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  2. Repeating decimal - Wikipedia

    en.wikipedia.org/wiki/Repeating_decimal

    A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.

  3. Midy's theorem - Wikipedia

    en.wikipedia.org/wiki/Midy's_theorem

    In mathematics, Midy's theorem, named after French mathematician E. Midy, [1] is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period (sequence A028416 in the OEIS). If the period of the decimal representation of a/p is 2n, so that

  4. Periodic continued fraction - Wikipedia

    en.wikipedia.org/wiki/Periodic_continued_fraction

    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 irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.

  5. 0.999... - Wikipedia

    en.wikipedia.org/wiki/0.999...

    In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. [46] Examples include: = 0. 142857 and 142 + 857 = 999. = 0. 01369863 and 0136 + 9863 = 9999.

  6. 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]

  7. Decimal representation - Wikipedia

    en.wikipedia.org/wiki/Decimal_representation

    Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating. Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, 36 ⁄ 25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".

  8. Cyclic number - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number

    Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...

  9. Fraction - Wikipedia

    en.wikipedia.org/wiki/Fraction

    Let x = the repeating decimal: x = 0.1523 987; Multiply both sides by the power of 10 just great enough (in this case 10 4) to move the decimal point just before the repeating part of the decimal number: 10,000x = 1,523. 987; Multiply both sides by the power of 10 (in this case 10 3) that is the same as the number of places that repeat:

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