Congruence Formula
The formula is:
or,
To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1.
The digital root is the value modulo 9 because and thus so regardless of position, the value mod 9 is the same – – which is why digits can be meaningfully added. Concretely, for a three-digit number,
To obtain the modular value with respect to other numbers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of modulo n, or analogously for for different bases. This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8).
Also of note is since and thus taking the alternating sum of digits yields the value modulo 11.
Read more about this topic: Digital Root
Famous quotes containing the words congruence and/or formula:
“As for butterflies, I can hardly conceive
of one’s attending upon you; but to question
the congruence of the complement is vain, if it exists.”
—Marianne Moore (1887–1972)
“Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
—Pierre Simon De Laplace (1749–1827)