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PrasadDigital Roots1 VEDIC MATHEMATICS : Digital Roots/Sums T. K. Prasad http://www.cs.wright.edu/~tkprasad
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Definition Digital root of a number is the single digit obtained by repeatedly summing all the digits of a number. Example: Digital root of 2357 = 8 because (2 + 3 + 5 + 7 = 17) and (1 + 7 = 8) Digital root of 89149 = 4 because (8 + 9 + 1 + 4 + 9 = 31) and (3 + 1 = 4) PrasadDigital Roots2
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Some facts we already know A number is divisible by 3 if its digital root is divisible by 3 (that is, it is 0, 3, 6, or 9). 1236 is divisible by 3 because 3 is divisible by 3. Note (1+2+3+6 = 12) and (1+2 = 3). Recall: 1x(999+1) + 2x(99+1) + 3x(9+1) + 6 A number is divisible by 9 if its digital root is divisible by 9 (that is, it is 0 or 9). PrasadDigital Roots3
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(cont’d) The digital root of a number is the remainder obtained by dividing it by 9. 1236 divided by 9 = … R 3 Recall: 1x(999+1) + 2x(99+1) + 3x(9+1) + 6 Note that 9 is treated similar to 0. 36 divided by 9 = … R 0 PrasadDigital Roots4
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(cont’d) Digital roots can be calculated quickly by casting out 9s. 12173645 => (1+2+1+7+3+6+4+5) = (2+9) = (1+1) = 2 12173645 => (1+1)=2 PrasadDigital Roots5
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VEDIC SQUARE Table of digital root of single digit product PrasadDigital Roots6
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9 point circle PrasadDigital Roots7
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Digital root pattern for 4x 1x 4 = 4 2 8 3 3 4 7 5 2 6 6 7 1 8 5 9 9 PrasadDigital Roots8
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Properties of digital roots Digital root of a square is 1, 4, 7, or 9square Digital root of a perfect cube is 1, 8 or 9perfect cube Digital root of a prime number (except 3) is 1, 2, 4, 5, 7, or 8prime number Digital root of a power of 2 is 1, 2, 4, 5, 7, or 8power of 2 PrasadDigital Roots9
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Justification for digital roots of a prime number Recall that digital root of 3, 6, or 9 implies it is divisible by 3. The digital root of 1, 2, 4, 5, 7, and 8 are realizable by the prime numbers 19, 2 (11), 13, 5 (23), 7 (43), and 8 (17), respectively. –This is a necessary (but not sufficient) condition for a number to be prime. PrasadDigital Roots10
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Additive Persistence of a Number Additive persistence of a number is the number of steps required to reach the digital root. Additive persistence of 52 = One, because (5 + 2) =One=> (7) Additive persistence of 5243 = Two, because (5 + 2 + 4 + 3) =One=> (14) =Two=> (5) PrasadDigital Roots11
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(cont’d) The smallest number for additive persistence 0 through 4 are: 0 step => 0 1 step => 10 2 steps => 19 3 steps => 199 4 steps => 19999999999999999999999 19999999999999999999999 PrasadDigital Roots12
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(cont’d) 0 step => 0 1 step => 10 2 steps => 19 (quotient 19-1 divide 9 = 2 ) 3 steps => 199 (2 9’s + 1) (quotient 199-1 divide 9 = 22) 4 steps => 1999999999999999999999 (22 9’s + 1) Hint: It is the number of 9’s we add to get (the previous number in the sequence – 1)? PrasadDigital Roots13
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(cont’d) 4 steps => 1999999999999999999999 (22 9’s + 1) 5 steps => 1 followed by (quotient 19999999999999999999998 divide 9) 9’s => 1 followed by 2222222222222222222222 9’s PrasadDigital Roots14
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How big is the last number? Larger than the number of stars in the universe? 10^21 (10 followed by 21 zeros) YES. Larger than the number of atoms in the universe? 10^80 YES. Larger than googol 10^100? YES. Larger than googolplex 10 followed by 10^100 0’s? NO, we have at last found a match! PrasadDigital Roots15
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Application: Parity and Checksum (Digression of sorts) Bit (Binary Digit) : 0, 1 Numbers in binary: 000, 001, 010, 011, 100, … Numbers in decimal: 0, 1, 2, 3, 4, … PrasadDigital Roots16
