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Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K .
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions [citation needed]. The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer ...
Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model. Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator.
The Decimal class in the standard library module decimal has user definable precision and limited mathematical operations (exponentiation, square root, etc. but no trigonometric functions). The Fraction class in the module fractions implements rational numbers. More extensive arbitrary precision floating point arithmetic is available with the ...
In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a monic polynomial. [8]
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.