What does e mean in math?

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Lowercase ‘e’ is also known as Euler’s number named after the famous mathematician Leonhard Euler. This number is an irrational number because it has a non – recurring decimal that stretches to infinity. Euler’s number ‘e’ is a numerical constant just like pi $(\pi )$ and is one of the key pillars in mathematical calculations.

The value of e is 2.71828182845…………

As e is an irrational number, it is usually used under logarithm concepts. e plays a key role in problem solving in both mathematics as well as physics. The symbol e is also known as Napier’s Constant.

The value of e raised to the power 1 will give the same value as e but value of e raised to the power 0 is equal to 1 and e raised to the power gives the value as infinity. The reason is that e when results in a very big number or infinity. Similarly, when e is raised to negative power of infinity, the number turns out to be very small, which is almost equal to zero.

\begin{array}{l}{{e}^{1}}\,\,=\,e\\{{e}^{0}}\,\,=\,1\\{{e}^{\infty }}\,=\,{{(2.71)}^{\infty }}\\{{e}^{{-\infty }}}=\,{{(2.71)}^{{-\infty }}}\,=0\end{array}

Important things to note about e

e is an irrational number i.e. digits after decimal are non-recurring. So, it cannot be expressed in the form of a fraction.
e has its application in many areas, concepts and subjects such as advanced mathematics and physics.
e is the base of all natural logarithms (This was discovered by John Napier).
e is transcendental or non-algebraic. This means that e cannot be obtained by performing calculations of any non-zero polynomial having finite degree as well as rational coefficients.
It’s the base to all natural logarithms i.e. Log (e) or Log_e e = 1
Area under the curve y = 1/x from x=1 to x=e is 1 sq. units.
The slope of the graph of $y={{e}^{x}}$ lies between $y={{2}^{x}}$ and $y={{3}^{x}}$ .

Euler’s number (e):

The Euler’s number ‘e’ is the limit of ${{(1+1/n)}^{n}}$ as n approaches infinity. It can also be expressed as the summation of the following series.

e=\,\sum\nolimits_{{n=0}}^{\infty }{{}}1/n!

e = 1/0! + 1/1! + 1/2! + 1/3! + ………………….and so on (here, ! means factorial)

e = 2.71828182845904523536028747135266249775724709369995…. which is an irrational number upto 50 decimal places.

Also, ${{e}^{x}}=1+\,\,\frac{x}{{1!}}\,+\,\,\frac{{{{x}^{2}}}}{{2!}}\,+\frac{{{{x}^{3}}}}{{3!}}\,+\,...,\,-\infty <x<\infty$

e is also equal to $\underset{{x\to \infty }}{\mathop{{\lim }}}\,\,\left( {1+\frac{1}{n}} \right)n$ where n tends to infinity.

Let’s check the value of e for some numbers!

n
1	2.00000
2	2.25000
5	2.48832
10	2.59374
100	2.70481
1,000	2.71692
10,000	2.71815
100,000	2.71827

For complex numbers the formula becomes:

eⁱ^π + 1 = 0

This number is used in various mathematical concepts and calculations. Similarly, like other mathematical constants such as beta, gamma, pi etc the value of e plays an important role.

Applications of e

Let’s check some areas where e plays a vital role not only in academics but in real life world as well.

Compound Interest
Probability, probability theorem, binomial distribution, binomial theorem and Pascal’s triangle.
Standard Normal Distribution and Derangements
Optical planning problems
Asymptotics
Differential and integral calculus
Exponential and logarithmic functions

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