VEDIC MATHEMATICS : Digital Roots/Sums

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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)

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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 (1236 12) and (12 3).
• Recall 1x(9991) 2x(991) 3x(91) 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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(contd)

• The digital root of a number is the remainder
  obtained by dividing it by 9.
• 1236 divided by 9 R 3
• Recall 1x(9991) 2x(991) 3x(91) 6
• Note that 9 is treated similar to 0.
• 36 divided by 9 R 0

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(contd)

• Digital roots can be calculated quickly by
  casting out 9s.
• 12173645 gt (12173645)
• (29)
  (11) 2
• 12173645 gt (11)2

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VEDIC SQUARE Table of digital root of single
digit product
7
9 point circle
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Digital root pattern for 4x
1 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) Onegt (7)
• Additive persistence of 5243 Two,
  because (5 2 4 3) Onegt (14) Twogt (5)

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(contd)

• The smallest number for additive persistence 0
  through 4 are
• 0 step gt 0
• 1 step gt 10
• 2 steps gt 19
• 3 steps gt 199
• 4 steps gt 19999999999999999999999
• 19999999999999999999999

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(contd)

• 0 step gt 0
• 1 step gt 10
• 2 steps gt 19
• (quotient 19-1 divide 9 2 )
• 3 steps gt 199 (2 9s 1)
• (quotient 199-1 divide 9 22)
• 4 steps gt 1999999999999999999999
• (22 9s 1)
• Hint It is the number of 9s we add to get
• (the previous number in the sequence 1)?

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(contd)

• 4 steps gt 1999999999999999999999
• (22 9s 1)
• 5 steps gt 1 followed by
• (quotient 19999999999999999999998
• divide 9) 9s
• gt 1 followed by
• 2222222222222222222222 9s

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How big is the last number?

• Larger than the number of stars in the universe?
• 1021 (10 followed by 21 zeros)
• YES.
• Larger than the number of atoms in the universe?
• 1080
• YES.
• Larger than googol 10100?
• YES.
• Larger than googolplex 10 followed by 10100 0s?
• 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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(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.

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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 gt 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 gt 0 and returns
  its additive persistence.

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• (define (digitalRoot n)
• (if (lt 0 n 9) n
• (digitalRoot
• (accumulate 0
• (map (lambda (d) (- d 48))
• (map char-gtinteger
• (string-gtlist
  (number-gtstring n)))
• )))
• ))
• (digitalRoot 10) "should be" 1
• (digitalRoot 1237890) "should be" 3

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• (define (add-persist n)
• (if (lt 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 gt 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 gt 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 2s, that is, 22222 22222 22222
  22222 22
• Total number of atoms in the universe is
• only 80 digits long!
• Googol 10100
• Googolplex 10(10100)