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Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days. [42] In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of π.
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
Julian Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context. [2] The purpose of the proof is not primarily to convince its readers that 22 / 7 (or 3 + 1 / 7 ) is indeed bigger than π. Systematic methods of computing the value of π ...
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.
In English and many other languages (including many that are written right-to-left), the integer part is at the left of the radix point, and the fraction part at the right of it. [ 24 ] A radix point is most often used in decimal (base 10) notation, when it is more commonly called the decimal point (the prefix deci- implying base 10 ).
Pi Approximation Day is observed on July 22 (22/7 in the day/month date format), since the fraction 22 ⁄ 7 is a common approximation of π, which is accurate to two decimal places and dates from Archimedes. [33] In Indonesia, a country that uses the DD/MM/YYYY date format, some people celebrate Pi Day every July 22. [34]
Approximating an irrational number by a fraction π: 22/7 1-digit-denominator Approximating a rational number by a fraction with smaller denominator 399 / 941 3 / 7 1-digit-denominator Approximating a fraction by a fractional decimal number: 5 / 3 1.6667: 4 decimal places: Approximating a fractional decimal number by one with fewer digits 2.1784
Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. [3]