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VEDIC MATHEMATICS : Arithmetic Operations T. K. Prasad http://www.cs.wright.edu/~tkprasad Prasad Arithmetic Operations Revisited 1 Positional Number System TENTHOUSANDS THOUSANDS HUNDREDS TENS UNITS 43210 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 +0 Prasad Arithmetic Operations Revisited 2 Two Digit Multiplication (above the base) using Vedic Approach 1. Prasad Method : Vertically and Crosswise Sutra 2. Correctness and Applicability Arithmetic Operations Revisited 3 Method: Multiply 13 * 12 • Write the first number to be multiplied and excess over 10 in the first row, and the second number to be multiplied and excess over 10 in the second row. 13 12 Prasad 3 2 Arithmetic Operations Revisited 4 13 3 12 2 • To determine the 3-digit product: – add crosswise to obtain the left digits • (13 + 2) = (12 + 3) = 15 – and – multiply the excess vertically to obtain the right digit. • (3 * 2) = 6 • 13 * 12 = 156 Prasad Arithmetic Operations Revisited 5 Another Example • 12 * 14 = • 12 • 14 • 16 2 4 8 • 12 * 14 = 168 Prasad Arithmetic Operations Revisited 6 Questions • Why do both crosswise additions yield the same result? • Why does this method yield the correct answer for this example? • Does this method always work for any pair of numbers? Prasad Arithmetic Operations Revisited 7 Proof Sketch • (12 + 4) = (14 + 2) = 16 • Why are they same? • That is, the sum of first number and excess over 10 of the second number, and …. • (12 + (14 – 10)) = (12+14 – 10) = (26 – 10) = 16 • (14 + (12 – 10)) = (14+12 – 10) = (26 – 10) = 16 Prasad Arithmetic Operations Revisited 8 Correctness Argument: Two possibilities • 12 = (10 + 2) • 14 = (10 + 4) • 12 * 14 = (10 + 2) * 14 = 10 * 14 + 2 * 14 = 10 * 14 + 2 * (10 + 4) = 10 * 14 + 2 * 10 + (2 * 4) = 10 * (14+2) + 8 Right digit = 10 * 16 + 8 [Vertical Left digits Product] = 168 [Crosswise Addition] Prasad • 12 = (10 + 2) • 14 = (10 + 4) • 12 * 14 = 12 * (10 + 4) = 12 * 10 + 12 * 4 = 12 * 10 + (10 + 2) * 4 = 12 * 10 + 10 * 4 + (2 * 4) = 10 * (12 + 4) + 8 Right digit = 10 * 16 + 8 [Vertical Left digits Product] = 168 [Crosswise Addition] Arithmetic Operations Revisited 9 Another Example • 15 * 12 15 12 17 18 Prasad 5 2 10 0 Arithmetic Operations Revisited 10 Yet Another Example • 17 * 15 17 15 22 22+3 25 7 5 35 5 5 Need proof to feel comfortable! Prasad Arithmetic Operations Revisited 11 Method: Multiply 113 * 106 • Write the first number to be multiplied and excess over 100 in the first row, and the second number to be multiplied and excess over 100 in the second row. 113 106 Prasad 13 6 Arithmetic Operations Revisited 12 113 13 106 6 • To determine the 5-digit product: – add crosswise to obtain the left digits • (113 + 6) = (106 + 13) = 119 – and – multiply the excess vertically to obtain the right digits. • (13 * 6) = 78 • 113 * 106 = 11978 Prasad Arithmetic Operations Revisited 13 Questions • Why do both crosswise additions yield the same result? • Why does this method yield the correct answer for this example? • Does this method always work for any pair of 3 digit numbers? Prasad Arithmetic Operations Revisited 14 Proof Sketch • (113 + 6) = (106 + 13) = 119 • Why are they same? • (113 + (106 – 100)) = (113 + 106 – 100) = 119 • (106 + (113 – 100)) = (106 + 113 – 100) = 119 Prasad Arithmetic Operations Revisited 15 Correctness of Product : Two possibilities • 113 = (100 + 13) • 106 = (100 + 6) • 113 * 106 = (100 + 13) * 106 = 100 * 106 + 13 * (100 + 6) = 100 * 106 + 13 * 100 + (13 * 6) = 100 * (106 + 13) + 78 = 100 * 119 + 78 = 11978 Left digits Right digits [Crosswise Addition] Prasad [Vertical Product] • 113 = (100 + 13) • 106 = (100 + 6) • 113 * 106 = 113 * (100 + 6) = 113 * 100 + (100 + 13) * 6 = 113 * 100 + 100 * 6 + (13 * 6) = 100 * (113 + 6) + 78 = 100 * 119 + 78 = 11978 Left digits Right digits Arithmetic Operations Revisited [Crosswise Addition] [Vertical Product] 16 Another Example • 160 * 180 160 60 Breakdown?! 