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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 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 7 752 = (4½)2 = 43 x 47 = 88 x 46 = Academy of Vedic Mathematics 8 9 . . . . 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 38 5030 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 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, 10 9 9 1 8 8 2 7 3 6 5 Academy of Vedic Mathematics 4 1 7 2 3 6 5 4 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 Groups adding to 9 Any group of figures in a number that add up to 9 can be "cast out“ 7312 Academy of Vedic Mathematics 42134 61395 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 Left to Right So:1. Mental calculations are easier 2. We can get at the most significant figures in a calculation first, not last 3. We can combine the operations 2 2 314 + 432 = 534.06.... 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 66 2) 4 3 7 871 Academy of Vedic Mathematics 3) 6 5 4 727 4) 2 4 6 8 3865 www.vedicmaths.org The Digit Sum Check for Addition Find 32 + 39 and check the answer using digit sums. 32 39+ Academy of Vedic Mathematics 77 124 + 279 121 + 490 www.vedicmaths.org Mental Multiplication 7 8 3x 67 7x 88 6x 63 8x Practice Multiply from left to right: 1) 6 6 6x Academy of Vedic Mathematics 2) 5 3 5x 3) 8 6 7x 4) 6 4 6x 5) 4 9 4x 5 3 4 6 8 7 7x 9 6 www.vedicmaths.org Longer Multiplications 3 8 5 3x 627 4x 5347 7x Practice Multiply from left to right: 1) 7 2 7 6x 2) 4 3 2 7x Academy of Vedic Mathematics 3) 6 5 4 3x 4) 2 4 6 8 5x www.vedicmaths.org The Digit Sum Check for Multiplication 3 8 3x Academy of Vedic Mathematics 627 4x www.vedicmaths.org Subtraction from left to right 6 2 5 1 6 7 3 7 – 5 7 – 5 1 6 7 Academy of Vedic Mathematics 6 2 5 5 6 1 – 8 3 3 5 5 7 – 4 2 – 7 3 7 2 3 3 5 2 3 8 6 7 3 8 – 4 1 – www.vedicmaths.org Checking Subtractions Method 1: Add the second and third rows to get the first row. 8 5 6 5 3 – 1 7 7 – 5 3 8 8 Practice Which is/are wrong? 1) 7 2 4 6 7 8 Academy of Vedic Mathematics 3 8 – 5 2) 8 0 5 4 5 8 – 3 5 7 3) 7 8 3 1 4 7 – 6 3 6 www.vedicmaths.org 9 1 8 Checking Subtractions 2 7 3 6 5 Method 2: Use the digit sums. 5 1 3 6 7 8 7 3 3 5 7 7 – 6 1 8 1 4 6 3 2 7 – 5 6 6 8 4 3 4 3 2 7 – 2 9 6 Practice Use digit sums to check these: 1) 4 2 1 5 7 7 Academy of Vedic Mathematics 6 8 – 8 2) 7 3 3 0 6 3 7 8 – 9 3) 9 3 8 1 8 7 – 7 4 1 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 = Academy of Vedic Mathematics 6) 1000 – 730 = 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 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 Subtraction 7312 3765– 905 447– 64332 28655– 2) 8 1 1 3) 7 6 4 2 3 Practice 1) 5 1 3 2 4 18745– Academy of Vedic Mathematics 345– 23567– www.vedicmaths.org Subtraction 908 456– 8393 2865– 64935 38657– Split the number where a bar digit is followed by a positive digit Practice 1) 6 1 3 8 1 48747– Academy of Vedic Mathematics 2) 6 1 9 135– 3) 8 6 8 2 9 38564– www.vedicmaths.org Multiplying Numbers Near a Base 100 98 – 02 97 – 3 95 / 06 89 – 11 97 – 3 86 / 33 89 – 11 89 – 11 78 / 21 = 79/21 1 Academy of Vedic Mathematics www.vedicmaths.org Practice: Multiply 89 96 97 94 91 95 99 88 98 98 88 88 Academy of Vedic Mathematics www.vedicmaths.org Proofs 100 89 – 11 97 – 3 86 / 33 Academy of Vedic Mathematics Arithmetic proof: 89 × 97 = 89×100 – 89×3 = 8900 – 100×3 + 11×3 = 8900 – 300 + 11×3 = 8600 + 33 = 8633 www.vedicmaths.org Proofs 100 89 – 11 97 – 3 86 / 33 Academy of Vedic Mathematics