There's a pair of recurrences related to successive numerators and denominators (I can refer you to some pages on this, easily) that is used to prove the fact that continued fractions, even in the case of irrationals such as pi, e, and various sqrt()s, MUST "converge" towards that particular value. The numerator sequence is often denoted by p(i) or P(i) and the denominator sequence is often denoted by q(i) or Q(i) in literature I've seen. It turns out that the even pairs are always less and the odd pairs always more, but in exactly this way:
This fact is developed through mathematical induction. I can describe it here, but most folks wouldn't care. (However, I did post something about it a week ago or so in sci.electronics.design.) The main point, though, is that the value is bounded by the even and odd convergents and trapped tighter and tighter as i goes towards infinity. Which, to me, makes the term very appropriate. Approximant just doesn't carry the meaning so well, to me.
25.4 is 254/10 AIUI the value 25.4mm was chosen for the inch in 1959 because it was close to and between the values the Brittish and Americans were using for the inch
prior to this US and Canadian miles were different sizes which was fun for surveyors.
"This left NGS in the position of not wanting to mandate which foot (U.S. Survey or International) a state should use. So, NGS left that decision to the individual states. Currently, NGS publishes SPCs for 7 states using the U.S. Survey Foot conversion factor, 1 state using the International Foot conversion factor, and 42 states using only meters, not feet, for SPCs. Based on STATE legislation we have or know about, 24 states have legislated the U.S. Survey Foot, 8 states have legislated the International Foot, and 18 states have no legislation on which conversion factor must be used."
What, you think someone had a warehouse full of 127-tooth gears they wanted to get rid of? :-). 127 is a prime number.
25.6mm/inch would match 256/10 and 256 is not a prime number, it is divisible by 2 and 4 and 8 and 16 and 32 and 64 and 128.
25.5mm/inch would match 255/10 and 255 is not a prime number, it is divisible by 3 and 5 and 17.
Those would've been more inspired choices if you wanted to use commonly-available gears.
No, there's no significance at all. The 25.4 value is a close approximation, but not exact. A meter originally was some even fraction of the distance between the equator and the north pole (I think), based on the crude measurements available at the time. The foot is based on the length of some famous person's foot. A meter is approximately 39.37 inches, and the inverse of that, with some scaling is ~25.4 mm/inch
10,000 kilometers, or 1E7 meters, from the equator to the north pole along the Paris meridian. Later measurements and a better understanding of the shape of this lump have refined the distance somewhat but that's still a good approximation to three sig figs.
International inch Effective July 1, 1959, the United States and countries of the British Commonwealth defined the length of the international yard to be 0.9144 meters. [1] Consequently, the international inch is defined to be equal to
25.4 millimeters.
The international standard symbol for inch is in (see ISO 31-1, Annex A). In some cases, the inch is denoted by a double prime, which is often approximated by double quotes, and the foot by a prime, which is often approximated by an apostrophe.
On what basis is one inch exactly equal to 25.4 mm? Has the imperial inch been adjusted to give this exact fit and if so when? (FAQ - Length) The (international) inch has been exactly 25.4 mm since July 1959. At this point in time the (international) yard was redefined as 0.9144 metre - until this time the ratio between the US yard and the metre was different to the ratio between the UK yard and the metre. For more information, see Engineering Metrology by K J Hume (2 ed) Macdonald London 1967. The American inch changed by 2 millionths of an inch and the UK inch by 1.7 millionths of an inch. The international inch falls mid way between the old UK and US inch.
References:
a.. The Yard Unit of Length Nature Vol. 200 No 4908 pp 730-732 23 Nov 1963 b.. The United Kingdom standards of the yard in terms of the metre (British Applied Journal of Physics)
------ Standardization of the Inch Don Hillger, PhD Most people do not realize that the standardization of the inch for worldwide use did not occur until 1959. Prior to that the inch had been defined differently among the major inch-using countries: the U.S., Great Britain, and Canada. Each of those countries had their own definition of the inch, and in each case the inch was defined in terms of metric units, the only set of internationally-accepted standards of length, mass, etc.
In the U.S. the metric system was made legal for all purposes, by the Metric Act of 1866, long before any law defined our common U.S. measures. Later, the Mendenhall Order of 1893 defined our common non-metric units in terms of metric units. That law regarded metric units as the fundamental and internationally-accepted standards for the U.S. It was this law that formally defined the inch based on the conversion factor of 39.37 inches = 1 meter as stated in the Act of 1866. This ratio gives an inch approximately equal to 25.40005 mm.
In Great Britain the National Physical Laboratory made comparisons of the Imperial Standard Yard to the International Meter, which yielded differing values for the inch over the years. The 1922 value of 25.399956 mm per inch by was arbitrarily selected for use in calibrating the most precise measuring devices.
The Canadian Parliament in 1951 established their inch based on a legal definition of the yard as 0.9144 m. This ratio defined the inch as 25.4 mm, a third definition of the inch. The Canadian inch was about 2 parts in 106 smaller than the U.S. standard and about 2 parts in 106 larger than the British standard.
The differences in definitions of the inch were enough to cause confusion, inefficiencies, and difficulties during World War II in attempts to interchange various precision products. It was not until later, in 1959, that the definition of the inch was standardized worldwide as 25.4 millimeters exactly.
But that agreement has not completely solved all the problems caused by differing values for the inch. A problem still exists for the foot, where the international foot (based on the 25.4 mm inch) and the survey foot (based on the 25.40005 mm inch) are both still in use. The Coast and Geodetic Survey continues to use the survey foot, whereas the rest of industry uses the 25.4 mm inch. This leaves us with two definitions of the mile, one based on the international foot and the other based on the survey foot. Although this may not seem like much, it causes the two miles to differ by about 3.2 mm (1/8 inch), or in 100 miles to differ by about 32 cm (over one foot)!
The way i was told, that a number of men coming out of a church were lined up toe-to-heel and the total length divided by the number of men became the foot; that number may have been ten.
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