# Circle number

The Greek letter pi is the symbol of the circle number.
A circle with the diameter 1 has the circumference .${\ displaystyle \ pi}$

The circle number ( Pi ), also Ludolph's number, Ludolf's number or Archimedes constant , is a mathematical constant that is defined as the ratio of the circumference of a circle to its diameter . This ratio is independent of the size of the circle. is a transcendent and therefore also irrational number and occurs in numerous sub-areas of mathematics, also outside of geometry . The decimal expansion of the circle number starts with being in practical calculations often only three significant digits are used: . ${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi = 3 {,} 1415926 \ ldots,}$${\ displaystyle \ pi}$${\ displaystyle \ pi \ approx 3 {,} 14}$

## History of the designation

The circle number and some of its properties were already known in ancient times.

The designation with the Greek letter Pi ( )${\ displaystyle \ pi}$ (after the first letter of the Greek word περιφέρεια  - Latin peripheria , "edge area" or περίμετρος  - perimetros , "circumference") was first used by William Oughtred in his Theorematum in libris Archimedis de , published in 1647 Sphæra & Cylyndro Declaratio used. In it he expressed the ratio of half the circumference (semiperipheria) to the radius (semidiameter) , i.e. H.${\ displaystyle {\ tfrac {\ pi} {\ delta}}}$${\ displaystyle {\ tfrac {\ pi} {\ delta}} = 3 {,} 1415 \ ldots}$

The English mathematician Isaac Barrow used the same names around 1664 .

David Gregory took (1697) for the ratio of circumference to radius. ${\ displaystyle {\ tfrac {\ pi} {\ rho}}}$

59 years later than Oughtred, namely in 1706, the Welsh mathematician William Jones was the first to use the Greek lowercase letter in his Synopsis Palmariorum Matheseos to express the ratio of circumference to diameter . ${\ displaystyle \ pi}$

It was only popularized in the 18th century by Leonhard Euler . He first used it for the circle number in 1737 , having previously used it. Since then, due to the importance of Euler, this name has been common. ${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle p}$

## definition

Circle with drawn center radius and diameter${\ displaystyle M,}$${\ displaystyle r}$${\ displaystyle d}$

There are several equivalent approaches to defining the circle number  . ${\ displaystyle \ pi}$

The first (classic!) Definition in geometry is that according to which the circle number is a ratio that numerically corresponds to the quotient formed from the circumference of a circle and the associated diameter . The second approach to geometry is related and consists in understanding the number of circles as the quotient that is formed from the area of a circle and the area of ​​a square over a radius ( length ) . (This radius is called the radius of a circle. ) This second definition is summarized in the motto that a circular area is related to the surrounding square area as . ${\ displaystyle U}$ ${\ displaystyle d}$ ${\ displaystyle \ pi = {\ tfrac {U} {d}}}$${\ displaystyle \ pi = {\ tfrac {A} {r ^ {2}}}}$ ${\ displaystyle A}$ ${\ displaystyle r}$${\ displaystyle \ pi: 4}$

In analysis (according to Edmund Landau ) one often proceeds as follows: first of all, the real cosine function is defined via its Taylor series and then the circle number is defined as twice the smallest positive zero of the cosine. Further analytical approaches go back to John Wallis and Leonhard Euler . ${\ displaystyle \ cos (x)}$

## properties

### Irrationality and transcendence

The number is an irrational number , i.e. a real , but not a rational number . This means that it cannot be represented as a ratio of two whole numbers , i.e. not as a fraction . This was proven by Johann Heinrich Lambert in 1761 (or 1767) . ${\ displaystyle \ pi}$ ${\ displaystyle p, q \ in \ mathbb {Z}}$ ${\ displaystyle {\ tfrac {p} {q}}}$

In fact, the number is even transcendent , which means that there is no polynomial with rational coefficients that has as a zero . This was first demonstrated by Ferdinand von Lindemann in 1882. The consequence of this is that it is impossible to express only with whole numbers or fractions and roots, and that the exact squaring of the circle with a compass and ruler is not possible. ${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$

### The first 100 decimal places

Since is an irrational number , its representation cannot be fully specified in any place value system : The representation is always infinitely long and not periodic . At the first 100 decimal places in the decimal fraction development ${\ displaystyle \ pi}$

${\ displaystyle \ pi}$ = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 9 ...

no regularity is evident. Statistical tests for randomness are also sufficient for further decimal places. See also the section on normality .

### Continued fraction development

An alternative way to represent real numbers is the continued fraction expansion . Since is irrational, this representation is infinitely long. The regular continued fraction of the circle number begins like this: ${\ displaystyle \ pi}$

${\ displaystyle \ pi = 3 + {\ frac {1} {7 + {\ frac {1} {15 + {\ frac {1} {1 + {\ frac {1} {292 + {\ frac {1} {1 + {\ frac {1} {1+ \ ddots}}}}}}}}}}}}}}$

A development of related to the regular continued fraction expansion is that of a negative regular continued fraction (sequence A280135 in OEIS ): ${\ displaystyle \ pi}$

${\ displaystyle \ pi = 4 - {\ frac {1} {2 - {\ frac {1} {2 - {\ frac {1} {2 - {\ frac {1} {2 - {\ frac {1} {2 - {\ frac {1} {2 - {\ frac {1} {17 - {\ frac {1} {294- \ ddots}}}}}}}}}}}}}}}}}}$

In contrast to Euler's number , no patterns or regularities could be found in the regular continued fraction representation of . ${\ displaystyle e}$${\ displaystyle \ pi}$

However, there are non-regular continued fraction representations of in which simple laws can be recognized: ${\ displaystyle \ pi}$

${\ displaystyle \ pi = 3 + {\ frac {1 ^ {2}} {\ scriptstyle 6 + {\ frac {3 ^ {2}} {6 + {\ frac {5 ^ {2}} {6+ { \ frac {7 ^ {2}} {6 + {\ frac {9 ^ {2}} {6 + {\ frac {11 ^ {2}} {6+ \ ddots}}}}}}}}}}} }} = {\ frac {4} {1 + {\ frac {1 ^ {2}} {2 + {\ frac {3 ^ {2}} {2 + {\ frac {5 ^ {2}} {2 + {\ frac {7 ^ {2}} {2 + {\ frac {9 ^ {2}} {2+ \ ddots}}}}}}}}}}}}} = {\ frac {4} {\ scriptstyle 1 + {\ frac {1 ^ {2}} {3 + {\ frac {2 ^ {2}} {5 + {\ frac {3 ^ {2}} {7 + {\ frac {4 ^ {2 }} {9 + {\ frac {5 ^ {2}} {11+ \ ddots}}}}}}}}}}}}}}$

### Approximate fractions of the circle number

From their regular continued fraction representation, the following are the best approximate fractions of the circle number (numerator sequence A002485 in OEIS or denominator sequence A002486 in OEIS ):

step Chain fraction Approximation break Decimal notation
(wrong digits in red)
Absolute error
when calculating the circumference of a circle
with a diameter of 1000 km
${\ displaystyle {\ frac {p_ {0}} {q_ {0}}}}$ ${\ displaystyle [3]}$ ${\ displaystyle {\ frac {3} {1}}}$ ${\ displaystyle 3 {,} {\ color {red} 0}}$ - 141.59 km
${\ displaystyle {\ frac {p_ {1}} {q_ {1}}}}$ ${\ displaystyle [3; 7]}$ ${\ displaystyle {\ frac {22} {7}}}$ ${\ displaystyle 3 {,} 14 {\ color {red} 2 \; 85 \; \ ldots}}$ + 1.26 km
${\ displaystyle {\ frac {p_ {2}} {q_ {2}}}}$ ${\ displaystyle [3; 7.15]}$ ${\ displaystyle {\ frac {333} {106}}}$ ${\ displaystyle 3 {,} 141 \; 5 {\ color {red} 09 \; 4 \ ldots}}$ - 83.22 m
${\ displaystyle {\ frac {p_ {3}} {q_ {3}}}}$ ${\ displaystyle [3; 7,15,1]}$ ${\ displaystyle {\ frac {355} {113}}}$ ${\ displaystyle 3 {,} 141 \; 592 \; {\ color {red} 920 \; \ ldots}}$ + 26.68 cm
${\ displaystyle {\ frac {p_ {4}} {q_ {4}}}}$ ${\ displaystyle [3; 7,15,1,292]}$ ${\ displaystyle {\ frac {103993} {33102}}}$ ${\ displaystyle 3 {,} 141 \; 592 \; 653 \; {\ color {red} 011 \; \ ldots}}$ - 0.58 mm
${\ displaystyle {\ frac {p_ {5}} {q_ {5}}}}$ ${\ displaystyle [3; 7,15,1,292,1]}$ ${\ displaystyle {\ frac {104348} {33215}}}$ ${\ displaystyle 3 {,} 141 \; 592 \; 653 \; {\ color {red} 921 \; \ ldots}}$ + 0.33 mm
${\ displaystyle \ vdots}$
${\ displaystyle {\ frac {p_ {10}} {q_ {10}}}}$ ${\ displaystyle [3; 7,15,1,292,1,1,1,2,1,3]}$ ${\ displaystyle {\ frac {4272943} {1360120}}}$ ${\ displaystyle 3 {,} 141 \; 592 \; 653 \; 589 \; {\ color {red} 389 \; \ ldots}}$ - 0.4 µm
(wavelength of blue light )
${\ displaystyle \ vdots}$
${\ displaystyle {\ frac {p_ {20}} {q_ {20}}}}$ ${\ displaystyle [3; 7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1]}$ ${\ displaystyle {\ frac {21053343141} {6701487259}}}$ ${\ displaystyle 3 {,} 141 \; 592 \; 653 \; 589 \; 793 \; 238 \; 462 \; {\ color {red} 381 \; \ ldots}}$ - 2.6 · 10 −16  m
(smaller than a proton )

The absolute error in practice quickly becomes negligible: With the 20th approximation, 21 decimal places correspond to those of the circle number . With this approximate fraction, the circumference of a circle with a diameter of around 3.8 trillion km (which corresponds to the distance to the North Star ) would be incorrectly calculated by one millimeter (too short here). ${\ displaystyle \ left ({\ tfrac {p_ {20}} {q_ {20}}} \ right)}$${\ displaystyle \ pi}$

### Spherical geometry

The term circle number is not used in spherical geometry, since the ratio of circumference to diameter in this case is no longer the same for all circles, but depends on their size. For a circle with a much smaller diameter than that of the sphere on whose surface it is "drawn" (for example a circle with a diameter of 1 m on the spherical surface of the earth), the curvature of the spherical surface compared to the Euclidean circular plane is usually negligibly small, at It must be taken into account in larger circles or with high precision requirements.

