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What does e mean in math?

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 Loge 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!

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:

eiπ + 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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