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In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable is denoted or , with the two notations used interchangeably.
An Egyptian fraction is the sum of a finite number of reciprocals of positive integers. According to the proof of the Erdős–Graham problem, if the set of integers greater than one is partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of 1.
A rational fraction is the quotient (algebraic fraction) of two polynomials. Any algebraic expression that can be rewritten as a rational fraction is a rational function. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero.
This rational will be best in the sense that no other rational in (x, y) will have a smaller numerator or a smaller denominator. [14] [15] If x is rational, it will have two continued fraction representations that are finite, x 1 and x 2, and similarly a rational y will have two representations, y 1 and y 2.
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Given a formal Laurent series = =, the corresponding Hankel operator is defined as [2]: [] [[]]. This takes a polynomial [] and sends it to the product , but discards all powers of with a non-negative exponent, so as to give an element in [[]], the formal power series with strictly negative exponents.