### normality

A particularly topical mathematical question at the moment is whether it is a normal number , i.e. whether it contains every possible finite binary or other group of digits in a binary (or any other n-ary ) number representation - just like it statistics would be expected if a number were generated entirely at random . ${\ displaystyle \ pi}$

Ultimately, this would mean, for example, that all books written so far and in the future must be contained somewhere in coded binary form (analogous to the Infinite Monkey Theorem ). ${\ displaystyle \ pi}$

In 2000, Bailey and Crandal used the Bailey-Borwein-Plouffe formula to show that the normality from base 2 can be reduced to a conjecture of chaos theory . ${\ displaystyle \ pi}$

In 2005, physicists at Purdue University examined the first 100 million decimal places of for their randomness and compared them with commercial random number generators . The researcher Ephraim Fischbach and his colleague Shu-Ju Tu could not discover any hidden patterns in the number . According to Fischbach, the number is actually a good source of randomness. However, some random number generators performed even better than . ${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$

So far it is not even known whether, for example, only the digits 5 and 6 appear from one point onwards.

## History of calculation

The necessity of determining the circumference of a circle from its diameter or vice versa arises in very practical everyday life - you need such calculations for fitting a wheel, for fencing in round enclosures, for calculating the area of ​​a round field or the volume of a cylindrical granary. Therefore, people looked for the exact circle number early on and made more and more accurate estimates.

Finally, the Greek mathematician Archimedes succeeded around 250 BC. Chr. To limit the number mathematically. In the rest of the story, the attempts to get as close as possible to a real record hunt, which at times took on bizarre and self-sacrificing features. ${\ displaystyle \ pi}$

### First estimates

The oldest known arithmetic book in the world, the ancient Egyptian arithmetic book of Ahmes (also Papyrus Rhind, 16th century BC), names the value.  As an approximation for the Babylonians used just 3 or also${\ displaystyle \ left ({\ tfrac {16} {9}} \ right) ^ {2} \ approx 3 {,} 1605.}$${\ displaystyle \ pi}$${\ displaystyle 3 + {\ tfrac {1} {8}} = 3 {,} 125.}$

The rough Babylonian value of 3 can also be found in the biblical description of the water basin that was created for the Jerusalem temple :

“Then he made the sea. It was cast from bronze and measured ten cubits from edge to edge; it was perfectly round and five cubits high. A cord of 30 cubits could span it all around. "

- 1 Kings 7.23  EU

The value 3 was also used in ancient China . In India one took for the circle number in the Sulbasutras , the string rules for the construction of altars, the value  and a few centuries BC. Chr. In astronomy the approximate value Indian mathematician and astronomer Aryabhata is 498 n. Chr., The ratio of the circumference to the diameter with at. ${\ displaystyle \ left ({\ tfrac {26} {15}} \ right) ^ {2} \ approx 3 {,} 0044}$ ${\ displaystyle {\ sqrt {10}} \ approx 3 {,} 1623.}$${\ displaystyle {\ tfrac {62832} {20000}} = 3 {,} 1416}$

### Archimedes of Syracuse

#### The approach: constant ratio for area and perimeter calculation

The sum of the areas of the gray moon corresponds to the area of ​​the right triangle

Archimedes of Syracuse proved that the circumference of a circle is related to its diameter in the same way as the area of ​​the circle is related to the square of the radius. The respective ratio therefore gives the circle number in both cases. For Archimedes and for many mathematicians after him, it was unclear whether the calculation of would not come to an end at some point, i.e. whether it was a rational number, which makes the centuries-long hunt for the number understandable. Although the Greek philosophers knew the existence of such numbers because of the irrationality of , Archimedes had no reason to exclude a rational representation of the area calculation for a circle from the outset . Because there are certainly areas that are curvilinearly delimited on all sides, which are even enclosed by parts of a circle, which can be represented as a rational number like the little moon of Hippocrates .${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle {\ sqrt {2}}}$

It was not until 1761/1767 that Johann Heinrich Lambert was able to prove the long suspected irrationality of . ${\ displaystyle \ pi}$

#### Approximation by polygons

Approaching a circle by inscribing pentagons, hexagons and octagons around and around

Archimedes, like other researchers, tried to approximate the circle with regular polygons and in this way to gain approximations for . With circumscribed and inscribed polygons, starting with hexagons, by repeatedly doubling the number of corners up to 96 corners, he calculated upper and lower bounds for the circumference. He came to the conclusion that the ratio had to be a little smaller than , but larger than :${\ displaystyle \ pi}$${\ displaystyle 3 + {\ tfrac {10} {70}}}$${\ displaystyle 3 + {\ tfrac {10} {71}}}$

${\ displaystyle 3 {,} 1408450 \ approx 3 + {\ frac {10} {71}} <\ pi <3 + {\ frac {10} {70}} \ approx 3 {,} 1428571}$

According to Heron , Archimedes had an even more accurate estimate, but it has been reported incorrectly:

${\ displaystyle 3 + {\ frac {9552} {67441}} <\ pi <3 + {\ frac {10835} {62351}} \ qquad (3 {,} 1416349 <\ pi <3 {,} 1737743)}$

Wilbur Knorr corrected:

${\ displaystyle 3 + {\ frac {8915} {62991}} <\ pi <3 + {\ frac {9552} {67441}} \ qquad (3 {,} 1415281 <\ pi <3 {,} 1416349)}$

### More precisely and precisely - 3rd to 18th centuries

As in many other areas of society and culture, mathematics in western cultures experienced a very long period of standstill after the end of antiquity and during the Middle Ages. During this time, progress in approaching was made primarily by Chinese and Persian scientists. ${\ displaystyle \ pi}$

In the third century, Liu Hui determined the limits 3.141024 and 3.142704 from the 192-corner and later the approximate value 3.1416 from the 3072-corner.

The Chinese mathematician and astronomer Zu Chongzhi (429–500) calculated around 480 for the circle number , ie the first 7 decimal places. He was also familiar with the almost equally good approximate fraction (this is the third approximate fraction of the continued fraction expansion of ), which was only found in Europe in the 16th century ( Adriaan Metius , therefore also called the Metius value). In the 14th century, Zhao Youqin calculated the number of circles to six decimal places using a 16384 corner. ${\ displaystyle 3 {,} 1415926 <\ pi <3 {,} 1415927}$${\ displaystyle {\ tfrac {355} {113}}}$${\ displaystyle \ pi}$

In his treatise on the circle , which he completed in 1424, the Persian scientist Jamschid Masʿud al-Kaschi calculated a 3 × 2 28 corner with an accuracy of 16 digits. ${\ displaystyle 2 \ pi}$

In Europe , Ludolph van Ceulen succeeded in calculating the first 35 decimal places of in 1596 . Allegedly he sacrificed 30 years of his life for this calculation. Van Ceulen has not yet contributed any new ideas to the calculation. He simply continued to calculate according to Archimedes' method, but while Archimedes stopped at the 96-corner, Ludolph continued the calculations up to the inscribed corner . The name Ludolphsche number is a reminder of his achievement. ${\ displaystyle \ pi}$${\ displaystyle 2 ^ {62}}$

The French mathematician François Viète varied the Archimedean exhaustion method in 1593 by approximating the area of ​​a circle using a series of inscribed corners . From this he was the first to derive a closed formula for in the form of an infinite product : ${\ displaystyle 2 ^ {n}}$${\ displaystyle \ pi}$

${\ displaystyle {\ frac {2} {\ pi}} = {\ frac {\ sqrt {2}} {2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2}}}} { 2}} \ cdot {\ frac {\ sqrt {2 + {\ sqrt {2 + {\ sqrt {2}}}}}}} {2}} \ cdot \ dots}$

The English mathematician John Wallis developed the Valais product named after him in 1655 :

${\ displaystyle {\ frac {\ pi} {2}} = {\ frac {2} {1}} \ cdot {\ frac {2} {3}} \ cdot {\ frac {4} {3}} \ cdot {\ frac {4} {5}} \ cdot {\ frac {6} {5}} \ cdot {\ frac {6} {7}} \ cdot {\ frac {8} {7}} \ cdot { \ frac {8} {9}} \ cdot \ dots}$

In 1655 Wallis showed this series to Lord Brouncker , the first President of the “ Royal Society ”, who represented the equation as a continued fraction as follows:

