 5 1 3 2 4 1 8 7 4 5 – 2) 8 1 1 3 4 5 – 3) 7 6 4 2 3 2 3 5 6 7 – Academy of Vedic Mathematics www.vedicmaths.org

Split the number where a bar digit is followed by a positive digit Subtraction 9 0 8 4 5 6 – 8 3 9 3 2 8 6 5 – 6 4 9 3 5 3 8 6 5 7 – Split the number where a bar digit is followed by a positive digit Practice 1) 6 1 3 8 1 4 8 7 4 7 – 2) 6 1 9 1 3 5 – 3) 8 6 8 2 9 3 8 5 6 4 – Academy of Vedic Mathematics www.vedicmaths.org

Multiplying Numbers Near a Base 100 89 97 – 11 98 97 – 02 – 3 – 3 86 / 33 95 / 06 89 – 11 – 11 / 21 1 78 = 79/21 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice: Multiply 8 9 9 6 9 7 9 4 9 1 9 5 9 9 8 8 9 8 8 8 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proofs 100 89 97 – 11 Arithmetic proof: 89 × 97 = 89×100 – 89×3 = 8900 – 100×3 + 11×3 = 8900 – 300 + 11×3 = 8600 + 33 = 8633 – 3 86 / 33 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proofs 100 89 97 – 11 Geometric proof: – 3 86 / 33 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proofs 100 89 97 – 11 (x – a)(x – b) = x(x – a – b) + ab – 3 86 / 33 Algebraic proof: (100 – a)(100 – b) = 100(100 – a – b) + ab Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Mentally 89 97 – 11 – 3 86 / 33 96 x 93 97 x 94 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Numbers Over 100 103 104 + 03 123 103 + 23 + 4 + 3 107 / 12 126 / 69 102 103 + 02 + 3 105 / 06 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice: Multiply 106 x 107 104 x 108 102 x 103 111 x 112 Academy of Vedic Mathematics www.vedicmaths.org

One Number Under and One Number Over 100 105 97 + 5 104 93 + 4 – 3 – 7 = 101 / 85 = 96 / 72 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proportionately 204 x 107 48 x 97 98 x 206 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proportionately 200 189 197 –11 304 307 + 4 300 – 3 + 7 2x 186 / 33 3x 311 / 28 = 372 / 33 = 933 / 28 Academy of Vedic Mathematics www.vedicmaths.org

Practice: Multiply Using the Base Method 307 308 2 0 1 2 3 4 1 9 8 1 8 8 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Base Multiplication – 2 6 – 4 12 13 + 2 8 7 – 3 – 4 + 3 5 / 6 2 / 6 = 3 / 6 15 / 6 1 Practice Multiply: 8 12 14 13 14 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proportionately 20 30 21 22 + 1 28 27 – 2 + 2 – 3 2x 23 / 2 3x 25 / 6 = 46 / 2 = 75 / 6 Academy of Vedic Mathematics www.vedicmaths.org

Practice: Multiply Using the Base Method 4 1 4 2 3 3 3 2 5 1 5 6 1 9 1 8 Academy of Vedic Mathematics www.vedicmaths.org

Numbers Near Large Bases 879 x 997 10003 x 12331 69978 x 99997 Academy of Vedic Mathematics www.vedicmaths.org

“Vertically and Crosswise” 2 3 4 1 × 3 3 4 4 × 6 7 5 2 × Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice Multiply: 5 3 3 2 × 7 2 7 3 × 4 4 4 4 × Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org The Vedic Method Can multiply numbers of any size in one line The simple pattern makes it easy to remember Easy to explain Can multiply from left to right or from right to left Reversible Algebraic products can be done the same way Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Division ? 8 × 2 4 3 x 8 = 24 24 ÷ 8 = 3 3 1 × 9 9 2 4 2 × 2 3 5 2 Academy of Vedic Mathematics www.vedicmaths.org

Find the Missing Number 5 4 × 3 4 5 6 7 3 × 2 3 3 7 Academy of Vedic Mathematics www.vedicmaths.org

Comparison with Conventional Methods Vedic Conventional 6 7 5 2 × 6 7 5 2 × Multiplication 5 2 × 3 4 8 4 52) 3 4 8 4 Division Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Algebra 2x + 3 3x + 5 × 3x – 2 x + 7 × + 19x + 15 + 19x – 14 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Algebraic Division 3x + 5 × + 19x + 15 Academy of Vedic Mathematics www.vedicmaths.org

Extending to 3-figure Numbers 10 4 5 3 × 104 x 53 11 2 11 3 × 112 x 113 Academy of Vedic Mathematics www.vedicmaths.org

Answers 2-Figures at a Time 11 3 20 3 × 1 13 2 03 × 113 x 203 113 x 203 12 02 11 03 × 1202 x 1103 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Bar Numbers 39 x 32 Bar numbers are extremely useful Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Extending the Pattern 5 0 4 3 2 1 4 1 3 5 2 3 504 x 321 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice Multiply: 4 3 2 5 1 3 × 3 4 5 2 0 7 × 4 4 4 4 4 4 × Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Extending the Pattern 3-figure numbers 4-figure numbers Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org 4-Figure Numbers 2 4 3 2 3 5 1 3 × 3 4 5 2 2 0 3 4 × Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice Multiply: 4 1 3 2 5 4 1 3 × 4 2 4 4 4 3 4 4 × Academy of Vedic Mathematics www.vedicmaths.org

