Circles space all similar, and also "the circumference divided by the diameter" to produce the same value regardless of your radius. This worth is the ratio of the circumference of a circle to its diameter and is referred to as π (Pi). This consistent appears in the calculation of the area that a circle, and is a type of an irrational number recognized as a transcendental number that deserve to be expressed neither by a portion nor by any type of radical authorize such together a square root, nor your combination. The number has actually an infinite variety of decimal places, namely, 3.1415926535..., and it has actually now been computed to 5 trillion decimal locations by computers.

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


As because that the value of π, ancient civilizations provided their own. Together a regular hexachathamtownfc.netn that is inscribed in a circle v a radius of 1 has actually a perimeter of 6, the is revealed that Pi has actually a value better than 3. In the old Egypt, they acquired an approximation of

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(approximately, 3.16)

by place a constant octachathamtownfc.netn ~ above a circle, and in old Babylonia they used

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.

Archimedes pertained to the conclusion in his work-related Kyklu metresis (measure the a circle) that Pi satisfies

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Enri (en method a circle and also ri means a theory), in which much more accurate values for Pi to be calculated, began 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 completed results that could be contrasted to european mathematics.

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

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

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the Grechathamtownfc.netry (1638-1675) and also Leibniz (1646-1716) Formula:

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Moreover, Newton (1642-1727) and Euler (1707-1783) discovered a series that converged faster, which enabled them to calculate worths of Pi to an ext decimal places. If we usage the relation

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found by J. Machin (1680-1752),

we can achieve a worth of 3.14159 because that π precise to five decimal places with the an initial 4 regards to the Taylor development of tan-1.In a recent computer calculation, the adhering to equations to be 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 end of Sanpo shojo, a an approach for calculating Pi appears. To summary what is explained in the book, the an approach is as follows: i think the original number=3,

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

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When incrementing n for

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(the amount of the powers of the herbal numbers),

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holds true; Hasegawa uses this to attain the an outcome of

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We perform not recognize anything around the number"s regularity from this an outcome alone. In fact, however, over there is a relationship between the terms. Every term is chose by multiplying its previous term by a regular portion as follows:

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

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Therefore, Kikuchi verified in the next record that the calculate

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in Enri shinko by Wada Yasushi was tantamount to

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Hasegawa"s calculate

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was indistinguishable to

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and that Matsunaga"s

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was identical to

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.

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

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, acquired by Takebe, is the very first formula to evaluate Pi in the history of Wasan. Takebe calculated π come 41 decimal areas with this formula. In the following treatise, Kikuchi derived

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to to express the square of s or the arc of a circle with sagitta c and also diameter d, i m sorry 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 also Diameter of a Circle

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

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In the critical paper, he showed that

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derived by Ajima Naonobu in Kohai jutsukai might be streamlined to

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In the following year, Kikuchi also wrote a record to present a technique of calculating the length of an arc derived by Takebe Katahiro in Tokyo Sugaku Butsuri Gakkai Kiji Vol. 8 (1897). This collection of papers was intended to introduce to the people the fact that theories he had found in calculations of Pi in Wasan were equivalent to calculus in the West.