Definition
For natural numbers the arithmetic derivative is defined as follows:
- for any prime .
- for any (Leibniz rule).
To coincide with the Leibniz rule is defined to be, as is . Explicitly, assume that
where are distinct primes and are positive integers. Then
The arithmetic derivative also preserves the power rule (for primes):
where is prime and is a positive integer. For example,
The sequence of number derivatives for k = 0, 1, 2, ... begins (sequence A003415 in OEIS):
- 0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ....
E. J. Barbeau was the first to formalize this definition. He extended it to all integers by proving that uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers,showing that the familiar quotient rule gives a well-defined derivative on Q:
Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents are allowed to be arbitrary rational numbers.
The logarithmic derivative is a totally additive function.
Read more about this topic: Arithmetic Derivative
Famous quotes containing the word definition:
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)