El number e, Euler's number or the well-known Napier constant is one of the most relevant and important irrational numbers in the fields of mathematics and algebra. A fundamental number in an exponential function that cannot be represented by a natural number. This number has great applications in the world of mathematics.
For this reason, we are going to dedicate this article to telling you everything you need to know about the number e, its characteristics and importance.
what is number e
It is an irrational number and we cannot know its exact value because it has infinite decimal places, so it is considered an irrational number. In mathematics, we can define the number e as the base of a natural exponential function, sometimes called neper base because neper mathematicians were the first to use it.
This number is called an irrational number because it cannot be represented as a ratio of two integers, its decimal number is infinite, and it is also a transcendental number because it cannot be represented as the root of an algebraic equation with rational coefficients. The number e is fundamental in many aspects, including the study of perfect numbers and in the analysis of natural phenomena, as well as in the history of mathematics.
Key features
Among the main features we can mention the following:
- This is a nondescript number whose numbers cannot be repeated regularly.
- The digits of the number e do not follow any kind of pattern.
- It is often called Napier's constant or Euler's number.
- It can be used in different branches of mathematics.
- It cannot be represented with two integers.
- It also cannot be represented as an exact decimal number or repeating decimals.
The famous and important mathematician Leonhard Euler, one of the most prolific mathematicians of all time, used the symbol e in the theory of logarithms in 1727. The coincidence between the first letter of your last name and the name of our number is purely coincidental. The first record or approximation of the number e found in mathematical papers dates back to 1614, when John Napier's Mirifici Logarithmorun Canonis was published. However, the first approximation to the numbers was obtained by Jacob Bernoulli when solving the problem of long-term interest in initial fixed quantities, which led him to understand and study the fundamental algebraic limit, and its value was fixed at 2,7182818.
Leonard Euler was the first to start recognizing numbers with the current symbol, which corresponds to the letter e, but he managed to introduce it about 10 years later in his Mathematical Mechanics. In fact, the number was first discovered by Leonhard Euler, but the man who discovered it in 1614 was a Scotsman named John Napier. Thanks to his discovery, multiplication can be replaced by addition, division by subtraction and multiplication by product, simplifying the manual execution of mathematical calculations.
Properties and applications of the number e
The following properties can also be used as definitions of e.
- e is the sum of the reciprocals of the factorials.
- e is the limit of the general sequence of terms.
- The fractional expansion of e has no regularity, but in normalized continued fractions, there may or may not be normalized continued fractions.
- e is irrational and transcendent.
Some applications in which this number can be used are the following:
- In economics, this is actually the first area of ​​compound interest calculation.
- In biology, being able to describe cell growth is very important.
- The discharge of a capacitor is described in electronics.
- Describes the development of ionic concentrations or reactions in the field of chemistry.
- Management of complex numbers, mainly Euler's formula.
- Carbon 14 dating of fossils in paleontology.
- Measure heat loss from inert objects in forensic medicine to determine the time of death.
- In statistics, probability theory and exponential functions
- In golden ratio and logarithmic spiral.
Because it appears in exponential functions that simulate growth, its presence is important when we study rapid growth or decline, such as bacterial populations, the spread of disease, or radioactive decay, and is also useful in dating fossils. On the other hand, the analysis of mayan numbers It also provides fascinating context on the importance of numbers in ancient cultures.
Importance and curiosities
The number e is roughly equivalent to 2.71828 and is usually written as ≈2718. This number is very important in mathematics and many other fields related to production, science and everyday life. This number plays a very important role in the field of calculus. and is part of many fundamental results such as limits, derivatives, integrals, series, etc. Furthermore, it has a set of properties that allow its use to define expressions that have important applications in many domains of human knowledge.
Some curiosities related to the number e are the following:
- The number e serves as the base of the natural or natural logarithmic system.
- The number is represented by lnx = t, where x is a positive real number, t is positive for x>1 and negative for x <1.
- It exists in the definition of a function y(x) = ex or y(x) = exp(x) whose CVA set of allowed values ​​is the set R of all real numbers.
Some history
The first indirect reference to this number occurs in John Napier's famous 1614 work, Mirifici Logarithmorum Canonis Descriptio, in which his ideas on logarithms, antilogarithms, results, and their calculation tables are first elaborated; however, Jacob Bernoulli will obtain the first approximation by solving the problem of the initial fixed amount of long-term interest, which takes you to the now known limit after successive iterations.
Set its value to 2,7182818. The mathematician and philosopher Gottfried Leibniz later took advantage of this value in letters to Christian Huygens in 1690 and 1691, denoting it with the letter b. Leonard Euler began to identify numbers in 1727 with the current symbol: the letter e, but it was not until a decade later that he presented the number to the mathematical community in his book Mechanics.
Later experts would use a, b, c and e until the latter wins for irrational numbers. Charles Hermite proved that this was a momentous number in 1873. Their approximation started with the work of Bernoulli, then Euler made an approximation of 18 positions after the comma, so they produced, as for determining the position of pi, the latest version of a competition was in 2010 Shigeru Kondo and Alexander J. Yee determined e to a billion exact decimal places.
I hope that with this information you can learn more about the e number and its characteristics.