Pi

${\displaystyle \pi }$ (π) (/p/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. This produces a number, and that number is always the same. However, the number is rather strange. The number starts as 3.141592653589793 and continues without end. Numbers like this are called irrational numbers.[1][2][3]

The diameter is the largest chord which can be fitted inside a circle. It passes through the center of the circle. The distance around a circle is known as the circumference. Even though the diameter and circumference are different for different circles, the number ${\displaystyle \pi }$ remains constant: its value never changes. This is because the relationship between the circumference and diameter is always the same.[4]

Fundamentals

Definition

${\displaystyle \pi }$ is defined as the ratio of a circle's circumference ${\displaystyle c}$ to its diameter ${\displaystyle d}$:[5]

${\displaystyle \pi ={\frac {c}{d}}}$

Approximate value

${\displaystyle \pi }$ is often written as π. It is also an irrational number, meaning it cannot be written as a fraction ${\displaystyle {\bigg (}{\frac {a}{b}}{\bigg )}}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are integers (whole numbers).[2][3] This basically means that the digits of ${\displaystyle \pi }$ that are to the right of the decimal go forever—without repeating in a pattern, and that it is impossible to write the exact value of ${\displaystyle \pi }$ as a number. ${\displaystyle \pi }$ can only be approximated, or measured to a value that is close enough for practical purposes.[6]

A value close to ${\displaystyle \pi }$ is 3.141592653589793238462643[7] A common fraction approximation of ${\displaystyle \pi }$ is ${\displaystyle 22/7}$, which yields approximately 3.14285714. This approximation is 0.04% away from the true value of ${\displaystyle \pi }$. While this approximation is accepted for most of its use in real life, the fraction ${\displaystyle 355/113}$ is more accurate (giving about 3.14159292), and can be used when a value closer to ${\displaystyle \pi }$ is needed.[8] Computers can be used to get better approximations of ${\displaystyle \pi }$.

In March 2019, Emma Haruka Iwao calculated the value of ${\displaystyle \pi }$ to 31.4 trillion digits.[9][10]

History

Mathematicians have known about ${\displaystyle \pi }$ for thousands of years, because they have been working with circles for the same amount of time. Civilizations as old as the Babylonians have been able to approximate ${\displaystyle \pi }$ to many digits, such as the fraction ${\displaystyle 25/8}$ and ${\displaystyle 256/81}$. Most historians believe that ancient Egyptians had no concept of ${\displaystyle \pi }$, and that the correspondence is a coincidence.[11]

The first written reference to ${\displaystyle \pi }$ dates to 1900 BCE.[12] Around 1650 BCE, the Egyptian Ahmes gave a value in the Rhind Papyrus. The Babylonians were able to find that the value of ${\displaystyle \pi }$ was slightly greater than 3, by simply making a big circle and then sticking a piece of rope onto the circumference and the diameter, taking note of their distances, and then dividing the circumference by the diameter.

Knowledge of the number ${\displaystyle \pi }$ passed back into Europe and into the hands of the Hebrews, who made the number important in a section of the Bible called the Old Testament. After this, the most common way of trying to find ${\displaystyle \pi }$ was to draw a shape of many sides inside any circle, and use the area of the shape to find ${\displaystyle \pi }$. The Greek philosopher Archimedes, for example, used a polygon shape that had 96 sides in order to find the value of ${\displaystyle \pi }$, but the Chinese in 500 CE were able to use a polygon with 16,384 sides to find the value of ${\displaystyle \pi }$. The Greeks, like Anaxagoras of Clazomenae, were also busy with finding out other properties of the circle, such as how to make squares of circles and squaring the number ${\displaystyle \pi }$. Since then, many people have been trying to find out more and more precise values of ${\displaystyle \pi }$.[13]

A history of ${\displaystyle \pi }$
Philosopher Date Approximation
Claudius Ptolemy around 150 CE 3.1416
Zu Chongzhi 430-501 CE 3.1415929203
al-Khwarizmi around 800 CE 3.1416
al-Kashi around 1430 3.14159265358979
Viète 1540–1603 3.141592654
Roomen 1561–1615 3.14159265358979323
Van Ceulen around 1600 3.14159265358979323846264338327950288

In the 16th century, better and better ways of finding ${\displaystyle \pi }$ became available, such as the complicated formula that the French lawyer François Viète developed. The first use of the Greek symbol "π" was in an essay written in 1706 by William Jones.

