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2. Fraction - Wikipedia

   en.wikipedia.org/wiki/Fraction

   The reciprocal of ⁠ 3 / 7 ⁠, for instance, is ⁠ 7 / 3 ⁠. The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not ...

3. Solving quadratic equations with continued fractions - Wikipedia

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

   By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the ...

4. Continued fraction - Wikipedia

   en.wikipedia.org/wiki/Continued_fraction

   A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.

5. Partial fraction decomposition - Wikipedia

   en.wikipedia.org/wiki/Partial_fraction_decomposition

   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. [1]

6. Periodic continued fraction - Wikipedia

   en.wikipedia.org/wiki/Periodic_continued_fraction

   The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. [4]

7. Mathematics - Wikipedia

   en.wikipedia.org/wiki/Mathematics

   [2] [3] Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . [ 4 ] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when ...