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

## Definition

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.

## Source

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.** **

## History

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** .

## Properties

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**.

## Applications

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.**

## Importance

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.**

## Curiosities

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**.