1. PrasadDigital Roots1 VEDIC MATHEMATICS : Digital Roots/Sums T. K. Prasad http://www.cs.wright.edu/~tkprasad

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) PrasadDigital Roots2

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). PrasadDigital Roots3

4. (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

5. (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

6. VEDIC SQUARE Table of digital root of single digit product PrasadDigital Roots6

7. 9 point circle PrasadDigital Roots7

8. 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

9. 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

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. PrasadDigital Roots10

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) PrasadDigital Roots11

12. (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

13. (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

14. (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

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 0’s? NO, we have at last found a match! PrasadDigital Roots15

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, … PrasadDigital Roots16

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. PrasadDigital Roots17

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). PrasadDigital Roots18

19. (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

20. Appendix : DrScheme Code PrasadDigital Roots20

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 PrasadDigital Roots21

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. PrasadDigital Roots22

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 PrasadDigital Roots23

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 PrasadDigital Roots24

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 PrasadDigital Roots25

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. PrasadDigital Roots26

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 PrasadDigital Roots27

28. 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