${\ displaystyle {\ frac {4} {\ pi}} = 1 + {\ frac {1 ^ {2}} {2+ \ textstyle {\ frac {3 ^ {2}} {2+ \ textstyle {\ frac {5 ^ {2}} {2+ \ textstyle {\ frac {7 ^ {2}} {2+ \ textstyle {\ frac {9 ^ {2}} {\; \, \ ddots}}}}}} }}}}}$

Gradually the calculations became more complicated, Gottfried Wilhelm Leibniz contributed the following series representation in 1682 :

${\ displaystyle {\ frac {\ pi} {4}} = \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n}} {2n + 1}} = {\ frac {1} {1}} - {\ frac {1} {3}} + {\ frac {1} {5}} - {\ frac {1} {7}} + {\ frac {1} {9 }} \ mp \ cdots}$

This was already known to Indian mathematicians in the 15th century. Leibniz rediscovered it for European mathematics and proved the convergence of this infinite sum. The above series is also a special case ( ) of the series expansion of the arctangent that the Scottish mathematician James Gregory found in the 1670s: ${\ displaystyle \ arctan 1 = {\ tfrac {\ pi} {4}}}$${\ displaystyle \ theta = 1}$

${\ displaystyle \ arctan \ theta = {\ frac {\ theta ^ {1}} {1}} - {\ frac {\ theta ^ {3}} {3}} + {\ frac {\ theta ^ {5} } {5}} - {\ frac {\ theta ^ {7}} {7}} \ pm \ cdots}$

In the following time it was the basis of many approximations of , all of which have a linear speed of convergence . ${\ displaystyle \ pi}$

In 1706, William Jones described in his work Synopsis palmariorum matheseos the series developed by him, with which he determined 100 decimal places (decimal places) of . ${\ displaystyle \ pi}$

William Jones designated 1706, like William Oughtred 1647 before, the circle number with ${\ displaystyle \ pi}$

"Let [...] Then  & c." ${\ displaystyle \ alpha = 2 {\ sqrt {3}}.}$${\ displaystyle \ alpha - {\ frac {1} {3}} {\ frac {3 \ alpha} {9}} + {\ frac {1} {5}} {\ frac {\ alpha} {9}} - {\ frac {1} {7}} {\ frac {3 \ alpha} {9 ^ {2}}} + {\ frac {1} {9}} {\ frac {\ alpha} {9 ^ {2 }}} - {\ frac {1} {11}} {\ frac {3 \ alpha} {9 ^ {3}}} + {\ frac {1} {13}} {\ frac {\ alpha} {9 ^ {3}}},}$

Also in 1706, John Machin calculated the first 100 decimal places of with his formula . His equation ${\ displaystyle \ pi}$

${\ displaystyle {\ frac {\ pi} {4}} = 4 \ arctan {\ frac {1} {5}} - \ arctan {\ frac {1} {239}}}$

can be used together with the Taylor series expansion of the arctangent function for fast calculations. In the real world, this formula can be obtained using the arctangent addition theorem; it is simpler by considering the argument of the complex number

${\ displaystyle (5 + {\ rm {i) ^ {4} \ cdot (239 - {\ rm {i) = 114244 + 114244 \; {\ rm {i = (1 + {\ rm {i) \ cdot 114244.}}}}}}}}}$

Over time, more formulas of this type have been found. An example comes from FCW Størmer (1896):

${\ displaystyle {\ frac {\ pi} {4}} = 44 \, \ arctan {\ frac {1} {57}} + 7 \, \ arctan {\ frac {1} {239}} - 12 \, \ arctan {\ frac {1} {682}} + 24 \, \ arctan {\ frac {1} {12943}},}$

which is equivalent to the fact that the real and imaginary parts of the Gaussian number

${\ displaystyle (57+ \ mathrm {i}) ^ {44} \ cdot (239+ \ mathrm {i}) ^ {7} \ cdot (682- \ mathrm {i}) ^ {12} \ cdot (12943 + \ mathrm {i}) ^ {24} = (1+ \ mathrm {i}) \ cdot n}$ With ${\ displaystyle n \ in \ mathbb {Z}}$

are the same.

Leonhard Euler (pastel by Emanuel Handmann, 1753)

In his Introductio in Analysin Infinitorum , published in 1748, Leonhard Euler cited 148 places in the first volume . Formulas discovered by Euler (see also Riemann ζ function ): ${\ displaystyle \ pi}$

${\ displaystyle \ zeta (2) = {\ frac {1} {1 ^ {2}}} + {\ frac {1} {2 ^ {2}}} + {\ frac {1} {3 ^ {2 }}} + {\ frac {1} {4 ^ {2}}} + \ cdots = {\ frac {\ pi ^ {2}} {6}}}$
${\ displaystyle \ zeta (4) = {\ frac {\ pi ^ {4}} {90}}, \ quad \ zeta (6) = {\ frac {\ pi ^ {6}} {945}}, \ quad \ cdots}$
${\ displaystyle {\ frac {\ pi ^ {2}} {8}} = {\ frac {1} {1 ^ {2}}} + {\ frac {1} {3 ^ {2}}} + { \ frac {1} {5 ^ {2}}} + {\ frac {1} {7 ^ {2}}} + {\ frac {1} {9 ^ {2}}} + \ cdots}$
${\ displaystyle {\ frac {\ pi -3} {4}} = {\ frac {1} {2 \ cdot 3 \ cdot 4}} - {\ frac {1} {4 \ cdot 5 \ cdot 6}} + {\ frac {1} {6 \ cdot 7 \ cdot 8}} \ mp \ cdots}$
Lambert: Contributions to the use of mathematics and its application. Volume 2, Issue 1, 1792, page 156.

Johann Heinrich Lambert published a continued fraction in 1770, which today mostly takes the form

${\ displaystyle {\ frac {4} {\ pi}} = 1 + {\ frac {1 ^ {2}} {3+ \ textstyle {\ frac {2 ^ {2}} {5+ \ textstyle {\ frac {3 ^ {2}} {7+ \ textstyle {\ frac {4 ^ {2}} {9+ \ textstyle {\ frac {5 ^ {2}} {11+ \ textstyle {\ frac {6 ^ {2 }} {\; \, \ ddots}}}}}}}}}}}}}}$

is written. There is an average of 0.76555 decimal places per step, which is relatively high compared to other continued fractions, so that this continued fraction is particularly suitable for calculating . ${\ displaystyle \ pi}$

### The craft practice yesterday and today

In times before slide rules and pocket calculators, craftsmen used the approximation and used it to calculate a lot in their heads. The error opposite is about 0.04%. That was completely sufficient for everyday practical situations. ${\ displaystyle {\ tfrac {22} {7}} \ approx 3 {,} 142857}$${\ displaystyle \ pi}$

Another often used approximation was the fraction , accurate to seven places. All these rational approximate values ​​for have in common that they correspond to partial evaluations of the continued fraction expansion of , e.g. B .: ${\ displaystyle {\ tfrac {355} {113}} \ approx 3 {,} 1415929}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$

${\ displaystyle {\ frac {22} {7}} = [3; 7], \ quad {\ frac {355} {113}} = [3; 7,15,1]}$

None of the formulas developed so far could be used to efficiently calculate approximate values ​​for , and the astonishing discovery by the Indian Srinivasa Ramanujan in 1914, based on studies of elliptical functions and modular functions , was not yet suitable for this: ${\ displaystyle \ pi}$

${\ displaystyle {\ frac {1} {\ pi}} = {\ frac {\ sqrt {8}} {9801}} \ cdot \ sum _ {n = 0} ^ {\ infty} {\ frac {(4n )! \ cdot (1103 + 26390n)} {(n!) ^ {4} \ cdot 396 ^ {4n}}}}$

This formula delivers 15 correct decimal places even with two iterations (up to ). ${\ displaystyle n = 1}$

More efficient methods, the implementation of which is only interesting if long number arithmetic is available, are iteration methods with quadratic or even higher convergence .

## Efficient method for numerical calculation

### BBP series

In 1995, Simon Plouffe, together with Peter Borwein and David Harold Bailey, discovered a new type of series representation for : ${\ displaystyle \ pi}$

${\ displaystyle \ pi = \ sum _ {k = 0} ^ {\ infty} {\ dfrac {1} {16 ^ {k}}} \ left ({\ dfrac {4} {8k + 1}} - { \ dfrac {2} {8k + 4}} - {\ dfrac {1} {8k + 5}} - {\ dfrac {1} {8k + 6}} \ right)}$

This series (also called Bailey-Borwein-Plouffe formula) enables the -th digit of a binary, hexadecimal or any representation to be calculated to a power of two without first having to calculate the previous digits. ${\ displaystyle n}$${\ displaystyle \ pi}$${\ displaystyle n-1}$

Later the following were found for other BBP series: ${\ displaystyle \ pi}$

{\ displaystyle {\ begin {aligned} \ pi & = {\ frac {1} {2}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {16 ^ {k}}} \ left ({\ frac {8} {8k + 2}} + {\ frac {4} {8k + 3}} + {\ frac {4} {8k + 4}} - {\ frac {1} {8k +7}} \ right) \\ & = {\ frac {1} {4}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {16 ^ {k}}} \ left ({\ frac {8} {8k + 1}} + {\ frac {8} {8k + 2}} + {\ frac {4} {8k + 3}} - {\ frac {2} {8k + 5 }} - {\ frac {2} {8k + 6}} - {\ frac {1} {8k + 7}} \ right) \\ & = \; \; \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {4 ^ {k}}} \ left ({\ frac {2} {4k + 1}} + {\ frac {2} {4k + 2} } + {\ frac {1} {4k + 3}} \ right) \ end {aligned}}}

