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Cube root. In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other ...
Square root. Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 52 (5 squared). In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1] For example, 4 and −4 are square roots of 16 ...
Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+ √ 2)/2) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic ...
Pell's equation for n = 2 and six of its integer solutions. Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever ...
A generalized continued fraction is an expression of the form. where the an ( n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas :
Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this ...
Nested radical. In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include. which arises in discussing the regular pentagon, and more complicated ones such as.
Solving quadratic equations with continued fractions. In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is. where a ≠ 0. The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots ...
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