Circles are all similar, and also "the circumference split by the diameter" produce the same value regardless of your radius. This value is the proportion of the one of a circle to its diameter and is called π (Pi). This continuous appears in the calculate of the area the a circle, and is a type of an irrational number well-known as a transcendental number that have the right to be expressed no by a fraction nor by any radical authorize such together a square root, nor your combination. The number has an infinite number of decimal places, namely, 3.1415926535..., and it has actually now been computed come 5 trillion decimal places by computers.

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The circumference is better than 6 indigenous the figure. As the diameter the the one is 2, Pi is greater than 3.


As because that the worth of π, ancient civilizations supplied their own. As a constant hexacouchsurfingcook.comn the is inscriptions in a circle v a radius that 1 has a perimeter the 6, the is revealed that Pi has actually a value better than 3. In the ancient Egypt, they derived an approximation of

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(approximately, 3.16)

by put a regular octacouchsurfingcook.comn on a circle, and in old Babylonia castle used

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.

Archimedes concerned the conclusion in his work Kyklu metresis (measure of a circle) that Pi satisfies

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Enri (en means a circle and also ri means a theory), in which much more accurate values for Pi to be calculated, started to evolve. Wasan scholars such as Muramatsu Shigekiyo, Seki Takakazu, Kamata Toshikiyo, Takebe Katahiro, and also Matsunaga Yoshisuke calculated more accurate values of Pi, and achieved results that could be compared to european mathematics.

In Europe, Viete (1540-1603) uncovered the an initial formula the expresses π:

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After that, the Wallis (1616-1703) Formula:

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the Grecouchsurfingcook.comry (1638-1675) and also Leibniz (1646-1716) Formula:

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Moreover, Newton (1642-1727) and also Euler (1707-1783) uncovered a series that converged faster, which allowed them to calculate values of Pi to an ext decimal places. If we use the relationship

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discovered by J. Machin (1680-1752),

we can attain a value of 3.14159 for π precise to 5 decimal places with the very first 4 regards to the Taylor development of tan-1.In a recent computer system calculation, the complying with equations were used:

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or

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* tan-1 : Arc tangent. The inverse duty of tangent.

Calculation that Pi in Wasan


At the finish of Sanpo shojo, a an approach for calculating Pi appears. To summary what is described in the book, the technique is as follows: i think the original number=3,

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Continue this until distinction 100 is created. Then, Pi is obtained by adding the original number, distinction 1, distinction 2, difference 3...and for this reason on. Rewriting this together a math expression, it is presented to have following regularity:

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When incrementing n for

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(the amount of the strength of the organic numbers),

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holds true; Hasegawa uses this to attain the result of

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We carry out not know anything around the number"s regularity indigenous this an outcome alone. In fact, however, over there is a relationship between the terms. Each term is chose by multiplying its previous hatchet by a regular fraction as follows:

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Kikuchi noticed that such a series was what K. F. Gauss (1777-1855) named a hypergeometric series. A hypergeometric series is characterized as follows:

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Therefore, Kikuchi confirmed in the next record that the calculation

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in Enri shinko by Wada Yasushi was indistinguishable to

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Hasegawa"s calculation

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was equivalent to

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and the Matsunaga"s

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was equivalent to

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.

In Wasan, Seki Takakazu, Takebe Katahiro, etc., sought calculation formulas for π2.

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, derived by Takebe, is the first formula to advice Pi in the background of Wasan. Takebe calculate π to 41 decimal locations with this formula. In the following treatise, Kikuchi acquired

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to refer the square of s or the arc of a circle v sagitta c and diameter d, which was explained by Yamaji Nushizumi in Kenkon no maki (c 1765), and proved

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because

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as soon as

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.


Arc, Sagitta, and Diameter of a Circle

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In the figure,
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,a part of the circumference, is dubbed ko (arc), the segment abdominal muscle is referred to as gen (chord), and the segment PR is ya (sagitta).The diameter PQ is referred to as kei in Japanese.When we draw a chord because that the arc PB and also a sagitta because that the chord, and continue come repeat this process with much shorter chords, the shape obtained by connecting this chords viewpoints that of a circle. This way, Yamaji calculates s, the size of the arc, once the diameter is d and the length of the sagitta is c.

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In the last paper, he verified that

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derived by Ajima Naonobu in Kohai jutsukai might be simplified to

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In the complying with year, Kikuchi additionally wrote a document to present a method of calculating the size of one arc acquired by Takebe Katahiro in Tokyo Sugaku Butsuri Gakkai Kiji Vol. 8 (1897). This collection of papers was plan to introduce to the civilization the reality that theory he had uncovered in calculations the Pi in Wasan were equivalent to calculus in the West.