### Drip algorithm

Closely related to the digit extraction methods are drip algorithms in which the digits are calculated one after the other. Stanley Rabinowitz found the first such algorithm for calculating . Since then, more drip algorithms have been found to calculate . ${\ displaystyle \ pi}$${\ displaystyle \ pi}$

### Method of Gauss, Brent and Salamin

The calculation of the arc length of a lemniscate using elliptic integrals and their approximation using the arithmetic-geometric mean according to Gauss provides the rapidly converging method of Salamin and Brent for numerical calculation. The basis for this is the following representation, initially assumed by Gauss : ${\ displaystyle \ pi}$

${\ displaystyle {\ frac {1} {\ pi}} = \ mathrm {AGM} (1, {\ sqrt {2}}) \ int _ {0} ^ {1} {\ frac {2 \ mathrm {d } x} {\ sqrt {1-x ^ {4}}}}.}$

The latter integral is also known as the lemniscatic constant . It then applies

${\ displaystyle \ pi = {\ frac {4 \ mathrm {AGM} (1, {\ frac {1} {\ sqrt {2}}}) ^ {2}} {1- \ sum _ {j = 1} ^ {\ infty} 2 ^ {j + 1} c_ {j} ^ {2}}}}$

being the arithmetic-geometric mean over the iteration

${\ displaystyle a_ {n} = {\ frac {a_ {n-1} + b_ {n-1}} {2}}, \ qquad b_ {n} = {\ sqrt {a_ {n-1} b_ { n-1}}}}$

is calculated with two initial arguments and is set. ${\ displaystyle a_ {0}, b_ {0}> 0}$${\ displaystyle c_ {n} ^ {2} = a_ {n} ^ {2} -b_ {n} ^ {2}}$

## Further calculation methods

### Calculation using the area formula

Circle inscribed in a square for calculation using the area formula

This calculation uses the relationship that the area formula of the circle contains, but not the area formula of the circumscribing square. ${\ displaystyle \ pi}$

The formula for the area of ​​the circle with radius is ${\ displaystyle r}$

${\ displaystyle A_ {K} = \ pi r ^ {2}}$,

the area of ​​the square with side length is calculated as ${\ displaystyle 2r}$

${\ displaystyle A_ {Q} = (2r) ^ {2}}$.

For the ratio of the area of ​​a circle and its circumscribing square, we get

${\ displaystyle {\ frac {A_ {K}} {A_ {Q}}} = {\ frac {\ pi r ^ {2}} {(2r) ^ {2}}} = {\ frac {\ pi} {4}}}$.

This can be written as four times this ratio: ${\ displaystyle \ pi}$

${\ displaystyle \ pi = 4 \, {\ frac {A_ {K}} {A_ {Q}}}}$.

#### program

Quarter circle, approximated with a surface grid of 10 × 10

As an example, an algorithm is given in which the area formula is demonstrated, which can be used for approximate calculations. ${\ displaystyle \ pi}$

To do this, you put a grid over the square and calculate for each individual grid point whether it is also in the circle. The ratio of the grid points within the circle to the grid points within the square is multiplied by 4. The accuracy of the approximation obtained in this way depends on the grid width and is controlled by. With you get z. B. 3.16 and already 3.1428. For the result 3.14159, however, one has to set what is reflected in the number of necessary calculations in square form due to the two-dimensional solution approach. ${\ displaystyle \ pi}$${\ displaystyle r}$${\ displaystyle r = 10}$${\ displaystyle r = 100}$${\ displaystyle r = 10000}$

 r = 10000
kreistreffer = 0
for i = 0 to r - 1
x = i + 0.5
for j = 0 to r - 1
y = j + 0.5
if x ^ 2 + y ^ 2 <= r ^ 2 then
kreistreffer = kreistreffer + 1
return 4 * kreistreffer / quadrattreffer


Note: The above program has not been optimized for the fastest possible execution on a real computer system, but has been formulated as clearly as possible for reasons of clarity. Furthermore, the circular area is determined imprecisely in that it is not the coordinates of the center that are used for the respective area units, but the area edge. By considering a full circle, the area of ​​which tends to zero for the first and last lines, the deviation for large ones is marginal.${\ displaystyle r}$

The constant Pi is typically already pre-calculated for everyday use in computer programs; the associated value is usually given with slightly more digits than the most powerful data types of this computer language can accept.

#### Alternative program

Circle area integration

This program adds up the area of ​​the circle from strips that are very narrow in relation to the radius. It uses the equations and as well .
${\ displaystyle y = \ pm {\ sqrt {r ^ {2} -x ^ {2}}}}$${\ displaystyle \ pi = {\ frac {A_ {K}} {r ^ {2}}}}$${\ displaystyle \ pi = \ int _ {- 1} ^ {1} 2 {\ sqrt {1-x ^ {2}}} \, \ mathrm {d} x}$

n := 1000000 // Halbe Anzahl der Streifen
s := 0       // Summe der Flächeninhalte

for x := -1 to +1 step 1/n:
// Flächeninhalt des Streifens an der Stelle x hinzuaddieren.
// Die Höhe des Streifens wird exakt in der Mitte des Streifens gemessen.
// Die 2 steht für die obere plus die untere Hälfte.
// Der Faktor 1/n ist die Breite des Streifens.
s += 2 * sqrt(1 - x*x) * 1/n

pi := s


The x-coordinates of the examined area go from to . Since circles are round and this circle has its center on the coordinates , the y-coordinates are also in the range from to . The program divides the area to be examined into 2 million narrow strips. Each of these strips has the same width, viz . However, the top edge of each strip is different and results from the above formula , in the code this is written as . The height of each strip goes from the top edge to the bottom edge. Since the two edges of circles are equidistant from the center line, the height is exactly twice the edge length, hence the 2 in the code. ${\ displaystyle -1}$${\ displaystyle +1}$${\ displaystyle 0,0}$${\ displaystyle -1}$${\ displaystyle +1}$${\ displaystyle 1 / n}$${\ displaystyle {\ sqrt {1-x ^ {2}}}}$sqrt(1 - x*x)

After passing through the for loop the area of the circle with radius is in the variable s 1. To find out this number the value of Pi, this number must according to the formula nor shared. In this example , it is omitted from the program code. ${\ displaystyle A = \ pi \ cdot r ^ {2}}$${\ displaystyle r ^ {2}}$${\ displaystyle r = 1}$

### Statistical determination

Quarter circle, the area of ​​which is approximated by the Monte Carlo method

One method of determining is the statistical method . For the calculation you let random points "rain" on a square and calculate whether they are inside or outside an inscribed circle. The proportion of internal points is${\ displaystyle \ pi}$${\ displaystyle \ approx {\ tfrac {\ pi} {4}}.}$

This method is a Monte Carlo algorithm ; the accuracy of the approximation of achieved after a fixed number of steps can therefore only be specified with a probability of error . However, due to the law of large numbers , the average accuracy increases with the number of steps. ${\ displaystyle \ pi}$

The algorithm for this determination is:

function approximiere_pi(tropfenzahl)

innerhalb := 0   // Zählt die Tropfen innerhalb des Kreises

// So oft wiederholen, wie es Tropfen gibt:
for i := 1 to tropfenzahl do

// Zufälligen Tropfen im Quadrat [0,0] bis (1,1) erzeugen
x := random(0.0 ..< 1.0)
y := random(0.0 ..< 1.0)

// Wenn der Tropfen innerhalb des Kreises liegt ...
if x * x + y * y <= 1.0
innerhalb++   // Zähler erhöhen

return 4.0 * innerhalb / tropfenzahl


The 4.0code in the code results from the fact that in the droplet simulation only the number for a quarter circle was calculated. To get the (extrapolated) number for a whole circle, the calculated number has to be multiplied by 4. Since the number Pi is the ratio between the area of ​​the circle and the square of the radius, the number obtained in this way has to be divided by the square of the radius. The radius in this case is 1, so splitting can be omitted.

#### Buffon's needle problem

Another unusual and probabilistic method is the Buffon needle problem by Georges-Louis Leclerc de Buffon (presented in 1733, published in 1777). Buffon tossed sticks over his shoulder onto a tiled floor. Then he counted the number of times they hit the joints. Jakow Perelman described a more practicable variant in the book Enthaltsame Geometrie. Take a needle about 2 cm long - or some other metal pencil of similar length and diameter, preferably without a point - and draw a series of thin parallel lines on a piece of paper, spaced twice the length of the needle. Then you let the needle fall very often (several hundred or thousand times) from an arbitrary but constant height onto the sheet and note whether the needle intersects a line or not. It doesn't matter how you count the touch of a line through the end of a needle. Divide the total number of needle throws by the number of times the needle has cut a line ${\ displaystyle N}$${\ displaystyle P}$

${\ displaystyle {\ frac {N} {P}} = {\ frac {\ pi} {2}} {\ frac {d} {\ ell}}}$,

where denotes the length of the needles and the distance between the lines on the paper. This easily gives an approximation for . The needle can also be bent or kinked several times, in which case more than one intersection point per throw is possible and accordingly has to be counted several times. In the middle of the 19th century, the Swiss astronomer Rudolf Wolf achieved a value of by throwing 5000 needles . ${\ displaystyle \ ell}$${\ displaystyle d}$${\ displaystyle \ pi}$${\ displaystyle \ pi \ approx 3 {,} 159}$

## Geometric constructions

Due to the transcendence of , it is not possible to use a compass and ruler to create a line with the exact length of a length . However, there are a number of compass-and-ruler constructions that provide very good approximations, as well as constructions that allow an exact construction thanks to another aid in addition to compasses and ruler. As a further aid of this kind, in particular curves called quadratizes are used, which can be drawn with the aid of a special drawing device or a template. ${\ displaystyle \ pi}$${\ displaystyle \ pi}$

### Approximate constructions

For the geometric construction of the number there is the approximate construction by Kochański from 1685, with which one can determine an approximate value of the circle number with an error of less than 0.002 percent. So it is an approximation construction for the (exactly not possible) quadrature of the circle . ${\ displaystyle \ pi}$

Circle number π, approximate construction according to CG Specht, 1828

143 years later, namely in 1828, CG Specht published his Second Approximation Construction of Circular Circumference in the Journal for Pure and Applied Mathematics. For the approach he found the value

${\ displaystyle 5 \ cdot {\ sqrt {\ frac {439} {278}}} = 6 {,} 28318528 \ ldots \ ;.}$

Halving this value results in a decimal number with seven digits after the decimal point the same as those of the circle number : ${\ displaystyle \ pi}$

${\ displaystyle 3 {,} 141 \; 592 \; 6 {\ color {red} 40 \; 1 \ ldots} \; \ approx \ pi.}$

A graphic representation is not recorded in the journal cited above.

