Euler’s Number

Euler’s number, Eulerian number (after Leonhard Euler and pronounced as ‘Oiler’ ), or Napier’s Constant, denoted as ‘e,’ is a mathematical constant whose value can be written as 2.71828182845904523536028747135266 and so on. Euler proved it is an irrational number by showing that its simple continued fraction expansion is infinite. When its value is represented in decimals, it never terminates, and its digits never repeat.

It is the base of the natural logarithm, whose value equals its derivatives, and is used to solve exponential growth and decay problems. 

Formula

Mathematically, ‘e’ is expressed by the equation:

${e=\lim _{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}}$

Now, by substituting the values of ‘n,’ we get the values of Euler’s number as:

n	${\left( 1+\dfrac{1}{n}\right) ^{n}}$	e
1	${\left( 1+\dfrac{1}{1}\right) ^{1}}$	2.00000
5	${\left( 1+\dfrac{1}{5}\right) ^{5}}$	2.48832
10	${\left( 1+\dfrac{1}{10}\right) ^{10}}$	2.59374
100	${\left( 1+\dfrac{1}{100}\right) ^{100}}$	2.70481
1000	${\left( 1+\dfrac{1}{1000}\right) ^{1000}}$	2.71692
10000	${\left( 1+\dfrac{1}{10000}\right) ^{10000}}$	2.71815
100000	${\left( 1+\dfrac{1}{100000}\right) ^{100000}}$	2.71827

Here, we observe that if ‘n’ reaches to ‘∞,’ the value of ${\left( 1+\dfrac{1}{n}\right) ^{n}}$ reaches to the value of ‘e.’

Alternative Formula

Euler proved that ‘e’ is irrational by showing that its continued fractional expansion is infinite. Here, ‘e’ is also defined as the sum of infinite numbers. 

Mathematically, it is also written as:

${e=\sum ^{\infty }_{n=0}\dfrac{1}{n!}=\dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\ldots}$

Let us verify the value of ‘e’ by adding the first few terms.

Now, we have

${\dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+\dfrac{1}{5!}+\dfrac{1}{6!}}$

= ${1+1+\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{24}+\dfrac{1}{120}+\dfrac{1}{720}}$

= 2.718055…

≈ 2.71806, near ‘e,’ which implies that the more terms we have, the more accurate ‘e’ value we get.

Moreover, according to the Lindemann–Weierstrass theorem, ‘e’ is transcendental, which means it is not a solution for any non-constant polynomial equation with rational coefficients. 

Applications

Euler’s number is often used to represent a non-linear increase or decrease in a function, such as the growth or decay of a population in real life, exponential distributions, calculus, probability theory, or logarithmic bases.

The Base of Natural Logarithm

Euler’s number is the base of the natural logarithm, denoted as log_e(x) or ln(x), which means that e^x represents the inverse function of the natural logarithm, and ln(e) equals 1.

Solving Exponential Functions

The exponential function with base ‘e’ is unique since its derivative is itself.

In calculus, this is expressed as ${\dfrac{d}{dx}e^{x}=e^{x}}$, which helps to solve differential equations and modeling processes involving exponential growth or decay.

However, its antiderivative is found using the formula ${\int e^{x}dx=e^{x}+c}$, where ‘c’ is an integral constant.

Calculating Exponential Growth or Decay of a Function

Euler’s number is used to calculate the exponential growth and decay of a number, as shown.

Here, the slope of the curve at any point equals e^x

Also, the area of Euler’s number is:

Here, the area up to any x-value equals e^x

The exponential function y = e^x is used to model population growth, radioactive decay, and bacterial growth, among many others.

Calculating Compound Interest

Euler’s number is found in the formula of compound interest used to calculate the amount of money accumulated over time. The formula is given by:

A = Pe^rt

Here. 

‘A’ is the amount of money accumulated, 

‘P’ is the principal amount, 

‘e’ is Euler’s number,

‘r’ is the interest rate, 

and ‘t’ is the time in years.

Finding Standard Normal Distribution

Using Euler’s number, the probability density function gives the standard normal distribution

${\phi \left( x\right) =\dfrac{1}{\sqrt{2\pi }}e^{-\dfrac{1}{2}x^{2}}}$

Solving Inequalities

Euler’s number is found in inequalities, which is true for

 ${\left( 1+\dfrac{1}{x}\right) ^{x} <e <\left( 1+\dfrac{1}{x}\right) ^{x+1}}$, with all positive values of ‘x.’ 

⇒ e^x ≥ x + 1 (∀ x Є ℝ), if and only if x = 0.

Finding Complex Numbers

The exponential function e^x can be written as a Taylor series:

${e^{x}=1+\dfrac{x}{1!}+\dfrac{x^{2}}{2!}+\ldots =\sum ^{\infty }_{n=0}\dfrac{x^{n}}{n!}}$

This series converges for all complex values of x, making it useful for defining e^x for complex numbers. Combining with the Taylor series for sin and cos x, we can derive Euler’s formula as

e^ix = cosx + isinx, which holds for every complex value of ‘x.’

Here, in the special case when x = ℼ, we get Euler’s identity: e^iℼ + 1 = 0, from which it follows that, in the principal branch of the logarithm,

 ln(-1) = iℼ.

Now, using the laws of exponentiation, we get

(cosx + isinx)ⁿ = (e^ix)ⁿ = e^inx = cos(nx) + isin(nx), de Moivre’s formula.

The expression cosx + isinx is sometimes called cis(x). Here, sinx = ${\dfrac{e^{ix}-e^{-ix}}{2i}}$ and cosx = ${\dfrac{e^{ix}+e^{-ix}}{2}}$

Solving Differential Equations

The solution to the differential equation y = y’ is y(x) = Ce^x, where ‘C’ is a real number.

Solved Example