The Sum Of The Digits

Sum of a number'due south digits

In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For instance, the digit sum of the decimal number $9045$ would exist $nine+0+4+5=xviii$ .

Definition [edit]

Let $n$ be a natural number. We define the digit sum for base $b>one$ $F_{b}:\mathbb {N} \rightarrow \mathbb {North}$ to be the post-obit:

F_{b}(n)=\sum _{i=0}^{thou}d_{i}

where $k=\lfloor \log _{b}{north}\rfloor$ is the number of digits in the number in base $b$ , and

d_{i}={\frac {n{\bmod {b^{i+1}}}-northward{\bmod {b}}^{i}}{b^{i}}}

is the value of each digit of the number.

For case, in base x, the digit sum of 84001 is $F_{ten}(84001)=8+iv+0+0+ane=13$ .

For any 2 bases $2\leq b_{1}<b_{2}$ and for sufficiently big natural numbers $n$ ,

\sum _{yard=0}^{northward}F_{b_{one}}(k)<\sum _{k=0}^{n}F_{b_{two}}(k)

.^[1]

The sum of the base of operations ten digits of the integers 0, 1, two, ... is given by OEIS: A007953 in the On-Line Encyclopedia of Integer Sequences. Borwein & Borwein (1992) use the generating function of this integer sequence (and of the analogous sequence for binary digit sums) to derive several rapidly converging series with rational and transcendental sums.^[2]

Extension to negative integers [edit]

The digit sum tin can exist extended to the negative integers by utilise of a signed-digit representation to represent each integer.

Applications [edit]

The concept of a decimal digit sum is closely related to, simply not the same as, the digital root, which is the upshot of repeatedly applying the digit sum functioning until the remaining value is but a unmarried digit. The digital root of whatsoever not-zero integer will exist a number in the range 1 to ix, whereas the digit sum tin take any value. Digit sums and digital roots tin can exist used for quick divisibility tests: a natural number is divisible by three or 9 if and just if its digit sum (or digital root) is divisible by 3 or nine, respectively. For divisibility by 9, this examination is called the rule of nines and is the footing of the casting out nines technique for checking calculations.

Digit sums are also a common ingredient in checksum algorithms to check the arithmetics operations of early computers.^[3] Earlier, in an era of hand calculation, Edgeworth (1888) suggested using sums of 50 digits taken from mathematical tables of logarithms as a form of random number generation; if 1 assumes that each digit is random, and then by the primal limit theorem, these digit sums will have a random distribution closely approximating a Gaussian distribution.^[4]

The digit sum of the binary representation of a number is known as its Hamming weight or population count; algorithms for performing this operation have been studied, and information technology has been included equally a built-in performance in some figurer architectures and some programming languages. These operations are used in computing applications including cryptography, coding theory, and calculator chess.

Harshad numbers are defined in terms of divisibility by their digit sums, and Smith numbers are defined by the equality of their digit sums with the digit sums of their prime factorizations.

References [edit]

^ Bush, L. Eastward. (1940), "An asymptotic formula for the average sum of the digits of integers", American Mathematical Monthly, Mathematical Clan of America, 47 (3): 154–156, doi:10.2307/2304217, JSTOR 2304217 .
^ Borwein, J. M.; Borwein, P. B. (1992), "Strange series and loftier precision fraud" (PDF), American Mathematical Monthly, 99 (7): 622–640, doi:10.2307/2324993, hdl:1959.13/1043650, JSTOR 2324993 .
^ Bloch, R. M.; Campbell, R. V. D.; Ellis, M. (1948), "The Logical Blueprint of the Raytheon Calculator", Mathematical Tables and Other Aids to Ciphering, American Mathematical Society, iii (24): 286–295, doi:10.2307/2002859, JSTOR 2002859 .
^ Edgeworth, F. Y. (1888), "The Mathematical Theory of Banking" (PDF), Journal of the Royal Statistical Society, 51 (1): 113–127, archived from the original (PDF) on 2006-09-13 .