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VEDIC MATHEMATICS : Digital Roots

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). Prasad. Digital

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VEDIC MATHEMATICS : Digital Roots

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    1. Prasad Digital Roots 1 VEDIC MATHEMATICS : Digital Roots/Sums T. K. Prasad http://www.cs.wright.edu/~tkprasad Practical application : checksum => redundancy check (necessary condition) Vedic Mathematics for Schools Book 1 by J. T. Glover Practical application : checksum => redundancy check (necessary condition) Vedic Mathematics for Schools Book 1 by J. T. Glover

    2. 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) Prasad Digital Roots 2

    3. 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). Prasad Digital Roots 3 http://www.youtube.com/watch?v=2Auiwko1m10http://www.youtube.com/watch?v=2Auiwko1m10

    4. (contd) 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 Prasad Digital Roots 4 http://www.youtube.com/watch?v=9nXhnj7jNqk&feature=related http://www.youtube.com/watch?v=9nXhnj7jNqk&feature=related

    5. (contd) 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 Prasad Digital Roots 5 http://www.youtube.com/watch?v=9nXhnj7jNqk&feature=related http://www.youtube.com/watch?v=9nXhnj7jNqk&feature=related

    6. VEDIC SQUARE Table of digital root of single digit product Prasad Digital Roots 6

    7. 9 point circle Prasad Digital Roots 7

    8. Digital root pattern for 4x Prasad Digital Roots 8

    9. Properties of digital roots Digital root of a square is 1, 4, 7, or 9 Digital root of a perfect cube is 1, 8 or 9 Digital root of a prime number (except 3) is 1, 2, 4, 5, 7, or 8 Digital root of a power of 2 is 1, 2, 4, 5, 7, or 8 Prasad Digital Roots 9 Digital root of 3, 6, and 9 implies composite number divisible by 3.Digital root of 3, 6, and 9 implies composite number divisible by 3.

    10. 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. Prasad Digital Roots 10

    11. 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) Prasad Digital Roots 11 The persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point. The persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point.

    12. (contd) 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 Prasad Digital Roots 12 Unintuitive properties --- need to be careful => proof sought even if apparently obvious! Rationale: The smallest number with additive persistence (i) is the number whose sum of digits results in the smallest number with additive persistence (i-1). For 4 step case, we need to get a number that digit-sums to 199, that is, 1 + 198/9, that is 1 followed by 22 9s!!! Because we are seeking the smallest number of steps, we want to have as many 9s as possible. And to overflow it we tag on a 1 at the highest significance place. (Counterexample: If you replace 9 with 333, you will get a larger number.) --------------------------------------------------------------------------------------------------------------------------------- The next number in the sequence (the smallest number of additive persistence 5) is [1 followed by 22222,22222,22222,22222,22 9's]. Why? Because we want a number with strings of 9s which when added yields 22 9s, which is (quotient of) 1,99999,99999,99999,99999,98 divide by 9. For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm. Unintuitive properties --- need to be careful => proof sought even if apparently obvious! Rationale: The smallest number with additive persistence (i) is the number whose sum of digits results in the smallest number with additive persistence (i-1). For 4 step case, we need to get a number that digit-sums to 199, that is, 1 + 198/9, that is 1 followed by 22 9s!!! Because we are seeking the smallest number of steps, we want to have as many 9s as possible. And to overflow it we tag on a 1 at the highest significance place. (Counterexample: If you replace 9 with 333, you will get a larger number.) --------------------------------------------------------------------------------------------------------------------------------- The next number in the sequence (the smallest number of additive persistence 5) is [1 followed by 22222,22222,22222,22222,22 9's]. Why? Because we want a number with strings of 9s which when added yields 22 9s, which is (quotient of) 1,99999,99999,99999,99999,98 divide by 9. For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.

    13. (contd) 0 step => 0 1 step => 10 2 steps => 19 (quotient 19-1 divide 9 = 2 ) 3 steps => 199 (2 9s + 1) (quotient 199-1 divide 9 = 22) 4 steps => 1999999999999999999999 (22 9s + 1) Hint: It is the number of 9s we add to get (the previous number in the sequence 1)? Prasad Digital Roots 13 --------------------------------------------------------------------------------------------------------------------------------- Recurrence relation: N(0) = 0 N(1) = 10 N(i) = 1 ^ Nines((N(i-1) - 1) div 9) --------------------------------------------------------------------------------------------------------------------------------- Recurrence relation: N(0) = 0 N(1) = 10 N(i) = 1 ^ Nines((N(i-1) - 1) div 9)

    14. (contd) 4 steps => 1999999999999999999999 (22 9s + 1) 5 steps => 1 followed by (quotient 19999999999999999999998 divide 9) 9s => 1 followed by 2222222222222222222222 9s Prasad Digital Roots 14

    15. 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 0s? NO, we have at last found a match! Prasad Digital Roots 15

    16. 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, Prasad Digital Roots 16

    17. 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. Prasad Digital Roots 17

    18. 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). Prasad Digital Roots 18

    19. (contd) 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. Prasad Digital Roots 19

    20. Appendix : DrScheme Code Prasad Digital Roots 20

    21. ;; (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 Prasad Digital Roots 21

    22. ;; 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. Prasad Digital Roots 22

    23. (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 Prasad Digital Roots 23

    24. (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 Prasad Digital Roots 24

    25. ;; 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 Prasad Digital Roots 25

    26. ;; 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. Prasad Digital Roots 26

    27. (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 Prasad Digital Roots 27 http://www.unc.edu/~rowlett/units/large.html http://www.unc.edu/~rowlett/units/large.html

    28. Interesting Case! (min-add-persist 5) ; Number of 9s in this number is ; (quotient 19999999999999999999998 9) ; which is 22 2s, 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) Prasad Digital Roots 28

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