E number


The number e , Euler’s number or as it is also known, Napier’s constant is one of the irrational numbers that has the greatest relevance and importance in the area of mathematics and algebra . A basic number in exponential functions that cannot be expressed in natural numbers.


What is the number e?

The number e, is an irrational number of which we cannot find out the exact value that it has since they have an infinite number of decimal places and for this reason it is considered as irrational.

  • Definition
  • Characteristics the number e
  • Source
  • History
  • Who discovered the number e
  • Properties
  • Applications
  • How much does it cost
  • How the number e is represented
  • Importance
  • Curiosities


In the area of ​​mathematics, we can define the number e as the base of the natural exponential function , which is sometimes also known as the Neperian base , this because the mathematician was the first to use it. The number is known as an irrational number because it cannot be expressed by the ratio of two whole numbers , its decimal numbers are infinite and also, it is a transcendent number since it cannot be expressed as the root of algebraic equations with rational coefficients .


Characteristics the number e

Among the main characteristics of the number e, we can mention the following:

  • It is an inconspicuous number and its figures cannot be repeated periodically .
  • The figures of the number e do not follow any kind of pattern .
  • It is generally known by the name of Napier’s constant or Euler’s number.
  • It can be used in different branches of the area of ​​mathematics.
  • It cannot be expressed using two whole numbers.
  • It also cannot be expressed as an exact decimal number or a repeating decimal. 


The well-known and important mathematician named Leonhard Euler , one of the most prolific mathematicians of all time, used the e notation in 1727 in connection with the theory of logarithms . The coincidence between the first letter of your surname and the name of our number is mere chance. 


The first time a record or an approximation of the number e is found in a mathematical treatise dates from the year 1614, in which John Napier’s Mirifici Logarithmorun Canonis is published . Despite this, the first approximation that was made with respect to the number was obtained by means of Jacob Bernoulli in the solution of the problem of the long-term interest of an initial fixed quantity that led him to know and investigate the fundamental algebraic limit , and whose value was set at 2.7182818.  Leonard Euler was the one who began to identify the number with its current symbol, which corresponds to the letter e, but he managed to present it to the mathematical community in his book Mechanica , approximately 10 years later.

Who discovered the number e

Actually the number was first discovered by Leonhard Euler but it was the Scotsman named John Napier who discovered this number which is actually a mathematical tool in the year 1614 . In an appendix to his work, his base constant appears , the number e, which we can see on all calculators today . It was thanks to his discovery that  multiplications can be replaced by additions , divisions by subtractions and powers by products , simplifying the manual performance of mathematical calculations .


The following properties can also be taken as the definition of e.

  • e is the sum of the inverses of the factorials .
  • e is the limit of the general term sequence .
  • the decimal expansion of e does not have any regularity whatsoever but with the continued fractions, which can be normalized or not, we obtain, in a normalized continuous fraction.
  • e is irrational and transcendental .


Some of the applications in which the e number can be used are the following:

  • In economics , which was actually the first area where he went to calculate compound interest .
  • In biology where it is very important to be able to describe cell growth .
  • In electronics to make descriptions about the discharge of a capacitor .
  • In the area of chemistry to describe ion concentrations or the development of a reaction .
  • To work with complex numbers , mainly in Euler’s formulas .
  • In paleontology to date fossils using Carbon 14.
  • In forensic medicine to measure the loss of heat from an inert body and thus know the moment of death.
  • In statistics , in probability theory and in the exponential function
  • In the golden ratio and the logarithmic spiral.
  • Because it appears in the exponential function, which models growth, its presence is important when we study accelerated growth or decline , such as populations of bacteria , the spread of diseases or radioactive decay , which is also useful in dating. of fossils.

How much does it cost

The number e equals approximately 2.71828 which is generally written as ≈ 2.718.


The number e is very important within the area of mathematics and in many other sectors that are related to production , science and everyday life . The number e plays a very important role in the area of calculus , and is part of many of the fundamental results of limits , derivatives , integrals , series , etc. In addition, it has a series of properties that make it possible to use it in the definition of expressions of great application in many areas of human knowledge.


Some curiosities related to the number e are the following:

  • The number e functions as the basis of the system of natural or natural logarithms .
  • The number is denoted by lnx = t , where x is a positive real number and t is positive for x> 1 and negative for x <1.
  • It is present in the definition of the function y (x) = ex, or y (x) = exp (x), its set of admissible values ​​CVA being the set R of all real numbers .

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