Vachmi
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.
|
$$ i^0 = 1$$ |
$$ i^1 = i$$ |
$$ i^2 = -1$$ |
$$ i^3 = -i$$ |
|
$$ i^4 = 1$$ |
$$ i^5 = i$$ |
$$ i^6 = -1$$ |
$$ i^7 = -i$$ |
As can be seen, the positive powers of $i$ are periodic with period of 4.
To get the value of $i^n$ where n is any positive integer, divide the integer by 4 and use the remainder to get the value from above table.
For example, to get the value of $i^{217}$, we will divide 218 by 4 giving us a remainder of 1 $$218 = 54 * 4 + 2$$ $$i^2 = -1$$
Negative powers of $i$ also repeat with periodicity of 4.
|
$$ i^{-1} = -i$$ |
$$ i^{-2} = -1$$ |
$$ i^{-3} = i$$ |
$$ i^{-4} = 1$$ |
|
$$ i^5 = -i$$ |
$$ i^6 = -1$$ |
$$ i^7 = i$$ |
$$ i^8 = 1$$ |
$i$ 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 $\sqrt{2}$
Pythagoras' constant is the square root of 2. This was the first irrational number ever discovered.
The numberical value of $\sqrt{2}$ truncated to 5 decimal places is 1.41421
With the help of modern computers, the value of $\sqrt{2}$ has been calculated upto 10 trillion decimal places
|
The square root of 2 is equal to the length of the hypotenuse of a right triangle with sides of length 1 |
Archimedes' constant $π$
The ratio of the circumference of a circle to its diameter is called Archimedes' constant and is represented by symbol $π$.
It is pronouced as pi.
|
$$π = \frac{C}{d} = \frac{C}{2r}$$ |