A mathematician named Lambert also showed in 1761 that the number ${\displaystyle \pi }$ was irrational; that is, it cannot be written as a fraction by normal standards. Another mathematician named Lindeman was also able to show in 1882 that ${\displaystyle \pi }$ was part of the group of numbers known as transcendentals, which are numbers that cannot be the solution to a polynomial equation.[3][14]

${\displaystyle \pi }$ can also be used for figuring out many other things beside circles.[11] The properties of ${\displaystyle \pi }$ have allowed it to be used in many other areas of math besides geometry, the study of shapes. Some of these areas are complex analysis, trigonometry, and series.

Pi in real life

There are different ways to calculate many digits of ${\displaystyle \pi }$. This is of limited use though.

${\displaystyle \pi }$ can sometimes be used to work out the area or the circumference of any circle. To find the circumference of a circle, use the formula ${\displaystyle 2\pi r}$ (Radius). To find the area of a circle, use the formula ${\displaystyle \pi r^{2}}$ (radius squared). This formula is sometimes written as ${\displaystyle A=\pi r^{2}}$, where ${\displaystyle A}$ is the variable for the area.

To calculate the circumference of a circle with an error of 1 mm:

• 4 digits are needed for a radius of 30 meters
• 10 digits for a radius equal to that of the earth
• 15 digits for a radius equal to the distance from the earth to the sun.
• 20 digits for a radius equal to the distance from the earth to Polaris.

People generally celebrate March 14 as Pi Day, because March 14 is also written as ${\displaystyle 3/14}$, which represents the first three numbers 3.14 in the approximation of ${\displaystyle \pi }$.[6] ${\displaystyle \pi }$ day was started in 1988 by physicist Larry Shaw at the San Francisco Exploratorium. On March 11, 2009, almost 21 years later US House of Representatives passed a resolution proclaiming March 14 to be celebrated as National ${\displaystyle \pi }$ Day every year.

References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-10.
2. "Pi". www.mathisfun.com. Retrieved 2020-08-10.
3. Weisstein, Eric W. "Pi". mathworld.wolfram.com. Retrieved 2020-08-10.
4. Berggren, Lennart; Borwein, Jonathan M.; Borwein, Peter B. (2000). Pi, a Sourcebook. Springer Verlag. ISBN 978-0-387-98946-4.
5. Arndt, Jörg; Haenel, Christoph (2006), Pi Unleashed, Springer-Verlag, ISBN 978-3-540-66572-4, English translation by Catriona and David Lischka
6. "About Pi". 1994–2010. Retrieved 2010-06-05.
7. "How Many Decimals of Pi Do We Really Need?". jpl.nasa.gov. Retrieved February 19, 2018.
8. "Pi to 4 Million Decimals". Archived from the original (php) on 2008-03-09. Retrieved 2010-06-05.
9. Cajori, Florian (2007). A History of Mathematical Notations: Vol. II. Cosimo, Inc. pp. 8–13. ISBN 978-1-60206-714-1. the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented 3.14159... by δ:π, as did Oughtred more than a century earlier
10. Shaban, Hamza (2019). "Pi Day news: Google employee breaks record, calculates 31.4 trillion digits of Pi". chicagotribune.com. Chicago Tribune. Retrieved 2019-03-14.
11. Arndt, Jorg; Haenel, Christoph (2001). Pi - Unleashed. Springer Science & Business Media. ISBN 978-3-540-66572-4.
12. Beckmann, Petr 1971. A History of Pi. St. Martins Press, London.
13. J.J. O'Connor; E.F. Robertson (August 2001). "Pi History". Retrieved 2010-06-05.
14. "PI". 2000–2005. Retrieved 2010-06-06.