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The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2 n 5 m, where m and n are non-negative integers. Proof:
In his Essai sur la théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the simple continued fraction of a given real number. [4] A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows: [5 ...
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, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...
Addition, negation, and comparison are all defined the same way for both surreal numbers and games. Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 0 | 0} is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of ...
Engineering notation or engineering form (also technical notation) is a version of scientific notation in which the exponent of ten is always selected to be divisible by three to match the common metric prefixes, i.e. scientific notation that aligns with powers of a thousand, for example, 531×10 3 instead of 5.31×10 5 (but on calculator displays written without the ×10 to save space).
Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. [93] Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers.
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...