Publisher's Note:

"*) It will be easy for the reader to design the figure according to the description."

- CG Specht : 40. Second approximation construction of the circle perimeter.

The following description of the adjacent construction is based on the original of the construction description.

First draw the unit circle around the point and then draw a straight line from; it results . Then it is erected in a perpendicular to the straight line; it creates . Four semicircles follow one after the other on the straight line, each with the radius around the newly resulting intersection point, which creates the points and . After the tripartite division of the routes in and as well as in and that point is now with connected. The resulting distance is removed from the vertical . Connect also the point with and transfer the new route from to the vertical; it surrenders . It continues with the connections of the points with and with . When transferring the route to the route ab results . Finally, drawing from a parallel to the route , which in cuts. The resulting distance corresponds approximately to the value . ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle a}$${\ displaystyle A}$${\ displaystyle M}$${\ displaystyle a}$${\ displaystyle {\ overline {Aa}}}$${\ displaystyle m, p, q}$${\ displaystyle B}$${\ displaystyle {\ overline {mp}}}$${\ displaystyle n}$${\ displaystyle o}$${\ displaystyle {\ overline {qB}}}$${\ displaystyle r}$${\ displaystyle s}$${\ displaystyle M}$${\ displaystyle m}$${\ displaystyle {\ overline {Mm}}}$${\ displaystyle A}$${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle r}$${\ displaystyle {\ overline {Rr}}}$${\ displaystyle A}$${\ displaystyle C}$${\ displaystyle C}$${\ displaystyle o}$${\ displaystyle C}$${\ displaystyle B}$${\ displaystyle {\ overline {AB}}}$${\ displaystyle {\ overline {Co}}}$${\ displaystyle C}$${\ displaystyle c}$${\ displaystyle c}$${\ displaystyle {\ overline {AB}}}$${\ displaystyle {\ overline {CB}}}$${\ displaystyle d}$${\ displaystyle {\ overline {Cd}}}$${\ displaystyle 2 \ pi}$

The approximation to the number of circles can, for. B. be clarified in the following way: ${\ displaystyle \ pi = {\ frac {U} {d}}}$

If the diameter were a circle , its approximate circumference would only be approx. Shorter than its theoretical value. ${\ displaystyle d}$${\ displaystyle 100 \; \ mathrm {km}}$${\ displaystyle U = d \ pi}$${\ displaystyle 2 {,} 7 \; \ mathrm {mm}}$

Circle number as an exact construction using the quadratrix, radius${\ displaystyle \ pi}$${\ displaystyle r = 1}$

The illustration opposite shows the number of circles as a segment, created with the help of the Quadratrix des Hippias . ${\ displaystyle \ pi}$

It begins with a straight line from the point and a perpendicular to this straight line . Then the semicircle is drawn with the radius around ; this results in the intersections and . Now you construct the square with the side length 1. The construction of the quadratrix follows, without a "gap" on the X-axis , with the parameter curve : ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle r = 1}$${\ displaystyle A}$${\ displaystyle B, D}$${\ displaystyle E}$${\ displaystyle ABCD}$${\ displaystyle \ gamma: (- \ pi, \ pi) \ rightarrow \ mathbb {R} ^ {2}}$

${\ displaystyle \ gamma (t) = {\ begin {pmatrix} x (t) \\ y (t) \ end {pmatrix}}}$

With

{\ displaystyle {\ begin {aligned} x (t) & = {\ begin {cases} {\ frac {2} {\ pi}} t \ cot (t) \, &, t \ in (- \ pi, \ pi) \ setminus \ {0 \} \\ {\ frac {2} {\ pi}} \, &, t = 0 \ end {cases}} \\ y (t) & = {\ frac {2} {\ pi}} t \ end {aligned}}}

According to the Dinostratos theorem, the quadratrix cuts the side of its associated square at the point and thus generates the value on the straight line, now used as a number line . Establishing the vertical on the line from to the semicircle results in the intersection . After extending the distance beyond and drawing a straight line from through to the extension, the point of intersection results . One way u. a. is now to determine the length of the line with the help of the ray theorem. In the drawing it can be seen that the route corresponds. As a result, the proportions of the sections are after the first theorem of rays${\ displaystyle {\ overline {AB}}}$${\ displaystyle F}$${\ displaystyle {\ frac {2} {\ pi}}}$${\ displaystyle {\ overline {AB}}}$${\ displaystyle {\ frac {2} {\ pi}}}$${\ displaystyle G}$${\ displaystyle {\ overline {BC}}}$${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle {\ overline {AH}}}$${\ displaystyle {\ frac {2} {\ pi}}}$${\ displaystyle {\ overline {AF}}}$

${\ displaystyle | AF |: | AB | = | AG |: | AH |,}$

transformed and the corresponding values ​​inserted results

${\ displaystyle | AH | = {\ frac {\ frac {1} {1}} {\ frac {2} {\ pi}}} \ cdot 1 = {\ frac {\ pi} {2}}.}$

Now the arc with the radius is drawn up to the number line; the point of intersection is created . The final Thales circle above from the point gives the exact circle number . ${\ displaystyle {\ overline {AH}}}$${\ displaystyle A}$${\ displaystyle {\ frac {\ pi} {2}}}$${\ displaystyle {\ frac {\ pi} {2}}}$${\ displaystyle A}$${\ displaystyle \ pi}$

## Experimental construction

Circle number as experimental construction: 1. Cylinder with water level = [LE] 2. Cuboid with poured water, water level = [LE]${\ displaystyle \ pi}$
${\ displaystyle 1}$
${\ displaystyle \ pi}$

The following method uses the circle number "hidden" in the circular area in order to represent the value of as a measurable quantity with the help of experimental physics . ${\ displaystyle \ pi}$${\ displaystyle \ pi}$

A cylinder with the radius and the height of the vessel is filled with water up to the height . The amount of water determined in this way is then transferred from the cylinder into a cuboid , which has a square base with side length and a vessel height of . ${\ displaystyle r = 1}$${\ displaystyle {\ hat {h}} _ {\ text {Z}} \ approx 1 {,} 5}$${\ displaystyle h _ {\ text {Z}} = 1}$${\ displaystyle a = 1}$${\ displaystyle {\ hat {h}} _ {\ text {Q}} \ approx 4}$

Amount of water in the cylinder in units of volume [VE]: ${\ displaystyle V _ {\ text {Z}}}$

${\ displaystyle V _ {\ text {Z}} = r ^ {2} \ pi h _ {\ text {Z}} = 1 ^ {3} \ cdot \ pi = 3 {,} 14159 \ dotso}$ [VE]

Water level in the cuboid in units of length [LE]: ${\ displaystyle h _ {\ text {Q}}}$

${\ displaystyle V _ {\ text {Q}} = a ^ {2} h _ {\ text {Q}} = 1 ^ {2} h _ {\ text {Q}} = V _ {\ text {Z}}}$, from it ${\ displaystyle h _ {\ text {Q}}}$
${\ displaystyle h _ {\ text {Q}} = {\ frac {1 ^ {3} \ pi} {1 ^ {2}}} = \ pi = 3 {,} 14159 \ dotso}$ [LE]

The result shows: A quantity of water that has water level 1 [LE] in the cylinder delivers - when poured into the cuboid - the water level [LE]. ${\ displaystyle \ pi}$

## Formulas and Applications

### Formulas that contain π

#### Formulas of geometry

In geometry , the properties of emerge directly as a circle number. ${\ displaystyle \ pi}$

• Perimeter of a circle with a radius :${\ displaystyle r}$${\ displaystyle U = 2 \ pi r}$
• Area of ​​a circle with a radius :${\ displaystyle r}$${\ displaystyle A = \ pi r ^ {2}}$
• Volume of a sphere with a radius :${\ displaystyle r}$${\ displaystyle V = {\ tfrac {4} {3}} \ pi r ^ {3}}$
• Surface of a sphere with a radius :${\ displaystyle r}$${\ displaystyle A_ {O} = 4 \ pi r ^ {2}}$
• Volume of a cylinder with radius and height :${\ displaystyle r}$${\ displaystyle a}$${\ displaystyle V = r ^ {2} \ pi a}$
• Volume one by the rotation of the graph around the defined axis rotational body to the limits and :${\ displaystyle y = f (x)}$${\ displaystyle x}$${\ displaystyle a}$${\ displaystyle b}$ ${\ displaystyle V = \ pi \ int _ {a} ^ {b} f (x) ^ {2} \ mathrm {d} x}$

