E number


The number e is an irrational number that provides a constant base to the natural logarithm and can be understood as a limit of a given progression.

In other words, the number e is a decimal whose decimal part is non-periodic and is the only number that makes the natural logarithm equal to 1.


The number e owes its name to its discoverers since on some occasions we can find this number in the form of Euler's number (Leonhard Euler) or Napier's constant (John Napier).

Formula of number e

The number e is expressed mainly by the letter e. We can also find it as an exponential function when the variable is equal to 1:

Mathematic expression

The first 40 decimals of the number e are:

E number

The number e and the limits

We have to think of a limit of a given function when we want to understand the mathematical origin of the number e:

This function is a sequence such that:

Mathematical succession

We speak of succession because we can assign ordered values ​​to the variable n.

  • For n = 1 we will have f = 2
  • For n = 20 we will have f = 2.65329
  • For n = 100 we will have f = 2.7048

You can see that as we increase the value of n, that is, we increase the length of the sequence, the more the result of the function gets closer to the number e. What will happen for n = 10,000?

  • For n = 10 000 we will have f = 2.718146

The generalized mathematical expression of the sequence would be taking into account that n approaches infinity. In the following graph you can see how the sequence (black line) is approaching the limit (blue line), that is, the number e, as n increases.

Approximation of n

Therefore, as n approaches infinity, the function, that is, f (n), collapses into the number e, that is, 2.7181.

Mathematical sequence when n tends to infinity

So, we can understand the number e as a limit of a sequence:

The number e and the limit of the mathematical sequence


The number e appears numerous times in the fields of calculus, mathematical analysis, number theory, statistics, and geometry. It also appears in the discipline of finance, specifically, in the calculation of continuous profitability or continuous interest.

Applications of the e number


Write two formulas that contain the number e.

For example, we can think of the density function of the normal distribution and, in the domain of complex numbers, we can find the Euler identity, formulated through the number e:

Formulas containing the number e

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