Geometric proof: www.vedicmaths.org Proofs 100 89 – 11 97 – 3 86 / 33 (x – a)(x – b) = x(x – a – b) + ab Algebraic proof: (100 – a)(100 – b) = 100(100 – a – b) + ab Academy of Vedic Mathematics www.vedicmaths.org Mentally 89 – 11 97 – 3 86 / 33 96 x 93 97 x 94 Academy of Vedic Mathematics www.vedicmaths.org Numbers Over 100 103 + 03 104 + 4 107 / 12 123 + 23 103 + 3 126 / 69 102 + 02 103 + 3 105 / 06 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 + 5 97 – 3 102 / 15 = 101 / 85 Academy of Vedic Mathematics 104 + 4 93 – 7 97 / 28 = 96 / 72 www.vedicmaths.org Proportionately 204 x 107 Academy of Vedic Mathematics 48 x 97 98 x 206 www.vedicmaths.org Proportionately 200 189 –11 197 – 3 2x 186 / 33 = 372 / 33 Academy of Vedic Mathematics 300 304 307 3x 311 = 933 +4 +7 / 28 / 28 www.vedicmaths.org Practice: Multiply Using the Base Method 201 234 307 308 198 188 Academy of Vedic Mathematics www.vedicmaths.org Base Multiplication 8–2 7–3 5/ 6 6–4 6–4 2/ 6 =3/6 12 + 2 13 + 3 15 / 6 1 Practice Multiply: 8 8 Academy of Vedic Mathematics 12 14 13 14 www.vedicmaths.org Proportionately 20 21 + 1 22 + 2 2x 23 / 2 = 46 / 2 Academy of Vedic Mathematics 30 28 – 2 27 – 3 3x 25 / 6 = 75 / 6 www.vedicmaths.org Practice: Multiply Using the Base Method 41 42 33 32 51 56 19 18 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 4 3 1 × Academy of Vedic Mathematics 3 4 3 4 × 6 5 7 2 × www.vedicmaths.org Practice Multiply: 5 3 3 2 × Academy of Vedic Mathematics 7 7 2 3 × 4 4 4 4 × 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 Division 3 x 8 = 24 3 1 × 9 9 2 Academy of Vedic Mathematics 24 ÷ 8 = 3 ? 8 × 2 4 4 2 × 2 3 5 2 www.vedicmaths.org Find the Missing Number 5 4 × 3 4 5 6 Academy of Vedic Mathematics 7 3 × 2 3 3 7 www.vedicmaths.org Comparison with Conventional Methods Multiplication Division Academy of Vedic Mathematics Vedic 6 7 5 2 × 52 × 3484 Conventional 6 7 5 2 × 52) 3 4 8 4 www.vedicmaths.org Algebra 2x + 3 3x + 5 × 2 6x + 19x + 15 Academy of Vedic Mathematics 3x – 2 x + 7 × 2 3x + 19x – 14 www.vedicmaths.org Algebraic Division 3x + 5 × 2 6x + 19x + 15 Academy of Vedic Mathematics www.vedicmaths.org Extending to 3-figure Numbers 104 x 53 10 5 112 x 113 Academy of Vedic Mathematics 4 3 × 11 2 11 3 × www.vedicmaths.org Answers 2-Figures at a Time 113 x 203 11 3 20 3 × 1202 x 1103 113 x 203 1 13 2 03 × 12 02 11 03 × Academy of Vedic Mathematics www.vedicmaths.org Bar Numbers 39 x 32 Bar numbers are extremely useful Academy of Vedic Mathematics www.vedicmaths.org Extending the Pattern 504 x 321 Academy of Vedic Mathematics 5 0 4 4 1 3 3 2 1 5 2 3 www.vedicmaths.org Practice Multiply: 4 3 2 5 1 3 × Academy of Vedic Mathematics 3 4 5 2 0 7 × 4 4 4 4 4 4 × www.vedicmaths.org Extending the Pattern 3-figure numbers 4-figure numbers Academy of Vedic Mathematics www.vedicmaths.org 4-Figure Numbers 2 4 3 2 3 5 1 3 × Academy of Vedic Mathematics 3 4 5 2 2 0 3 4 × www.vedicmaths.org Practice Multiply: 4 1 3 2 5 4 1 3 × Academy of Vedic Mathematics 4 2 4 4 4 3 4 4 × 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 2 1 3 5 1 3 4 7 2 1 3 5 Academy of Vedic Mathematics www.vedicmaths.org Adding/subtracting Fractions 3 5 4 6 Academy of Vedic Mathematics 7 11 18 30 www.vedicmaths.org Practice: Find 3 2 7 5 4 2 5 3 5 1 6 9 11 7 15 25 Academy of Vedic Mathematics www.vedicmaths.org Three Fractions 1 1 2 2 3 5 1 2 4 3 5 7 Academy of Vedic Mathematics www.vedicmaths.org Greatest or Least Which fraction is greater? 