Search results
Results from the WOW.Com Content Network
Continued fractions can also be applied to problems in number theory, ... {x 1, x 3, x 5, ... which can be simplified by noting that 5 / 10 ...
This method was an improvement compared to other methods because it started from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and was numerically stable. The original algorithm uses algebra to bypass a zero in either the numerator or denominator. [5]
The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges. When the monic quadratic equation with real coefficients is of the form x 2 = c, the general solution described above is useless because division by zero is not well ...
Taniyama's problems [3] 36 – Yutaka Taniyama: 1955 Thurston's 24 questions [4] [5] 24 – William Thurston: 1982 Smale's problems: 18: 14: Stephen Smale: 1998 Millennium Prize Problems: 7: 6 [6] Clay Mathematics Institute: 2000 Simon problems: 15 < 12 [7] [8] Barry Simon: 2000 Unsolved Problems on Mathematics for the 21st Century [9] 22 ...
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]
The method of equating coefficients is often used when dealing with complex numbers. For example, to divide the complex number a + bi by the complex number c + di , we postulate that the ratio equals the complex number e+fi , and we wish to find the values of the parameters e and f for which this is true.
This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction. If we have an equation =, where x is a variable we are interested in solving for, we can use cross-multiplication to determine that =.
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value .