Constants

A list of mathematical constants from across the internet.

π (Pi)

3.1415926535897932384626433832795028841971693993751…

Possibly the most famous of constants, Pi is a constant that is believed to date to the time of the early Egyptians in 2560 BC. Some sources claim that it was first written during the Babylonian times. Since then, Pi has been computed to 31,415,926,535,897 digits, which took over 120 days.

Pi has become common among most people, however, not many people know what it is or why it’s so important. Pi is commonly described to be the ratio of the circumference of a circle to its diameter. However, it also appears when computing the volumes of circles, cylinders, and spheres, which are all similar to circles. It is also sometimes described by the equation \int_{-1}^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}} .

Pi is irrational, transcendental, and likely normal. Irrational means that it can not be described by the division of two integer values (a fraction). Transcendental means that it is never the solution to a polynomial equation with integer coefficients (for example, 3x^2+2x^2-x+1 ). Likely normal means that it is likely to contain every number in its digits. Normality of Pi has not been proven, however it is likely. While there are other numbers that share these properties, Pi is the most popular and useful one.

Approximations of Pi extend all the way back to its origins. The Ancient Egyptians are believed to have approximated Pi to be 4*\frac{8}{9}^2 \approx 3.16045… The next highest approximation was made in 250 BC, by famous mathematician Archimedes. He described Pi as \frac{223}{71} < \pi < \frac{22}{7}, which had only approximated about 2 digits. In 150 AD, Ptolemy created the approximation of \frac{377}{120} \approx 3.14166… As time went on, the approximations of Pi hadn’t become better for centuries. However, in 1706, a man named John Machin calculated Pi to 100 digits, the previous best being 71. He achieved this by using the equation \frac{\pi}{4} = 4\arctan{\frac{1}{5}}-\arctan{\frac{1}{239}}. Further large improvements came in 1853 with 440 digits, and in 1949 with 1,120 digits. The modern era came with very good calculations of Pi, with the technology of computers. In 1973, over a million digits of Pi had been calculated and it had only taken 24 hours. As mentioned, the highest number of digits computed is now in the trillions, with a computation by Emma Haruka Iwao being the most recent contributor on Pi Day, 2019 using y-cruncher.

Equations of calculating Pi used to just be fractions as math was still in a somewhat elementary phase. However, over time there are many ways of computing Pi. An early one is one by Nilakantha in the 15th century. The series goes: \pi = 3 + \frac{4}{2*3*4} - \frac{4}{4*5*6} + \frac{4}{6*7*8}… In publications by Indian mathematician Srinivasa Ramanujan, many formulae for Pi were given. A popular one is \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4(396^{4k})} . Another fast equation was given by the Chudnovsky brothers in 1987. It is given by: \frac{12}{640320^\frac{3}{2}} \sum_{k=0}^\infty \frac{(6k)!(13591409+545140134k)}{(3k)!(k!)^3(-640320)^{3k}} .

e (Euler’s number)

2.7182818284590452353602874713526624977572470936999…

The first references to the number e was in a publication by John Napier in 1618, he had described many limit equations that tended towards e, however he never gave a real value. The first person accredited to have discovered the value of e was Jacob Bernoulli in 1683, after he solved the limit \lim_{n\to\infty} (1+\frac{1}{n})^n. e has been calculated to over 8,000,000,000,000 digits, which took place over a time period of 53 days.

e is not very well known to the common people, however it is very important in almost any subject of math you can think of. It is usually explained to be the result of when compound interest is applied more frequently over a period of time, however the percentage of the interest goes down. If you have 1 dollar, and compound interest is applied every year, after one year you have $2 dollars. If interest is applied every month, you end up with approximately $2.633035 as the interest rate has gone down. If it is applied every week, $2.714567. It increases, however the rate of which it does decreases. Once you have constant interest applied, you end up with e.

Like Pi, e is irrational, transcendental, and believed to be normal.

Jacob Bernoulli, the person who first described e, only had one digit correct in 1690. In 1714, Roger Cotes was able to calculate 13 digits. Leonhard Euler, the person the constant is named after calculated 23 digits in 1748. William Shanks was able to produce 137 digits in 1853, and 205 in 1871. In 1949, John von Neumann calculated 2,010 digits. The constant has since been calculated to 8,000,000,000,000 digits by Gerald Hofmann on January 3rd, 2019, using y-cruncher.

Approximations of e are less common, but the common ones are simple and converge very quickly. A very common one is: e = \sum_{n=0}^\infty \frac{1}{n!}. Here, n! is the factorial function, multiplying all the numbers from 1 up to n. A lot of the definitions root back to the original paper by Jacob Bernoulli. Another limit other than the one mentioned above is \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}. Some faster methods exist such as [\sum_{k=0}^\infty\frac{4k+3}{2^{2k+1}(2k+1)!}].

√2

1.4142135623730950488016887242096980785696718753769…

The earliest record of the constant was in 1800-1600 BC during the Babylonian Era. The original approximation had 5 digits correct, being that \sqrt{2}\approx1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}=1.4142196.... Another early figure was given in a text during 800-200 BC in India. The text, the Sulbasutras, describes \sqrt{2}\approx1+\frac{1}{3}+\frac{1}{3*4}-\frac{1}{3*4*34}=1.41421568.... It has been calculated to 10,000,000,000,000 digits, which took 44 days.

The square root of 2 and other roots aren’t as common but are important in architecture, geometry, and many other subjects of math. It is used as the ratio of A-sized paper’s length to its width for all sizes. It also appears naturally, as when you have a right triangle with two congruent legs of integer lengths, the hypotenuse is always a multiple of the square root of 2.

The square root of 2 is irrational and possibly normal, however it is not transcendental as it is the solution to a polynomial with integer coefficients (one being x^2=2).

The first few mentions of the constant are not attributed to a single person due to lack of documentation. It was mentioned in Babylonian times and during Ancient India. Hippasus of Metapontum proved that the square root of 2 was irrational, however the exact time is not known. Vitruvius described how the constant was used in architecture at the time, called the ad quadratum technique. Not much is documented in between, however in the time of the computer era, approximation got better, which has eventually come to a calculation of 10,000,000,000,000 digits by Ron Watkins.

Approximations of the constant are very simple and converge very quickly. A common one is the Babylonian Method, which goes a_0>0, a_{n+1}=\frac{a_n}{2}+\frac{1}{a_n}. This converges quickly and roughly doubles the number of correct digits. Due to the very quick convergence, other approximations are infrequent and unused.