Mathematical Constants, e, pi, Euler's constant and the golden ratio

The several basic mathematical constants include Pi, e also known as Euler's number, Euler's constant or Euler-Mascheroni constant and the golden ratio.


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There are four major constants that appear within mathematical calculations. These math constants are used in a whole variety of equations and formulae and are repeatedly seen in a variety of areas.

The four major constants are Pi, Π, the natural log base which is often written as ‘e’ and is also called Euler’s number, then there is Euler’s constant which may also be referred to as the Euler Mascheroni constant and finally there is the Golden ratio.

These four math constants are seen in many mathematical calculations and they are widely used.

Table of the mathematical constants


Constant Letter value
Pi π 3.14159265358979323846264
Natural log base / Euler's number e 2.71828 18284 59045 23536....
Euler's constant
Euler-Mascheroni constant
g 0.57721 56649 01532 86060 65120 90082...
Golden ratio f 1.618033988749894...

Pi, π

Pi represented by the Greek letter p is a mathematical constant. It is the ratio between the circumference of a circle to its diameter as well as being the ratio between the area of a circle to the square of its radius.

Pi is an irrational number, i.e. it cannot be expressed as a fraction of two integers. The commonly used fraction 22/7 which is often used is only a rough approximation, although sufficient for many basic calculations where only accuracy is not required. In addition to this the decimal representation of Pi never ends or repeats. Beyond being irrational, Pi is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can ever produce it exactly.

Pi can be given in several representations:


Notation Number
Decimal 3.14159265358979323846
Hexadecimal 3.243F6A8885A308D31319
Binary 11.00100100001111110110


Natural log base

The mathematical number e, also known as Euler's number (not to be confused with the Euler-Mascheroni constant, sometimes called simply Euler's constant) is the unique real number that has the mathematical property that the function ex has the same value as the slope of the tangent line, for all values of x.

Euler's number, e, is transcendental, i.e. it is a number that does not arise from an ordinary algebraic expression. As a result, Euler's number, e is also irrational; its value cannot be given exactly as a finite or eventually repeating decimal.


Euler's constant

constant is also sometimes called the Euler-Mascheroni constant, and denoted by the Greek letter gamma. It is a less well known mathematical constant than pi or e, but it is still a very important one.

Euler's constant is defined as the limit, as n tends to infinity, of the sum of 1 + 1/2 + 1/3 + ... up to 1/n, minus the natural logarithm of n


Golden ratio

The golden ratio is an unusual number which exists in mathematics. It is a ratio that is said to have perfect dimensions and as a result it has also been used in artistic endeavour as well as mathematical calculations. Also, mathematicians down the years have studied the golden ratio because of its unique and interesting properties.

Two quantities are said to have the golden ratio if the ratio between the sum of the two quantities and the larger one is the same as the ratio between the larger one and the smaller.

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter (phi). The figure of a golden section illustrates the geometric relationship that defines this constant.

   



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