#### Formulas of analysis

In the area of analysis also plays a role in many contexts, for example in ${\ displaystyle \ pi}$

• the integral representation that Karl Weierstrass used in 1841 to define${\ displaystyle \ pi = \ int _ {- \ infty} ^ {\ infty} {\ frac {\ mathrm {d} x} {1 + x ^ {2}}} = 2 \ cdot \ int _ {- 1st } ^ {1} {\ frac {\ mathrm {d} x} {1 + x ^ {2}}}}$${\ displaystyle \ pi}$
• the infinite series : ( Euler , see also Riemann zeta function ),${\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {2}}} = {\ frac {\ pi ^ {2}} {6}}}$
• the Gaussian normal distribution : or other representation: ,${\ displaystyle \ int _ {- \ infty} ^ {\ infty} e ^ {- x ^ {2}} \ mathrm {d} x = {\ sqrt {\ pi}}}$${\ displaystyle \ int _ {\ mathbb {R} ^ {2}} e ^ {- | x | ^ {2}} \ mathrm {d} x = \ pi}$
• the Stirling formula as an approximation of the Faculty for large : ,${\ displaystyle n}$${\ displaystyle n! \ approx {\ sqrt {2 \ pi n}} \ left ({\ frac {n} {e}} \ right) ^ {n}}$
• the Fourier transform : .${\ displaystyle F (\ omega) = {\ mathcal {F}} \ {f \} (\ omega) = {\ frac {1} {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {\ infty} f (t) e ^ {- {\ rm {i \ omega t}}} \, \ mathrm {d} t}$
##### Function theory formulas

As for all sub-areas of analysis, the number of circles is of fundamental importance for function theory (and beyond that for the entire complex analysis ). As prime examples are here

• the Euler identity ${\ displaystyle e ^ {\ rm {i \ pi}} + 1 = 0}$

to call as well

• the integral formula of Cauchy .${\ displaystyle f (z) = {\ frac {1} {2 \ pi {\ rm {i}}}} \ oint _ {\ gamma} {\ frac {f (\ zeta)} {\ zeta -z} } \ mathrm {d} \ zeta}$

In addition, the meaning of the circle number is also evident in the formulas for the partial fraction decomposition of the complex-valued trigonometric functions , which are related to the Mittag-Leffler theorem . Here are above all

{\ displaystyle {\ begin {aligned} \ pi \ cot (\ pi z) & = \ sum _ {n = {- \ infty}} ^ {+ \ infty} {\ frac {1} {z + n}} \\ & = {\ frac {1} {z}} + \ sum _ {n = 1} ^ {\ infty} \ left ({\ frac {1} {zn}} + {\ frac {1} {z + n}} \ right) \\ & = {\ frac {1} {z}} + 2z \ cdot \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {z ^ {2} -n ^ {2}}} \\ & = z \ cdot \ sum _ {n = {- \ infty}} ^ {+ \ infty} {\ frac {1} {z ^ {2} -n ^ {2 }}} \ quad (z \ in {\ mathbb {C} \ setminus \ mathbb {Z}}) \\\ end {aligned}}}

to mention as well as those from it - among others! - to be won

{\ displaystyle {\ begin {aligned} \ left ({\ frac {\ pi} {\ sin (\ pi z)}} \ right) ^ {2} & = \ sum _ {n = {- \ infty}} ^ {+ \ infty} {\ frac {1} {(zn) ^ {2}}} \ quad (z \ in {\ mathbb {C} \ setminus \ mathbb {Z}}) \\\ left ({\ frac {\ pi} {\ cos (\ pi z)}} \ right) ^ {2} & = \ sum _ {n = {- \ infty}} ^ {+ \ infty} {\ frac {1} {( z - {\ tfrac {2n-1} {2}}) ^ {2}}} \ quad \ left (z \ in \ mathbb {C} \ setminus \ left \ {{\ frac {2n-1} {2 }} \ colon n \ in \ mathbb {Z} \ right \} \ right) \\\ end {aligned}}}

The above partial fraction to the sine then supplies by insertion of the well-known series representation${\ displaystyle z = {\ frac {1} {2}}}$

${\ displaystyle {\ frac {{\ pi} ^ {2}} {8}} = 1 + {\ frac {1} {9}} + {\ frac {1} {25}} + {\ frac {1 } {49}} + \ cdots = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {(2n-1) ^ {2}}},}$

which in turn directly to the Eulerian series representation

${\ displaystyle {\ frac {{\ pi} ^ {2}} {6}} = 1 + {\ frac {1} {4}} + {\ frac {1} {9}} + {\ frac {1 } {16}} + \ cdots = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {2}}}}$

In addition to these π formulas, which originate from the partial fraction series, function theory knows a large number of others which, instead of the representation with infinite series, have a representation using infinite products . Many of them go back to the work of Leonhard Euler ( see below ).

#### Formulas of number theory

• The relative frequency that two randomly chosen natural numbers , which are below a bound, are coprime , tends to counter this .${\ displaystyle M}$${\ displaystyle M \ rightarrow \ infty}$${\ displaystyle {\ frac {6} {\ pi ^ {2}}}}$

#### Formulas of physics

In physics plays alongside ${\ displaystyle \ pi}$

• the circular motion: (angular speed equal to the rotational frequency)${\ displaystyle \ omega = 2 \ pi f}$${\ displaystyle 2 \ pi}$

This is particularly important for waves , as the sine and cosine functions are included there; so for example ${\ displaystyle \ pi}$

also

• in the calculation of the buckling load ${\ displaystyle F_ {K} = {\ frac {\ pi ^ {2} EI} {s ^ {2}}}}$
• and in the friction of particles in liquids ( Stokes law )${\ displaystyle F_ {R} = 6 \, \ pi \, \ eta \, r \, v.}$

#### Product formulas from Leonhard Euler

• If the sequence of prime numbers is denoted by, as usual , then:${\ displaystyle {\ bigl (} p_ {k} {\ bigr)} _ {k \ in \ mathbb {N}} = {\ bigl (} 2,3,5,7, \ dots {\ bigr)}}$
{\ displaystyle {\ begin {aligned} {\ frac {{\ pi} ^ {2}} {6}} & = \ prod _ {k = 1} ^ {\ infty} {\ frac {{p_ {k} } ^ {2}} {{p_ {k}} ^ {2} -1}} & = {\ frac {2 ^ {2}} {2 ^ {2} -1}} \ cdot {\ frac {3 ^ {2}} {3 ^ {2} -1}} \ cdot {\ frac {5 ^ {2}} {5 ^ {2} -1}} \ cdot {\ frac {7 ^ {2}} { 7 ^ {2} -1}} \ cdot \; \ dots & = {\ frac {4} {3}} \ cdot {\ frac {9} {8}} \ cdot {\ frac {25} {24} } \ cdot {\ frac {49} {48}} \ cdot \; \ dots \\ {\ frac {{\ pi} ^ {4}} {90}} & = \ prod _ {k = 1} ^ { \ infty} {\ frac {{p_ {k}} ^ {4}} {{p_ {k}} ^ {4} -1}} & = {\ frac {2 ^ {4}} {2 ^ {4 } -1}} \ cdot {\ frac {3 ^ {4}} {3 ^ {4} -1}} \ cdot {\ frac {5 ^ {4}} {5 ^ {4} -1}} \ cdot {\ frac {7 ^ {4}} {7 ^ {4} -1}} \ cdot \; \ dots & = {\ frac {16} {15}} \ cdot {\ frac {81} {80} } \ cdot {\ frac {625} {624}} \ cdot {\ frac {2401} {2400}} \ cdot \; \ dots \\ {\ frac {{\ pi} ^ {8}} {9450}} & = \ prod _ {k = 1} ^ {\ infty} {\ frac {{p_ {k}} ^ {8}} {{p_ {k}} ^ {8} -1}} & = {\ frac {2 ^ {8}} {2 ^ {8} -1}} \ cdot {\ frac {3 ^ {8}} {3 ^ {8} -1}} \ cdot {\ frac {5 ^ {8} } {5 ^ {8} -1}} \ cdot {\ frac {7 ^ {8}} {7 ^ {8} -1}} \ cdot \; \ dots & = {\ frac {256} {255} } \ cdot {\ frac {6561} {6560}} \ cdot {\ frac {390625} {390624}} \ cdot {\ frac {5764801} {5764800}} \ cdot \; \ dots \\ \ end {aligned}}}
The first of the three following formulas is also known as Euler's supplementary theorem . The following two product formulas for sine and cosine are absolutely convergent products. Both product formulas result from the supplementary sentence, whereby the product formula of the cosine is itself a direct application of the product formula of the sine.${\ displaystyle \ cos (z) = {\ tfrac {\ sin (2z)} {2 \ sin (z)}}}$
{\ displaystyle {\ begin {aligned} \ Gamma (z) \ cdot \ Gamma (1-z) & = {\ frac {\ pi} {\ sin (\ pi z)}} \ quad (z \ in \ mathbb {C} \ setminus \ mathbb {Z}) \\\ sin (\ pi z) & = \ pi z \ prod _ {k = 1} ^ {\ infty} \ left (1 - {\ frac {z ^ { 2}} {k ^ {2}}} \ right) \ quad (z \ in \ mathbb {C}) \\\ cos (\ pi z) & = \ prod _ {k = 1} ^ {\ infty} \ left (1 - {\ frac {4z ^ {2}} {(2k-1) ^ {2}}} \ right) \ quad (z \ in \ mathbb {C}) \\\ end {aligned}} }
The product formula of the sine then leads to this interesting relationship (sequence A156648 in OEIS ): ${\ displaystyle z = \ mathrm {i}}$
{\ displaystyle {\ begin {aligned} \ prod _ {k = 1} ^ {\ infty} \ left (1 + {\ frac {1} {k ^ {2}}} \ right) & = {\ frac { e ^ {\ pi} -e ^ {- \ pi}} {2 \ pi}} & = {\ frac {\ sinh (\ pi)} {\ pi}} & \ approx 3 {,} 6760779103749 \\\ end {aligned}}}

