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A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains ... ("Exact" continued fraction for Pi)
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.
But every number, including π, can be represented by an infinite series of nested fractions, called a simple continued fraction: = + + + + + + + + Truncating the continued fraction at any point yields a rational approximation for π ; the first four of these are 3 , 22 / 7 , 333 / 106 , and 355 / 113 .
In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction /, where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus .
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
A simple or regular continued fraction is a continued fraction with numerators all equal one, ... OEIS sequence A133593 ("Exact" continued fraction for pi)
A History of Pi; In culture; Indiana pi bill; Pi Day; Related topics; Squaring the circle; ... For more on the fourth identity, see Euler's continued fraction formula.
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