“Vertically and Crosswise” Numbers of any size can be multiplied in one line Numbers of any size can be divided in one line The multiplication method simplifies for squaring numbers And this is reversible to do square roots (in one line) And quadratic and higher order equations Academy of Vedic Mathematics www.vedicmaths.org

Adding/subtracting Fractions Academy of Vedic Mathematics www.vedicmaths.org

Adding/subtracting Fractions Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Practice: Find Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Three Fractions Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Greatest or Least Which fraction is greater? Which fraction is the greatest? Academy of Vedic Mathematics www.vedicmaths.org

Unifying the Four Operations Addition Subtraction Multiplication Division + – × ÷ Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Algebraic Fractions Academy of Vedic Mathematics www.vedicmaths.org

Equation of a Line through Two Given Points Find the equation of the line through the points (7,4) and (5,1). O (5,1) (7,4) l Academy of Vedic Mathematics www.vedicmaths.org

Equation of a Line through Two given Points Find the equation of the line through the points (0,-4) and (-2,3). Practice 1 Find the equation of the line through the points:- 1) (4,9), (1,2) 2) (8,5), (-3,1) 3) (5,0), (-2,-5) Academy of Vedic Mathematics www.vedicmaths.org

Equation of a Line through a given Point and Parallel to a given Line Find the equation of the line through the point (5,7) and parallel to the line 2x + 3y = 5. O (5,7) l 2x + 3y = 5 Academy of Vedic Mathematics www.vedicmaths.org

Equation of a Line through a given Point and Parallel to a given Line Find the equation of the line through the point (-3,2) and parallel to the line x - 5y = 1. Practice 2 Find the equation of the line through the given point and parallel to the given line:- 1) (4,1), 2x + y = 3 2) (-2,5), 3x – 5y = 2 3) (0,-3), 5y + 2x = 2 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Equation of a Line through a given Point and Perpendicular to a given Line Find the equation of the line through the point (3,1) and perpendicular to the line 2x + 3y = 5. O (3,1) l 2x + 3y = 5 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Equation of a Line through a given Point and Perpendicular to a given Line Find the equation of the line through the point (2,5) and perpendicular to the line 3x - y = 2. Practice 3 Find the equation of the line through the given point and perpendicular to the given line:- 1) (7,2), 2x + y = 1 2) (-3,4), 5x – 2y = 3 3) (0,-1), 4y + x = 2 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Dividing by 19 Academy of Vedic Mathematics www.vedicmaths.org

Right to Left / 9-Point Circle 1 9,0 2 3 6 8 7 5 4 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Divisor Ending in 9 Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org A Short Cut Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org Proportionately (to 5 decimal places) (to 5 decimal places) Academy of Vedic Mathematics www.vedicmaths.org

Academy of Vedic Mathematics www.vedicmaths.org COMPOUND ANGLES If sinA = (A is obtuse) and cosB = (B is acute) find tan(A-B). A -3 4 5 B 5 12 13 A-B 33 56 65 Tan(A-B) = Academy of Vedic Mathematics www.vedicmaths.org

TRIGONOMETRIC EQUATIONS x c s 1 a -1 2 - x-a 2 1 + x -4 3 5 x 0 5 5 1 2 R a Vedic method from “Advanced Mathematics 1” by Celia, Nice & Elliott, Page 119. Academy of Vedic Mathematics www.vedicmaths.org

APPLICATIONS OF TRIPLES Trigonometry Coordinate geometry (2 & 3 dimensions) Transformations (2 & 3 dimensions) Simple Harmonic Motion Projectiles Complex numbers Hyperbolic functions Conics Astronomy Academy of Vedic Mathematics www.vedicmaths.org

SOLUTION OF A QUADRATIC EQUATION Find, correct to 2 decimal places, the roots of the equation 3x2 – 5x – 7 = 0. Comparing 3x2 – 5x – 7 = 0 with ax2 + bx + c = 0; a = 3, b = -5, c = -7. from “General Mathematics” Book 3, by Channon & Smith, Page 195. a = 3, D1 = 6x – 5 = 13 13 7 . 0 0 0 Vedic method Academy of Vedic Mathematics www.vedicmaths.org

INTEGRATION BY ‘PARTS’ Let let from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 313. { Vedic method Academy of Vedic Mathematics www.vedicmaths.org

SQUARE ROOT OF A COMPLEX NUMBER Equating real and imaginary parts gives: a2 – b2 = 15 (1) and 2ab = 8 (2) Vedic method Thus a2 – 16 = 0 or a2 + 1 = 0 But a is real so a2 + 1 = 0 gives no suitable values Referring to equation (2) we have a b 4 1 -4 -1 from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 536. Academy of Vedic Mathematics www.vedicmaths.org

Features of Vedic Mathematics Works the way the mind works More unified Can be a mental system which develops memory and mental agility Very flexible: choice of methods left to right or right to left Encourages creativity and innovation Academy of Vedic Mathematics www.vedicmaths.org