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Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root , and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function 3 y = x {\displaystyle {^{3}}y=x} , the two inverses are the cube super-root of ...
An exponential factorial is an operation recursively defined as =, = . For example, a 4 = 4 3 2 1 {\displaystyle \ a_{4}=4^{3^{2^{1}}}\ } where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.
This definition of exponentiation with negative exponents is the only one that allows extending the identity + = to negative exponents (consider the case =). The same definition applies to invertible elements in a multiplicative monoid , that is, an algebraic structure , with an associative multiplication and a multiplicative identity denoted 1 ...
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. [1]In his 1947 paper, [2] R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations.
Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can ...
Calculators generally perform operations with the same precedence from left to right, [1] but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
In mathematics, pentation (or hyper-5) is the fifth hyperoperation. Pentation is defined to be repeated tetration , similarly to how tetration is repeated exponentiation , exponentiation is repeated multiplication , and multiplication is repeated addition .
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...