Circles space all similar, and "the circumference split by the diameter" to produce the same value nevertheless of your radius. This value is the proportion of the one of a circle come its diameter and is dubbed π (Pi). This consistent appears in the calculation of the area of a circle, and is a form of one irrational number recognized as a transcendental number that have the right to be expressed neither by a portion nor by any radical authorize such together a square root, nor your combination. The number has an infinite variety of decimal places, namely, 3.1415926535..., and it has actually now been computed come 5 trillion decimal areas by computers.

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


As for the value of π, ancient civilizations used their own. Together a continuous hexausmam.orgn the is enrolled in a circle through a radius the 1 has actually a perimeter the 6, the is revealed that Pi has actually a value better than 3. In the ancient Egypt, they acquired an approximation of

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(approximately, 3.16)

by place a continual octausmam.orgn top top a circle, and also in ancient Babylonia castle used

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.

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

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Enri (en means a circle and ri means a theory), in which an ext accurate worths for Pi were calculated, began to evolve. Wasan scholar such together Muramatsu Shigekiyo, Seki Takakazu, Kamata Toshikiyo, Takebe Katahiro, and Matsunaga Yoshisuke calculated an ext accurate worths of Pi, and accomplished results that can be contrasted to europe 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 Greusmam.orgry (1638-1675) and also Leibniz (1646-1716) Formula:

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Moreover, Newton (1642-1727) and also Euler (1707-1783) found a collection that converged faster, which permitted 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 because that π precise to five decimal places with the first 4 terms of the Taylor growth of tan-1.In a recent computer calculation, the complying with equations to be used:

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or

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

Calculation that Pi in Wasan


At the finish of Sanpo shojo, a method for calculating Pi appears. To summary what is defined in the book, the an approach is together follows: assume the initial number=3,

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Continue this until difference 100 is created. Then, Pi is obtained by adding the initial number, difference 1, difference 2, difference 3...and for this reason on. Rewriting this together a mathematics expression, it is presented to have complying with 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 offers this to attain the an outcome of

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We carry out not understand anything about the number"s regularity indigenous this an outcome alone. In fact, however, there is a relationship between the terms. Every term is made decision by multiply its previous hatchet by a regular portion 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 identified as follows:

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Therefore, Kikuchi showed in the next document that the calculation

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in Enri shinko by Wada Yasushi was tantamount 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 indistinguishable to

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.

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

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

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to refer the square the s or the arc the a circle with sagitta c and also diameter d, which was defined 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 that a Circle

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In the figure,
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,a component of the circumference, is called ko (arc), the segment abdominal muscle is called gen (chord), and also the segment PR is ya (sagitta).The diameter PQ is dubbed kei in Japanese.When we attract a chord because that the arc PB and also a sagitta because that the chord, and continue to 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 also the size of the sagitta is c.

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In the last paper, he proved that

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derived through Ajima Naonobu in Kohai jutsukai might be streamlined to

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In the complying with year, Kikuchi also wrote a file to introduce a technique of calculating the length of one arc derived by Takebe Katahiro in Tokyo Sugaku Butsuri Gakkai Kiji Vol. 8 (1897). This collection of files was to plan to present to the world the fact that theory he had uncovered in calculations that Pi in Wasan were comparable to calculus in the West.