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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, a 10 cm (3.9 in) circular would have a maximum precision approximately equal to a 31.4 cm (12.4 in) ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be reoriented when results are near 1.0—the rule is always on scale.
13 cm = 1.3 dm – body length of a Goliath birdeater; 15 cm = 1.5 dm – approximate size of largest beetle species; 19 cm = 1.9 dm – length of a banana; 26.3 cm = 2.6 dm – length of average male human foot; 29.98 cm = 2.998 dm – distance light in vacuum travels in one nanosecond; 30 cm = 3.0 dm – maximum leg length of a Goliath birdeater
A football field is often used as a comparative measurement of length when talking about distances that may be hard to comprehend when stated in terms of standard units. An American football field is usually understood to be 100 yards (91 m) long, though it is technically 120 yards (110 m) when including the two 10 yd (9.1 m) long end zones.
Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 −5). Its simple continued fraction is periodic and given by:
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011.
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 ...
More precisely, the order of magnitude of a number can be defined in terms of the common logarithm, usually as the integer part of the logarithm, obtained by truncation. [ contradictory ] For example, the number 4 000 000 has a logarithm (in base 10) of 6.602; its order of magnitude is 6.