## Others

### Records and curiosities

Pi floor mosaic at the entrance of the mathematics building of the TU Berlin
• Friends of Numbers celebrate Pi Day on March 14th because of the American date notation 3/14.${\ displaystyle \ pi}$
• In 1897 were in the US - state of Indiana with the Indiana Pi Bill -Gesetzentwurf (for to act introducing a new mathematical truth) for the number by Act 4 and 3.2 proposed. The amateur mathematician Edwin J. Goodwin was sure to have found the square of the circle. He proposed to the Government of Commerce to forego all royalties from the use of his discovery in mathematical education and training if his discovery were made law. Only after the Enlightenment by Clarence A. Waldo , a mathematics professor at Purdue University , who discovered by chance during a visit to the Parliament of the draft law, the second chamber of the adjourned Parliament to the House of Representatives (parliament) already unanimously draft adopted indefinitely.${\ displaystyle \ pi}$
• In Germany, § 30b of the Road Traffic Licensing Regulations determines the displacement of an internal combustion engine (relevant for vehicle tax): "For pi, the value of 3.1416 is used."
• The version number of the typesetting program TeX by Donald E. Knuth has been incremented, contrary to the usual conventions of software development since the 1990s, so that it slowly approaches. The current version from 2014 has the number 3.14159265.${\ displaystyle \ pi}$
• The version name of the free geographic information system software QGIS is in version 3.14 "Pi". Additional decimal places are added for bugfix versions.
• Scientists use radio telescopes to send the circle number into space . They feel that other civilizations need to know this number if they can pick up the signal.
• The current record for reading Pi aloud is 108,000 decimal places in 30 hours. The world record attempt began on June 3, 2005 at 6:00 p.m. and was successfully completed on June 5, 2005 at 0:00 a.m. Over 360 readers read 300 decimal places each. The world record was organized by the Mathematikum in Gießen .

### Film, music, culture and literature

Pi in the Vienna Opera Passage . The number is in the middle of the mirror wall.
• In the novel The Magic Mountain by Thomas Mann , the narrator describes in the chapter The great stupidity on compassionate-belächelnde way as the minor character of the prosecutor Partition the "desperate break" Pi tries to unravel. Paravant believes that “providential planning” determined him to “tear the transcendent goal into the realm of earthly precise fulfillment”. He tries to awaken in his environment a “humane sensitivity to the shame of the contamination of the human spirit by the hopeless irrationality of this mystical relationship” and asks himself “whether it has not been too difficult for mankind to solve the problem since Archimedes' days and whether this solution is actually the simplest for children. ”In this context, the narrator mentions the historical Zacharias Dase , who calculated pi to two hundred places after the decimal point.
• In episode 43 of the science fiction series Spaceship Enterprise , The Wolf in Sheep's Clothing (original title Wolf in the Fold ), a strange being seizes the on-board computer. The chief officer Spock then orders the computer to calculate the number pi down to the last decimal place. The computer is so overwhelmed by this task that the being leaves the computer again.
• In 1981, Carl Sagan's book Contact was published. The book describes the SETI program for the search for extraterrestrial intelligence and related philosophical considerations. It ends with the fictitious answer to the question of whether the universe came into being by chance or was systematically created. The number  plays the central role for the exciting answer that is consistent in the context of the plot.${\ displaystyle \ pi}$
• 1988 initiated Larry Shaw the Pi Day on March 14 at the Exploratorium .
• In 1998 Darren Aronofsky (Requiem for a Dream) released the film Pi (1998), in which a mathematical genius (Sean Gullette as "Maximilian Cohen") wants to filter out the world formula .${\ displaystyle \ pi}$
• In the published 2005 double album Aerial of Kate Bush song of Pi is dedicated.
• The media installation Pi , which opened in November 2006 in the Wiener Opernpassage, is dedicated to the circle number, among other things.
• In the film Nights in Museum 2 (2009), the circle number is the combination for the Ahkmenrah table. The combination is solved with the help of Bobble-Head Einstein and opens the gate to the underworld in the film.
• The progressive deathcore band After the Burial released the song Pi (The Mercury God of Infinity) on their debut album Forging a Future Self . It consists of an acoustic guitar solo followed by a breakdown , the rhythm of which is based on the first 110 digits of the circle number.

### Pi sport

Memorizing the number pi is the most popular way to demonstrate remembering long numbers. So learning from pi has become a sport. The Indian Rajveer Meena is the official world record holder with a confirmed 70,000 decimal places, which he recited flawlessly on March 21, 2015 in a time of 10 hours. He is listed as a record holder in the Guinness Book of Records.

The unofficial world record was in October 2006 at 100,000 positions, set by Akira Haraguchi . The Japanese broke his also unofficial old record of 83,431 decimal places. Jan Harms holds the German record with 9140 jobs. Special mnemonic techniques are used to memorize Pi . The technique differs according to the preferences and talents of the memory artist and the number of decimal places to be memorized.

There are simple memory systems for memorizing the first digits of Pi, as well as Pi-Sport memorization rules .

### Development of the decimal places of π

mathematician year Decimal places method Computing time
Egypt, arithmetic book of Ahmes ( Papyrus Rhind ) approx. 16th century BC Chr. 1 example
Archimedes approx. 250 BC Chr. 2 96-sided
Liu Hui after 263 5 3072 corner
To Chongzhi approx. 480 6th
Jamjid Masʿud al-Kashi circa 1424 15th 3 · 2 28 corner
Ludolph van Ceulen 1596 20th
Ludolph van Ceulen 1610 35 2 62 corner
William Jones
John Machin
1706 100 Series Developments
William Jones: Let it then be John Machin:${\ displaystyle \ alpha = 2 {\ sqrt {3}},}$
${\ displaystyle \ pi = \ alpha - {\ frac {1} {3}} {\ frac {3 \ alpha} {9}} + {\ frac {1} {5}} {\ frac {\ alpha} { 9}} - {\ frac {1} {7}} {\ frac {3 \ alpha} {9 ^ {2}}} + {\ frac {1} {9}} {\ frac {\ alpha} {9 ^ {2}}} - {\ frac {1} {11}} {\ frac {3 \ alpha} {9 ^ {3}}} + {\ frac {1} {13}} {\ frac {\ alpha } {9 ^ {3}}} \ ldots}$

${\ displaystyle {\ frac {\ pi} {2}} = {\ frac {2} {1}} \ cdot {\ frac {2} {3}} \ cdot {\ frac {4} {3}} \ cdot {\ frac {4} {5}} \ cdot {\ frac {6} {5}} \ cdot {\ frac {6} {7}} \ cdot {\ frac {8} {7}} \ cdot { \ frac {8} {9}} \ dotsm}$
Jurij Vega 1794 126
William Shanks 1853 (527) Series expansion of and . Calculation of the first 707 decimal places by hand. In 1945 it was discovered that the last 180 digits were incorrect.${\ displaystyle \ arctan {\ tfrac {1} {5}}}$${\ displaystyle \ arctan {\ tfrac {1} {239}}}$
${\ displaystyle \ pi}$
Levi B. Smith, John W. Wrench 1949 1,120
G. Reitwiesner 1949 2,037 with the tube calculator ENIAC 70 h
Nicholson, Jaenel 1954 3,092 Naval Ordnance Research Calculator 0:13 h
George E. Felton 1957  7,480 Pegasus 33 h
F. Genuys 1958 10,000 with the magnetic core memory computer IBM 704 , using the Machin formula 10 h
George E. Felton 1958 10,021 Pegasus 33 h
Jean Guilloud  1959 16,167 IBM 704 4:18 h
Daniel Shanks , John W. Wrench 1961 100,265 with the transistor computer IBM 7090 8:43 h
Jean Guilloud, J. Filliatre 1966 250,000 IBM 7030 41:55 h
Jean Guilloud, M. Dichampt 1967 500,000 CDC 6600 28:10 h
Jean Guilloud, Martin Boyer 1973 1.001.250 CDC 7600 23:18 h
Kazunori Miyoshi , Yasumasa Canada 1981 2,000,036 FACOM M-200 137: 18 h
Jean Guilloud 1981 2,000,050
Yoshiaki Tamura 1982 2,097,144 MELCOM 900II 7:14 h
Yoshiaki Tamura, Yasumasa Canada 1982 4,194,288 HITAC M-280H 2:21 h
Yoshiaki Tamura, Yasumasa Canada 1982 8,388,576 HITAC M-280H 6:52 h
Yasumasa Canada, Sayaka Yoshino, Yoshiaki Tamura 1982 16,777,206 HITAC M-280H <30 h
Yasumasa Canada, Yoshiaki Tamura, Yoshinobu Kubo 1987 134.217.700
David and Gregory Chudnovsky 1989 1,011,196,691
Yasumasa Canada, Daisuke Takahashi 1997 51,539,600,000
Yasumasa Canada, Daisuke Takahashi 1999 206.158.430.000
Yasumasa Canada 2002 1,241,100,000,000 Calculation: Verification:
${\ displaystyle \ pi = 48 \ arctan {\ tfrac {1} {49}} + 128 \ arctan {\ tfrac {1} {57}} - 20 \ arctan {\ tfrac {1} {239}} + 48 \ arctan {\ tfrac {1} {110443}}}$

