EULER NUMBER OR E NUMBER: HOW MUCH IS IT WORTH, PROPERTIES, APPLICATIONS - MATHS - 2025

The Euler number or number e is a well-known mathematical constant that appears frequently in numerous scientific and economic applications, along with the number π and other important numbers in mathematics.

A scientific calculator returns the following value for the number e:

Figure 1. Euler's number appears frequently in Science. Source: F. Zapata.

e = 2.718281828…

But many more decimals are known, for example:

e = 2.71828182845904523536…

And modern computers have found trillions of decimal places for the number e.

It is an irrational number, which means that it has an infinite number of decimal places without any repeating pattern (the sequence 1828 appears twice at the beginning and is no longer repeated).

And it also means that the number e cannot be obtained as the quotient of two whole numbers.

History

The number e was identified by the scientist Jacques Bernoulli in 1683 when he was studying the problem of compound interest, but previously it had appeared indirectly in the works of the Scottish mathematician John Napier, who invented logarithms around 1618.

However, it was Leonhard Euler in 1727 who gave it the name number e and intensively studied its properties. This is why it is also known as the Euler number and also as a natural base for the natural logarithms (an exponent) currently used.

How much is the number e worth?

The number e is worth:

e = 2.71828182845904523536…

The ellipsis means that there are an infinite number of decimal places and in fact, with today's computers, millions of them are known.

Representations of the number e

There are several ways to define e that we describe below:

The number e as a limit

One of the various ways in which the number e is expressed is the one that the scientist Bernoulli found in his works on compound interest:

In which you have to make the value n a very large number.

It is easy to check, with the help of a calculator, that when n is very large, the previous expression tends to the value of e given above.

Of course we can ask ourselves how big n can be made, so let's try round numbers, like these for example:

n = 1000; 10,000 or 100,000

In the first case we obtain e = 2.7169239…. In the second e = 2.7181459… and in the third it is much closer to the value of e: 2.7182682. We can already imagine that with n = 1,000,000 or larger, the approximation will be even better.

In mathematical language, the procedure of making n get closer and closer to a very large value is called the limit to infinity and is denoted like this:

To denote infinity the symbol "∞" is used.

The number e as a sum

It is also possible to define the number e through this operation:

The figures that appear in the denominator: 1, 2, 6, 24, 120… correspond to the operation n !, where:

And by definition 0! = 1.

It is easy to check that the more addends added, the more precisely the number e is reached.

Let's do some tests with the calculator, adding more and more addends:

1 +1+ (1/2) + (1/6) = 2.71667

1 +1+ (1/2) + (1/6) + (1/24) = 2.75833

1 +1+ (1/2) + (1/6) + (1/24) + (1/120) = 2.76667

1 +1+ (1/2) + (1/6) + (1/24) + (1/120) + (1/720) = 2.71806

The more terms added to the sum, the more the result resembles e.

Mathematicians devised a compact notation for these sums involving many terms, using the summation symbol Σ:

This expression is read like this "sum from n = 0 to infinity of 1 between n factorial".

The number e from the geometric point of view

The number e has a graphical representation related to the area under the graph of the curve:

y = 1 / x

When the values ​​of x are between 1 and e, this area is equal to 1, as illustrated in the following figure:

Figure 2. Graphic representation of the number e: the area under the 1 / x curve, between x = 1 and x = e is worth 1. Source: F. Zapata.

Properties of the number e

Some of the properties of the number e are:

-It is irrational, in other words, it cannot be obtained simply by dividing two whole numbers.

-The number e is also a transcendent number, which means that e is not a solution to any polynomial equation.

-It is related to four other famous numbers in the field of mathematics, namely: π, i, 1 and 0, through the Euler identity:

-The so-called complex numbers can be expressed through e.

-It constitutes the base of the natural or natural logarithms of the present time (the original definition of John Napier differs a little).

-It is the only number such that its natural logarithm is equal to 1, that is:

Applications

Statistics

The number e appears very frequently in the field of probability and statistics, appearing in various distributions, such as normal or Gaussian, Poisson's and others.

Engineering

In engineering it is frequent, since the exponential function y = e ^x is present in mechanics and electromagnetism, for example. Among the many applications we can mention:

-A cable or chain that hangs held by the ends, adopts the shape of the curve given by:

y = (e ^x + e ^-x) / 2

-An initially discharged capacitor C, which is connected in series to a resistor R and a voltage source V to charge, acquires a certain charge Q as a function of time t given by:

Q (t) = CV (1-e ^{-t / RC})

biology

The exponential function y = Ae ^Bx, with A and B constant, is used to model cell growth and bacterial growth.

Physical

In nuclear physics, radioactive decay and age determination are modeled by radiocarbon dating.

Economy

In the calculation of compound interest the number e arises naturally.

Suppose that you have a certain amount of money P _o to invest at an interest rate of i% per year.

If you leave the money for 1 year, after that time you will have:

After another year without touching it, you will have:

And continuing in this way for n years:

Now let's remember one of the definitions of e:

It looks a bit like the expression for P, so there must be a relationship.

We are going to distribute the nominal interest rate i in n periods of time, in this way the compound interest rate will be i / n:

This expression looks a bit more like our limit, but it still isn't exactly the same.

However, after some algebraic manipulations it can be shown that by making this change of variable:

Our money P becomes:

And what is between the braces, even if it is written with the letter h, is equal to the argument of the limit that defines the number e, missing only the limit.

Let's make h → ∞, and what is between the braces becomes the number e. This does not mean that we have to wait an infinitely long time to withdraw our money.

If we look closely, by making h = n / i and tending to ∞, what we have actually done is spread the interest rate over very, very small periods of time:

i = n / h

This is called continuous compounding. In such a case the amount of money is easily calculated like this:

Where i is the annual interest rate. For example, when depositing € 12 at 9% per year, through continuous capitalization, after one year you have:

With a profit of € 1.13.

References

1. Enjoy math. Compound interest: Periodic composition. Recovered from: enjoylasmatematicas.com.
2. Figuera, J. 2000. Mathematics 1st. Diversified. CO-BO editions.
3. García, M. The number e in elementary calculus. Recovered from: matematica.ciens.ucv.ve.
4. Jiménez, R. 2008. Algebra. Prentice Hall.
5. Larson, R. 2010. Calculation of a variable. 9th. Edition. McGraw Hill.