Mathematics

Euler's Number 'e' (Napier's Constant)

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06 October 2020                

Read time: 5 minutes

Introduction

 The number e popularly known as Euler’s number is a significant mathematical constant approximately equal to 2.718281828. Although it was discovered many years ago, its impact in the modern era is an inevitable factor. It is quite interesting to learn about Euler’s number.

Learn about  Euler’s number


History

The number e is the base of the natural logarithm.

The great Mathematician John Napier is best known as the discoverer of logarithms,e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler, it is also called Euler's constant or Napier's constant.

 Though the history of Euler’s number comprises several mathematicians, namely John Napier, William Oughtred, Jacob Bernoulli, Euler made a great impact in discovering this number hence the name “Euler’s Number.”

 John Napier -The great Mathematician

Source: wikipedia.org


Mathematical origin of \(e^x\)

The numerical constant e equals 2.718281828.

The value of e is found in many mathematical formulas such as those describing a non-linear increase or decrease of certain constants.

For instance, calculating LIC policies that constitute compound interest, the statistical bell curve, the shape of a hanging cable, calculating the increase, or decreasing certain cities’ population.

It is defined as

\[e = 2.7172817284590 45\]

e is an irrational number, so the decimal number does not terminate, and it goes on.

Why “e”

Euler used the mathematical constant 'a' in his derivation and so it is not possible to use the letter again, hence he used the next vowel 'e' which became popular as Euler's number with the first letter "e".

Simple tricks to remember Euler’s constant

\[e = 2.717281728459045\]

Just remember 2.7. Repeat the number 1728 twice.

Then half of the right angle which is 45⁰ then the right angle 90⁰ and again 45⁰.

It can be easily read as \(e = 2.7 1728 1728 45 90 45\)


Graph of \(e^x\)

Graph


Properties of \(e^x\)

The letter e in Mathematics so-called Euler's number or constant is commonly used as a natural base for logarithmic and exponential series.

\[ x = \text{ln } x\]

The series in the Euler's number consists of the decimal number and it is a non-responding and non-terminating series.

One of the unique properties of e is that it helps in simplifying many mathematical equations and constants.

Application in Growth models in Real life

Suppose you deposit an amount, say, for example, Rs. 10000 every month. If the interest increases every six months, it is calculated with the help of 'e'. Here compound interest plays a significant role.

Appearance in Physics

  • It is used in statistical mechanics to find the Boltzmann distribution constant with the help of e.
  • To calculate the rate of a radioactive substance decay in Physics.
  • In astrophysics, Euler's number has a great role as well.

Role of Euler’s constant in chemistry

  • It is mainly used in chemistry to study the molecular division of the corresponding compounds which divides equally in particular seconds which is very well calculated with ‘e’.
  • According to the chemical variation, capsules are produced in a designated ratio for the pharmaceutical industry using Euler’s constant.

Role in fine arts

It is quite interesting to know that Euler’s constant is used in fine arts, whether you believe it or not Euler's number has a significant impact on the music industry.

To tell the musicians about the tones, notes are used on the musical scale. Euler’s Number is instrumental in explaining the high pitch, low pitch, and medium variation.

Here the ratio of keys is integrated with that of the musical note.

Musical note

In Economics

Economics is a critical factor that affects every country. Although it may look like a single term, Euler's number substantially impacts calculations of a country’s development, that is in economics, the inflation rate, GDP of a country, or a particular individual.

In various Mathematical field

Euler's number has fair, practical use, mathematicians use Euler’s number for different areas such as calculus, differential equation, Discrete Mathematics, Trignometry, Complex analysis, Statistics, etc.

Many Mathematicians recognized that Euler’s invariant possesses a wealth of exciting and non-trivial mathematical properties in the real life.

In miscellaneous areas

 Euler's number is used to deal with the banking and life insurance field, to examine a country’s population after a certain period.

To find the population of extinct species such as passenger pigeon, Dodo, Pyrenean ibex can be calculated with the help of Euler’s constant.


Summary

As the number has no end, it flows like a river. The concept of Euler's theorem and its application also comes under the never-ending category, the above description is just a brief review to know basic things about Euler's number. Many researchers and analyses are going on to date to know the true values and applications in real life and it is inexplicable in nature. Further study on this concept helps the reader to know further.


Examples

The population of a certain city is given. As the number of years varies as the Euler’s constant also varies, which is given in the following table.

n

Euler’s number

1

2.0000

2

2.2500

5

2.4883

10

2.5937


Frequently Asked Questions (FAQs)

Who founded Euler’s constant?

Leonhard Euler

Is Euler’s number a whole number?

No, it is an irrational number and the value is a never-ending process.

Tell few areas where Euler’s constant is used?

Cell division in microbiology and dealing compound interest in the banking sector.

Does Euler's number have any impact on a further mathematical study?

Of course, many mathematicians are working on it and it has a huge impact on the mathematical background.

Does the value Euler’s constant change while varying n?

Yes, the value of Euler's number increases or decreases based on n.

What is the value of e?

The value of \(e = 2.7182818459045.\)

What is Euler’s Number?

Euler’s number is a mathematical constant used as the base of the natural logarithm.

What is a natural log?

The natural log of a number is the logarithmic value of a number to the base e.

What is the natural log of e?

The natural log of e is 1. ln(e) = 1


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