Prasad Digital Roots 1

VEDIC MATHEMATICS : Digital Roots/Sums

T. K. Prasadhttp://www.cs.wright.edu/~tkprasad

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)

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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).

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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

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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

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VEDIC SQUARE Table of digital root of single digit product

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9 point circle

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Digital root pattern for 4x1 x 4 = 4

2 x 4 = 8

3 x 4 = 3

4 x 4 = 7

5 x 4 = 2

6 x 4 = 6

7 x 4 = 1

8 x 4 = 5

9 x 4 = 9

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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

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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.

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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)

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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

19999999999999999999999Prasad Digital Roots 12

(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)?

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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

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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!

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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, …

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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.

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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).

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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.

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Appendix : DrScheme Code

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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

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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.

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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

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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

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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

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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.

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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

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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)

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