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  2. Solving quadratic equations with continued fractions - Wikipedia

    en.wikipedia.org/wiki/Solving_quadratic...

    If this infinite continued fraction converges at all, it must converge to one of the roots of the monic polynomial x 2 + bx + c = 0. Unfortunately, this particular continued fraction does not converge to a finite number in every case. We can easily see that this is so by considering the quadratic formula and a monic polynomial with real ...

  3. Continued fraction - Wikipedia

    en.wikipedia.org/wiki/Continued_fraction

    Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. [9]

  4. Category:Continued fractions - Wikipedia

    en.wikipedia.org/wiki/Category:Continued_fractions

    In mathematics, regular continued fractions play an important role in representing real numbers, and have a rich general theory touching on a variety of topics in number theory. Moreover, generalized continued fractions have important and interesting applications in complex analysis

  5. Simple continued fraction - Wikipedia

    en.wikipedia.org/wiki/Simple_continued_fraction

    Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and the other coefficients are positive integers. These two representations agree except in their final terms.

  6. Periodic continued fraction - Wikipedia

    en.wikipedia.org/wiki/Periodic_continued_fraction

    Periodic continued fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part

  7. Euler's continued fraction formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_continued_fraction...

    Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...

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