${\ displaystyle \ pi = 176 \ arctan {\ tfrac {1} {57}} + 28 \ arctan {\ tfrac {1} {239}} - 48 \ arctan {\ tfrac {1} {682}} + 96 \ arctan {\ tfrac {1} {12943}}}$
Daisuke Takahashi 2009 2,576,980,370,000 Calculation: Gauss-Legendre algorithm
Fabrice Bellard 2010 2,699,999,990,000 Calculation: TachusPi software ( Chudnovsky formula , verification: Bellard's formula ) 131 days
Shigeru Kondo, Alexander Yee 2010 5,000,000,000,000 Calculation: y-cruncher software (Chudnovsky formula, verification: Plouffes and Bellards formula) 090 days
Shigeru Kondo, Alexander Yee 2011 10,000,000,000,050 Calculation: y-cruncher software (Chudnovsky formula, verification: Plouffes and Bellards formula) 191 days
Shigeru Kondo, Alexander Yee 2013 12.100.000.000.050 Calculation: y-cruncher software (Chudnovsky formula, verification: Bellard's formula) 082 days
Sandon Van Ness (Houkouonchi) 2014 13,300,000,000,000 Calculation: y-cruncher software (Chudnovsky formula, verification: Bellard's formula) 208 days
Peter Trüb / DECTRIS 2016 22.459.157.718.361 Calculation: y-cruncher software (Chudnovsky formula, verification: Bellard's formula) 105 days
Emma Haruka Iwao / Google LLC 2019 31,415,926,535,897 Calculation: y-cruncher software (Chudnovsky formula, verification: Plouffes and Bellards formula) 121 days
Timothy Mullican 2020 50,000,000,000,000 Calculation: y-cruncher software (Chudnovsky formula, verification: Plouffes and Bellards formula) 303 days

### Alternative circle number τ

The American mathematician Robert Palais suggested in an issue of the mathematics magazine The Mathematical Intelligencer in 2001 that instead of using the quotient of the circumference and diameter of a circle, the quotient of the circumference and radius (correspondingly ) should be used as the basic constant in the future . His reasoning is based on the fact that in many mathematical formulas the factor appears before the circle number. Another argument is the fact that the new constant in radians represents a full angle instead of a half angle, making it less arbitrary. The newly standardized circle number, for whose notation Michael Hartl and Peter Harremoës suggested the Greek letter (Tau), would shorten these formulas. With this convention shall be considered , therefore .${\ displaystyle \ pi}$${\ displaystyle 2 \ pi}$${\ displaystyle 2}$${\ displaystyle \ pi}$${\ displaystyle \ tau}$${\ displaystyle \ tau = 2 \ pi = 6 {,} 283185 \ ldots}$${\ displaystyle \ pi = {\ tfrac {\ tau} {2}}}$

## literature

• Jörg Arndt, Christoph Haenel: Π [Pi] . Algorithms, computers, arithmetic. 2nd, revised and expanded edition. Springer Verlag, Berlin 2000, ISBN 3-540-66258-8 (with CD-ROM , 1st edition. 1998 - without CD-ROM, ISBN 3-540-63419-3 ).
• Heinrich Behnke , Friedrich Sommer : Theory of the analytical functions of a complex variable (=  The basic teachings of the mathematical sciences in single representations . Volume 77 ). Springer-Verlag, Berlin / Heidelberg / New York 1965.
• Petr Beckmann: A History of π . St. Martin's Press, New York City 1976, ISBN 978-0-312-38185-1 (English).
• Ehrhard Behrends (ed.): Π [Pi] and co . Kaleidoscope of mathematics. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-77888-2 .
• David Blatner: Π [Pi] . Magic of a number. In: rororo non-fiction book (=  rororo . No. 61176 ). Rowohlt, Reinbek bei Hamburg 2001, ISBN 3-499-61176-7 (original title: The Joy of Π [pi] . Translated by Hainer Kober).
• Jonathan Borwein , Peter Borwein : Pi and the AGM . A Study in Analytic Number Theory and Computational Complexity. In: Canadian Mathematical Society Series of Monographs and Advan . 2nd Edition. Wiley, New York NY 1998, ISBN 0-471-31515-X (English).
• Egmont Colerus : From multiplication tables to integral . Mathematics for Everyone (=  Rowohlt-fiction book . No. 6692 ). Rowohlt, Reinbek near Hamburg 1974, ISBN 3-499-16692-5 .
• Jean-Paul Delahaye : Π [Pi] . The story. Birkhäuser, Basel 1999, ISBN 3-7643-6056-9 .
• Keith Devlin : Great moments of modern mathematics . famous problems and new solutions (=  dtv-Taschenbuch 4591 ). 2nd Edition. Deutscher Taschenbuch Verlag, Munich 1992, ISBN 3-423-04591-4 (Original title: Mathematics . Translated by Doris Gerstner, license from Birkhäuser-Verlag, Basel).
• Leonhard Euler: Introduction to the Analysis of the Infinite . Springer Verlag, Berlin / Heidelberg / New York 1983, ISBN 3-540-12218-4 (first part of the Introductio in Analysin Infinitorum - reprint of the Berlin 1885 edition).
• Eberhard Freitag , Rolf Busam : Function theory 1 (=  Springer textbook ). 3rd, revised and expanded edition. Springer Verlag, Berlin (inter alia) 2000, ISBN 3-540-67641-4 .
• Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 .
• Paul Karlson: On the magic of numbers . Fun math for everyone. In: The modern non-fiction book . 8th, revised edition. tape 41 . Ullstein, Berlin 1965 (without ISBN , previous title: You and the magic of numbers ).
• Karel Markowski: The calculation of the number Π [(Pi)] from sine and tangent intervals . 1st edition. Trigon, Potsdam 2007, ISBN 978-3-9810752-1-2 .
• Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin (inter alia) 1964, ISBN 3-540-03138-3 .
• Konrad Knopp: Function Theory II . Applications and continuation of the general theory (=  Göschen Collection . Volume 703 ). 11th edition. de Gruyter, Berlin 1965.
• Herbert Meschkowski : Infinite rows . 2nd, improved and enlarged edition. BI Wissenschaftsverlag, Mannheim (among others) 1982, ISBN 3-411-01613-2 .
• Jakow Perelman : Entertaining Geometry . People and Knowledge, Berlin 1962.
• Jürgen Petigk: Triangular circles or how you can determine Π [Pi] with a needle . Math puzzles, training for the brain. Komet, Cologne 2007, ISBN 978-3-89836-694-6 (published in 1998 as Mathematics in Leisure Time by Aulis-Verlag Deubner, Cologne, ISBN 3-7614-1997-X ).
• Karl Helmut Schmidt: Π [Pi] . History and algorithms of a number. Books on Demand GmbH, Norderstedt, ISBN 3-8311-0809-9 ([2001]).
• Karl Strubecker : Introduction to higher mathematics. Volume 1: Basics . R. Oldenbourg Verlag, Munich 1956.
• Heinrich Tietze : Mathematical Problems . Solved and unsolved math problems from old and new times. Fourteen lectures for amateurs and friends of mathematics. CH Beck, Munich 1990, ISBN 3-406-02535-8 (special edition in one volume, 1990 also as dtv paperback 4398/4399, ISBN 3-423-04398-9 - volume 1 and ISBN 3-423-04399-7 - Part 1).

Commons : Pi  - collection of pictures, videos and audio files
Wiktionary: Kreiszahl  - explanations of meanings, word origins, synonyms, translations

## Remarks

1. The number theorist Ivan Niven provided a simple proof of irrationality in 1947 . (Ivan Niven: A simple proof that π is irrational . In: Bulletin of the American Mathematical Society . Volume 53 , 1947, pp. 509 ( MR0021013 ). )
2. Here all partial counters are equal to 1.
3. Here all partial counters are equal to −1.
4. See Bailey's website for more details .
5. This is${\ displaystyle n = 2 {,} 84438 \ dotso \ cdot 10 ^ {226}.}$
6. The Euler identity is seen as a combination of the circle number , the likewise transcendent Euler number , the imaginary unit and the two algebraic base quantities and as one of the “most beautiful mathematical formulas”.${\ displaystyle \ pi}$ ${\ displaystyle e}$ ${\ displaystyle {\ rm {i}}}$ ${\ displaystyle 0}$${\ displaystyle 1}$
7. The song on YouTube with an explanation of the rhythm in the video description, written by one of the guitarists. Video on YouTube .

## Individual evidence

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5. William Jones: Synopsis Palmariorum Matheseos. Palmariorum Matheseos, p. 263, see below: “3.14159, & c. = [...] Whence in the Circle, any one of these three, [area] a, [circumference] c, [diameter] d, being given, the other two are found, as, d = c ÷ = ( a ÷ 1 / 4 ) 1/2 , c = d × = ( a × 4 ) 1/2 , a = 1/4 × d 2 = c 2 ÷ 4. "In: Göttinger Digitization Center. J. Matthews, London, 1706, accessed August 19, 2019 .${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$${\ displaystyle \ pi}$
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43. ^ Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 , pp. 140 .
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