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Mathematical constants


List of Mathematical Constants

Mathematical constants are some interesting numbers which occupy special place in mathematics.
While there are many constants, in this article we are going to restrict ourselves to only the following constants.

Constant

Denoted by

Value

Euler's number

$e$

~2.71828

Imaginary unit $i$

$i$

$\sqrt{-1}$

Pythagoras' constant

$\sqrt{2}$

~1.41421

Archimedes' constant

$π$

~3.14159




Euler's number $e$

Euler's number $e$ is a famous irrational number. It was named after the famous 18th century Swiss mathematician and physicist Leonhard Euler. As it is an irrational number, it cannot be expressed with complete accuracy.
The value of $e$ can approximately be given as 2.7182818284590452353602875..

There are many ways of calculating the value of e. One of the ways is to find the sum of the following series.
$$ e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + \frac{1}{8!} +... $$
In other words, $$ e = \sum_{n=0}^\infty \frac{1}{n!}$$
The value of $e$ is found in many mathematical formulas such as those describing a nonlinear increase or decrease such as growth or decay.
One of the interesting uses of $e$ is to calculate interest using continuous compounding. Do not forget to read up the article on continous compunding here.



Imaginary unit $i$

The imaginary number $i$ is defined as $i = \sqrt{-1}$. So an imaginary number gives a negative number -1 when it is squared.
$$ i × i = -1$$
There are other alternate notations for imaginary unit such as $j$ and are used where notation $i$ may create confusion such as in electrical engineering (where $i$ is typically used to represent current).

There are two complex square roots of -1, namely $i$ and -$i$.

The existance of $i$ can be pretty useful in some cases like solving some quadratic equations, in Fourier Transforms, in electrical and electronics and many more such cases. Hence, even though they are called imanginary, they are not really 'imaginary' in that sense.

The higher powers of $i$ repeat themselves with a fixed frequency.

Graphical representation of square

$i$ and all the complex numbers can be represented on a two dimensional complex plane in which horizontal axis represents real numbers and vertical axis represents imaginary axis.




Pythagoras' constant $\sqrt{2}$

A circle is a plane figure whose boundary (circumference) consists of points equidistant from a fixed point (called its centre)
Graphical representation of circle

Area of a circle = $π × radius^2$

Area of a circle with centre O and radius OA (r) = $π × r^2$

Perimeter (Circumference) of a rectangle = $2 × π × radius$

Perimeter (Circumference) of a circle with centre O and radius OA (r) = $2 × π × r$




Archimedes' constant $π$

A sector of a circle is an area enclosed by two radii and an arc of a circle.
Graphical representation of a Sector

Area of a circle = $π × radius^2 × C/360$

Area of a circle with centre O and radius OA (r) = $π × r^2 × C/360$
(when C is in degrees)

Perimeter of a sector = $(2 × radius) + $ (Length of arc AB)

Perimeter of a sector with radius r and arc AB = $(2 × r) + 2 π r × Angle (AOB)/360$
= $2r + π r × C/ 180$
(when C is in degrees)






Pentagon

A regular pentagon is a polygon with five equal sides. This also makes all the five angles in pentagon equal.
A pentagon can be considered as 5 equal triangles put together. Hence area of a pentagon is 5 times the area triangle.
Graphical representation of pentagon

Area of a pentagon = $ 5 × (1/2 × height × base)$

Area of a pentagon ABCDE = $ 5 × (1/2 × b × h)$

Perimeter of a pentagon = sum of the lengths of all the five sides

Perimeter of a pentagon ABCDE = $l(AB) + l(BC) + l(CD) + l(DE) + l(AE)$