Envir Issues Concerns Fin Sum13106a

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(sometimes written pi) is a mathematical constant whose value isthe ratio of any circle's circumference to its diameter in the Euclidean

plane; this is the same value as the ratio of a circle's area to the

square of its radius. It is approximately equal to 3.14159265 in theusual decimal notation. Many formulae from mathematics, science,

and engineeringinvolve , which makes it one of the most importantmathematical constants.[1]
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is an irrational number, which means that its value cannot be expressedexactly as a fractionm/n, where mand nare integers. Consequently, itsdecimal representation never ends or repeats. It is also a transcendentalnumber, which implies, among other things, that no finite sequence ofalgebraic operations on integers (powers, roots, sums, etc.) can be equal toits value; proving this was a late achievement in mathematical history and asignificant result of 19th century German mathematics. Throughout thehistory of mathematics, there has been much effort to determine moreaccurately and to understand its nature; fascination with the number haseven carried over into non-mathematical culture.Probably because of the simplicity of its definition, the concept of has

become entrenched in popular culture to a degree far greater than almostany other mathematical construct.[2] It is, perhaps, the most common groundbetween mathematicians and non-mathematicians.[3] Reports on the latest,most-precise calculation of (and related stunts) are common newsitems.[4][5][6]The current record for the decimal expansion of , if verified,stands at 5 trillion digits.[7]The Greek letter , often spelled out piin text, was first adopted for thenumber as an abbreviation of the Greek word for perimeter"" (oras an abbreviation for "perimeter/diameter") by William Jones in 1706. Theconstant is also known as Archimedes' Constant, afterArchimedes ofSyracuse who provided an approximation of the number, although this name

for the constant is uncommon in modern English-speaking contexts.
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The name of the Greek letter is pi.[8] The name pi is commonly used as an alternative

to using the Greek letter. As a mathematical symbol, the Greek letter is not capitalized() even at the beginning of a sentence, and instead the lower case () is used at the

beginning of a sentence. When referring to this constant, the symbol is always

pronounced "pie" in English, which is the conventional English pronunciation of the Greekletter. The constantis named "" because "" is the first letter of the Greek word (perimeter), probably referring to its use in the formula perimeter/diameter

which is constant for all circles, the word "perimeter" being synonymous here with"circumference." [9]William Jones was the first to use the Greek letter in this way, in1706,[10] and it was later popularized by Leonhard Euler in 1737.[11][12] William Joneswrote:There are various other ways of finding the Lengths or Areas of particular Curve Lines,or Planes, which may very much facilitate the Practice; as for instance, in the Circle, theDiameter is to the Circumference as 1 to ... 3.14159, &c. = [13]The capital letter pi () has a completely different mathematical meaning; it is used for

expressing products (notice that the word "product" begins with the letter "p" just like"perimeter/diameter" does). It can also refer to the osmotic pressure of a solution.
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The decimal representation of truncated to 50 decimal places is:[20]

= 3.14159265358979323846264338327950288419716939937510...Various online web sites provide to many more digits.[21] While the decimalrepresentation of has been computed to more than a trillion (1012) digits,[22] elementaryapplications, such as estimating the circumference of a circle, will rarely require morethan a dozen decimal places. For example, the decimal representation of truncated to

11 decimal places is good enough to estimate the circumference of any circle that fits

inside the Earth with an error of less than one millimetre, and the decimal representationof truncated to 39 decimal places is sufficient to estimate the circumference of any

circle that fits in the observable universe with precision comparable to the radius of ahydrogen atom.[23][24]Because is an irrational number, its decimal representation does not repeat, andtherefore does not terminate. This sequence of non-repeating digits has fascinated

mathematicians and laymen alike, and much effort over the last few centuries has beenput into computing ever more of these digits and investigating 's properties.[25] Despitemuch analytical work, and supercomputer calculations that have determined over 1trilliondigits of the decimal representation of , no simple base-10 pattern in the digitshas ever been found.[26]Digits of the decimal representation of are available on manyweb pages, and there is software for calculating the decimal representation of to

billions of digits on any personal computer.Estimating
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is an irrational number, meaning that itcannot be written as the ratio of twointegers. is also a transcendentalnumber, meaning that there is nopolynomial with rational coefficients for

which is a root.[16]

An importantconsequence of the transcendence of isthe fact that it is not constructible. Becausethe coordinates of all points that can beconstructed with compass and straightedgeare constructible numbers, it is impossible

to square the circle: that is, it is impossibleto construct, using compass andstraightedge alone, a square whose area isequal to the area of a given circle.[17] Thisis historically significant, for squaring acircle is one of the easily understood

elementary geometry problems left to usfrom antiquity; many amateurs in moderntimes have attempted to solve each ofthese problems, and their efforts aresometimes ingenious, but in this case,doomed to failure: a fact not always

understood by the amateur involved.[18][19]Decimal re resentation
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The Great Pyramid at Giza, constructed c.25892566 BC, was built with a perimeter of1760 cubitsand a height of 280 cubits; the ratio 1760/280 2. The same apotropaicproportions were used earlier at the Pyramid ofMeidum c.2613-2589 BC and later at thepyramid of Abysir c.2453-2422. Some Egyptologists consider this to have been theresult of deliberate design proportion. Verner wrote, "We can conclude that although the

ancient Egyptians could not precisely define the value of , in practice they used it".[39]

Petrie, author ofPyramids and Temples of Gizehconcluded: "but these relations of areasand of circular ratio are so systematic that we should grant that they were in thebuilders design".[40] Others have argued that the Ancient Egyptians had no concept of piand would not have thought to encode it in their monuments. The creation of thepyramid may instead be based on simple ratios of the sides of right angled triangles

(the seked).[41]The early history of from textual sources roughly parallels thedevelopment of mathematics as a whole.[42]Antiquity
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