Pi in numbers
π is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats. π is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.
Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Perhaps because of the simplicity of its definition, π has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of π are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.
The Greek letter π was first adopted for the number as an abbreviation of the Greek word for perimeter (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse who provided an approximation of the number during the 3rd century BC, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen, who computed a 35-digit approximation around the year 1600.
The Latin name of the Greek letter π is pi. When referring to the constant, the symbol π is pronounced like the English word "pie", the conventional English pronunciation of the Greek letter. The constant is named "π" because "π" is the first letter of the Greek word περιφέρεια "periphery" (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler in 1737. William Jones wrote:
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, the Diameter is to the Circumference as 1 to ... 3.14159, etc. = π ...
When used as a symbol for the mathematical constant, the Greek letter (π) is not capitalized at the beginning of a sentence. The capital letter Π (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference C to its diameter d:
The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.
This definition depends on results of Euclidean geometry, such as the fact that all circles are similar, which can be a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which the trigonometric function cos(x) equals zero
Irrationality and transcendence
π is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.
The decimal representation of π truncated to 50 decimal places is:
π = 3.14159265358979323846264338327950288419716939937510... (sequence A000796 in OEIS).
Various online web sites provide π to many more digits. While the decimal representation of π has been computed to more than a trillion digits, elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π 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 millimeter, and the decimal representation of π 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 a hydrogen atom.
Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties. Despite much analytical work, and supercomputer calculations that have determined over 10 trillion digits  of the decimal representation of π, no simple base-10 pattern in the digits has ever been found. Digits of the decimal representation of π are available on many web pages, and there is software for calculating the decimal representation of π to billions of digits on any personal computer.
Nobel-winning poet Wisława Szymborska once wrote a poem about pi:
The caravan of digits that is pi does not stop at the edge of the page, but runs off the table and into the air, over the wall, a leaf, a bird's nest, the clouds, straight into the sky, through all the bloatedness and bottomlessness. Oh how short, all but mouse-like is the comet's tail!
- 1120 digits of pi: 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959
- Pi to the 1st 1,000,000 digits:
Don't forget to celebrate Pi Day!