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A continued fraction is a mathematical expression that can be written as a fraction with a ... is a continuous ... Here are three of π 's best-known ...
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 ...
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
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.
For example, the fraction 22 / 7 is greater than π, but 22 / 7 − π is less than 1 / 7 × 106 = 1 / 742 (in fact, 22 / 7 − π is just more than 1 / 791 = 1 / 7 × 113 ).
mathematical constant π; 3.14159 26535 89793 23846 26433... Uses; Area of a circle; ... For more on the fourth identity, see Euler's continued fraction formula.
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 numeric value of π is approximately: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510... (sequence A000796 in the OEIS). Unusually good approximations are given by the fractions 22/7 and 355/113. Memorizing as well as computing increasingly more digits of π is a world record pursuit.