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  2. Exponentiation - Wikipedia

    en.wikipedia.org/wiki/Exponentiation

    For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below).

  3. Bernoulli's inequality - Wikipedia

    en.wikipedia.org/wiki/Bernoulli's_inequality

    2 Real exponent. 3 History. ... The exponent can be generalized to an arbitrary real number as follows: if >, then (+ ... World Scientific. p.

  4. Irrationality measure - Wikipedia

    en.wikipedia.org/wiki/Irrationality_measure

    The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.

  5. Transcendental function - Wikipedia

    en.wikipedia.org/wiki/Transcendental_function

    For example, (+ /) converges to the exponential function , and the infinite sum = ()! turns out to equal the hyperbolic cosine function ⁡. In fact, it is impossible to define any transcendental function in terms of algebraic functions without using some such "limiting procedure" (integrals, sequential limits, and infinite sums are just a few).

  6. Characterizations of the exponential function - Wikipedia

    en.wikipedia.org/wiki/Characterizations_of_the...

    One way of defining the exponential function over the complex numbers is to first define it for the domain of real numbers using one of the above characterizations, and then extend it as an analytic function, which is characterized by its values on any infinite domain set.

  7. Elementary function - Wikipedia

    en.wikipedia.org/wiki/Elementary_function

    In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).

  8. Hyperreal number - Wikipedia

    en.wikipedia.org/wiki/Hyperreal_number

    Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number ...

  9. Floating-point arithmetic - Wikipedia

    en.wikipedia.org/wiki/Floating-point_arithmetic

    The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB [27] as a hexadecimal number. An example of a layout for 32-bit floating point is and the 64-bit ("double") layout is similar.