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Parity Parity bit is added to ensure that the number of 1 bits in a given set or sequence of bits is always even or odd. Odd parity: 000 1, 001 0, 011 1, 100 0, etc Even parity: 000 0, 001 1, 011 0, 100 1, etc Parity is used to detect single bit errors in transmission. PrasadDigital Roots17
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Check digit (Checksum) Check digit is single digit computed from the digits of a number (usually representing a short message or identifying an object). Check digit (or more generally checksum) is used to detect errors (in message transmission or storage). PrasadDigital Roots18
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(cont’d) E.g., The final digit in UPC code (barcode) for products, ISBN number for books, etc are a form of check digit. E.g., Credit card numbers use check digits. E.g., Message/Data encoding techniques (for storing information on devices such as hard disk, CD, DVD, etc or transmitting information over the wire or by wireless means) use sophisticated checksums. PrasadDigital Roots19
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Appendix : DrScheme Code PrasadDigital Roots20
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;; (accumulate + 0 (1 2 13 4)) = 0 + 1 + 2 + 13 + 4 = 20 (define (accumulate f id lis) (if (null? lis) id (f (car lis) (accumulate f id (cdr lis))) ) (accumulate + 0 '(1 2 3 8 9 0)) "should be" 23 PrasadDigital Roots21
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;; DEFINITION: Digital sum/root of a number is obtained by ;; repeatedly summming its digits or of the sum so obtained ;; till it reduces to a single digit. ;; digitalRoot takes a number n >= 0 and returns its digital sum/root ;; DEFINITION: Additive persistance of a number is the number of ;; steps it takes to reduce the number to its digital sum/root. ;; add-persist takes a number n >= 0 and returns its additive persistence. PrasadDigital Roots22
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(define (digitalRoot n) (if (<= 0 n 9) n (digitalRoot (accumulate + 0 (map (lambda (d) (- d 48)) (map char->integer (string->list (number->string n))) ))) )) (digitalRoot 10) "should be" 1 (digitalRoot 1237890) "should be" 3 PrasadDigital Roots23
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(define (add-persist n) (if (<= 0 n 9) 0 (+ 1 (add-persist (digitalRoot n)) ) ) (add-persist 9) "should be" 0 (add-persist 10) "should be" 1 (add-persist 1237890) "should be" 3 PrasadDigital Roots24
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;; nines takes an number n > 0 and returns a number with n 9s (define (nines n) (if (eq? n 1) 9 (+ 9 (* (nines (- n 1)) 10)) ) (nines 1) "should be" 9 (nines 10) "should be" 9999999999 PrasadDigital Roots25
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;; Recall, additive persistance of a number is the number of steps ;; it takes to reduce the number to its digital sum/root. ;; min-add-persist takes an number n >= 0 and ;; returns the smallest number with additive persistence of n ;; LOGIC: To get min-add-persist of n, take the largest number with ;; additive persistance of (n-1) and generate a string of 9s that ;; add upto it, and then prepend it with 1, to get the smallest number ;; that overflows addtive persistance of n. PrasadDigital Roots26
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(define (min-add-persist n) (cond ((eq? n 0) 0) ((eq? n 1) 10) (else (let ((max-prev (quotient (- (min-add-persist (- n 1)) 1) 9))) (+ (expt 10 max-prev) (nines max-prev))) ) ) (map min-add-persist '(0 1 2 3 4)) “should be” (0 10 19 199 19999999999999999999999) (add-persist (min-add-persist 4)) "should be" 4 (add-persist (- (min-add-persist 4) 1)) "should be" 3 PrasadDigital Roots27
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Interesting Case! (min-add-persist 5) ; Number of 9s in this number is ; (quotient 19999999999999999999998 9) ; which is 22 2’s, that is, 22222 22222 22222 22222 22 ; Total number of atoms in the universe is ; only 80 digits long! ; Googol = 10^100 ; Googolplex = 10^(10^100) PrasadDigital Roots28
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