Mathematical constant

Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (π).

It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.

Its numerical value truncated to 50 decimal places is: Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers.

The constant π (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle.

It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, and Cauchy distributions in probability.

Memorizing as well as computing increasingly more digits of π is a world record pursuit.

The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth.

The number φ, also called the golden ratio, turns up frequently in geometry, particularly in figures with pentagonal symmetry.

This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).

Euler's constant or the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm: It appears frequently in mathematics, especially in number theoretical contexts such as Mertens' third theorem or the growth rate of the divisor function.

That includes the major open questions of whether it is a rational or irrational number and whether it is algebraic or transcendental.

is approximately: Apery's constant is defined as the sum of the reciprocals of the cubes of the natural numbers:

Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.

The numeric value of Apéry's constant is approximately: Catalan's constant is defined by the alternating sum of the reciprocals of the odd square numbers: It is the special value of the Dirichlet beta function

Catalan's constant appears frequently in combinatorics and number theory and also outside mathematics such as in the calculation of the mass distribution of spiral galaxies.

having been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven.

is approximately: Iterations of continuous maps serve as the simplest examples of models for dynamical systems.

The map was popularized in a seminal 1976 paper by the Australian biologist Robert May,[13] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

The Feigenbaum constants in bifurcation theory are analogous to π in geometry and e in calculus.

The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2.

[19] In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomly chosen Turing machine will halt, formed from a construction due to Argentine-American mathematician and computer scientist Gregory Chaitin.

It is common to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it).

Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries.

For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi.

Some constants differ so much from the usual kind that a new notation has been invented to represent them reasonably.

[23][24] It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis.

Many mathematical constants have an analytic form, that is they can be constructed using well-known operations that lend themselves readily to calculation.

Symbolizing constants with letters is a frequent means of making the notation more concise.

However, for more important constants, the symbols may be more complex and have an extra letter, an asterisk, a number, a lemniscate or use different alphabets such as Hebrew, Cyrillic or Gothic.

For example, American mathematician Edward Kasner's 9-year-old nephew coined the names googol and googolplex.

The circumference of a circle with diameter 1 is π .
The square root of 2 is equal to the length of the hypotenuse of a right-angled triangle with legs of length 1.
A diagram of a circle with a square coving the circle's upper right quadrant.
The area of the circle equals π times the shaded area. The area of the unit circle is π .
Exponential growth (green) describes many physical phenomena.
The imaginary unit i in the complex plane . Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
Golden rectangles in a regular icosahedron
The area between the two curves (red) tends to a limit, namely the Euler-Mascheroni constant.
Bifurcation diagram of the logistic map.
This Babylonian clay tablet gives an approximation of the square root of 2 in four sexagesimal figures: 1; 24, 51, 10, which is accurate to about six decimal figures. [ 15 ]
The universal parabolic constant is the ratio, for any parabola , of the arc length of the parabolic segment (red) formed by the latus rectum (blue) to the focal parameter (green).