5 7 or 7 10 Which fraction is the greatest? Academy of Vedic Mathematics 3 4 5 , , 8 9 11 www.vedicmaths.org Unifying the Four Operations Addition Subtraction Multiplication Division 4 1 + 5 3 4 1 – 5 3 4 1 × 5 3 4 1 ÷ 5 3 Academy of Vedic Mathematics www.vedicmaths.org Algebraic Fractions 2x x 8x + 3x 11x + = = 12 12 3 4 1 2 x + 3 + 2 x + 2 3x + 7 + = = x +2 x +3 x + 2 x + 3 x + 2 x + 3 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). l (7,4) l (5,1) O 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) Academy of Vedic Mathematics 2) (8,5), (-3,1) 3) (5,0), (-2,-5) 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. l (5,7) O 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 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. l (3,1) O 2x + 3y = 5 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 Dividing by 19 1 19 11 19 Academy of Vedic Mathematics www.vedicmaths.org Right to Left / 9-Point Circle . . 1 = 0. 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 1 1 1 1 1 1 1 1 1 19 1 19 9,0 1 8 7 2 3 6 5 Academy of Vedic Mathematics 4 www.vedicmaths.org Divisor Ending in 9 17 29 15 39 29 49 Academy of Vedic Mathematics www.vedicmaths.org A Short Cut . = 0.1 0 5 12 6 3 11 15 17 18 1 19 11 19 . =0.1 51 718 91 4 71316 8 . 4 2 1 10 5 12 6 31 1 9 1 4 7 13 16 8 4 2 1 17 29 . . =0.2 51 8 6 2202628191615 5 21 71 2 . 4 11 2327 9 3 110131 42 4 8221 7 6 39 Academy of Vedic Mathematics www.vedicmaths.org Proportionately 4 7 8 13 4 33 7 11 5 23 3 17 (to 5 decimal places) (to 5 decimal places) Academy of Vedic Mathematics www.vedicmaths.org COMPOUND ANGLES 4 5 If sinA = (A is obtuse) and cosB = (B is acute) 5 13 find tan(A-B). A -3 4 5 B 5 12 13 A-B 33 56 65 56 Tan(A-B) = 33 Academy of Vedic Mathematics www.vedicmaths.org TRIGONOMETRIC EQUATIONS Find the general solution of the equation 2sinx - cosx = 2 2sinx cosx Rsinxcos Rcosxsin Rsin(x ). If 2 Rcos and 1 Rsin ,i.e.R2 22 12 giving R 5 and cos 2 1 , sin giving 26 34'. 5 5 R 1 x x- x x c -1 2 -4 0 s 2 1 3 5 1 + 5 5 2 2sinx cosx 5 sin(x 26 34'). Vedic method Thus the equation 2sin x cosx 2 becomes 5 sin(x 26 34') 2 2 sin(x 26 34') 0.8945 5 x 26 34' 180n ( 1)n (63 26') x 180n ( 1)n (63 26') 26 34' 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. b b2 4ac Then x = . 2a So x = 5 2 5 4.3. 7 2.3 5 25 84 6 5 109 6 5 10.44 6 15.44 5.44 or 6 6 a = 3, D1 = 6x – 5 = 13 . 13 7 . 0 0 0 5 2 11 3. 4 3 x 2.57, 0.91. Vedic method 2.57 or 0.91. from “General Mathematics” Book 3, by Channon & Smith, Page 195. Academy of Vedic Mathematics www.vedicmaths.org INTEGRATION BY ‘PARTS’ Find x 2sinxdx. Let { v x2 du sin x dx dv 2x dx u cos x du dv Then v dx uv u dx dx dx gives x2sinxdx = (-cosx)(x2 ) - (-cosx)(2x)dx x sinxdx = -x cosx +2xsinx +2cosx +K 2 2 Vedic method x2 cos x 2 x cos xdx Thus, for x cos xdx, let { vx du cos x dx dv 1 dx u sin x giving x cos xdx (sin x)(x) (sin x)(1)dx xsinx cosx K Hence x2sinxdx = -x2cosx +2xsinx +2cosx +K from “Mathematics: The Core Course for A-level” by Bostock & Chandler, Page 313. Academy of Vedic Mathematics www.vedicmaths.org SQUARE ROOT OF A COMPLEX NUMBER Find 15 + 8i Let 15 8i = a + bi, where a and b are real 15 8i (a bi)2 a2 b2 2abi Equating real and imaginary parts gives: a2 – b2 = 15 and 2ab = 8 Using b (1) (2) 15 8i 4 i Vedic method 4 16 in (1) gives a2 2 15. a a a 4 15a2 16 0 (a2 - 16)(a2 + 1) = 0 Thus a2 – 16 = 0 or a2 + 1 = 0 But a is real so a2 + 1 = 0 gives no suitable values so a 4 Referring to equation (2) we have Hence 15 8i = 4 + i or -4 - i a b 4 1 -4 -1 4 i 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