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In number theory, a narcissistic number [1] [2] (also known as a pluperfect digital invariant (PPDI), [3] an Armstrong number [4] (after Michael F. Armstrong) [5] or a plus perfect number) [6] in a given number base is a number that is the sum of its own digits each raised to the power of the number of digits.
The earliest reference to a similar formula appears to be Armstrong (1985, p. 348), where it is called "adjusted MAPE" and is defined without the absolute values in the denominator. It was later discussed, modified, and re-proposed by Flores (1986). Armstrong's original definition is as follows:
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
A trimorphic number or spherical number occurs when the polynomial function is () =. [1] All automorphic numbers are trimorphic. The terms circular and spherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself.
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers =,,, … as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:
Whether or not a rational number has a terminating expansion depends on the base. For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating.
By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0, (2) H x = H x−1 + 1/x for all complex numbers x except the non-positive integers, and (3) lim m→+∞ (H m+x − H m) = 0 for all complex values x.