180 80 240 4800 288 00 • Note that, the product of the excess over 100 has more than two digits. However, the weight associated with 240 and 48 are both 100, and hence they can be combined. Prasad Arithmetic Operations Revisited 17 Yet Another Example • 190 * 199 190 199 289 289+89 378 90 99 8910 10 10 Breakdown?! This approach is valid with suggested modifications! Prasad Arithmetic Operations Revisited 18 More Shortcuts Prasad Arithmetic Operations Revisited 19 Quick squaring of numbers that end in 5 • 15 * 15 = 225 = (1*2) (5*5) • 75 * 75 = 5625 = (7*8) (5*5) • 95 * 95 = 9025 = (9*10) (5*5) Prasad • Proof: Let the two digit number be written as D5. • D5 * D5 = (D*10 + 5) * (D*10 + 5) = (D*D*100) + (D*2*50) + 5*5 = (D*(D+1))*100 + 25 Arithmetic Operations Revisited 20 Quick Multiplication : Special Case • Proof: Let two digit numbers be AB and AC. • AB * AC = (A*10 + B) * (A*10 + C) = (A*A*100) + (A*10*(B+C)) + B*C = (A*A)*100 + (A)*(B+C)*10 + (B*C) • For B+C=10, this reduces to A*(A+1)*100 + B*C • For A=12, B=8 and C=2, this reduces to (12)*(13)*100 + 16 = 15616 Prasad Arithmetic Operations Revisited 21 Quicking squaring of numbers that begin with 5 • 51 * 51 = (5*5+1)*100 + (1*1) = 2601 • 57 * 57 = (5*5+7) *100 + (7*7) = 3249 • 59 * 59 = (5*5+9) *100 + (9*9) =3481 Prasad • Proof: Let the two digit number be written as 5D. • 5D * 5D = (50 + D) * (50 + D) = (25 + D)*100 + (D*D) Arithmetic Operations Revisited 22 Quick squaring of two digit numbers • Proof: Let two digit numbers be AB. • AB * AB = (A*10 + B) * (A*10 + B) = (A*A)*100 + 2*(A*10)*B + B*B = (A*A)*100 + 20*(A*B) + (B*B) • For AB=79, this reduces to 4900+20*63+81 = 4981+1260 =6241 • For AB=116, this reduces to 12100+20*66+36 = 12136+1320 =13456 Prasad Arithmetic Operations Revisited 23 Generalized Multplication Using Working Base Prasad Arithmetic Operations Revisited 24 23 24 +3 +4 • To determine the product, choose working base as 20: – add crosswise to obtain the left digits with weight 20 • (23 + 4) = (24 + 3) = 27 – multiply the excess vertically to obtain the right digits. • (3 * 4) = 12 • 23 * 24 = 27 * 20 + 12 • = 540 + 12 23 * 24 = 552 Prasad Arithmetic Operations Revisited 25 723 724 +23 +24 • To determine the product, choose working base as 700: – add crosswise to obtain the left digits with weight 700 • (723 + 24) = (724 + 23) = 747 – multiply the excess vertically to obtain the right digits. • (23 * 24) = 552 • 723 * 724 = 747 * 700 + 552 • = 522900 + 552 723 * 724 = 523452 Prasad Arithmetic Operations Revisited 26 783 775 -17 -25 • To determine the product, choose working base as 800: – add crosswise to obtain the left digits with weight 800 • (783 - 25) = (775 - 17) = 758 – multiply the excess vertically to obtain the right digits. • (17 * 25) = 425 • 783 * 775 = 758 * 800 + 425 • = 606400 + 425 783 * 775 = 606825 Prasad Arithmetic Operations Revisited 27 532 472 +32 -28 • To determine the product, choose working base as 1000/2: – add crosswise to obtain the left digits with wt. 1000/2 • (532 - 28) = (472 + 32) = 504 – multiply the excess vertically to obtain the right digits. • (+32) * (-28) = 896 • 532 * 472= (504 / 2)*1000 + (104 -1000) • = 252000 + 104 - 1000 532 * 472= 251104 Prasad Arithmetic Operations Revisited 28