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Last modified on March 9th, 2024

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.

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.’

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.

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.

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.

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.

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.

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.

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

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

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.

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 = ${\dfrac{e^{ix}+e^{-ix}}{2}}$

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

**Jackson deposits $850 in an account at a rate of 4% that is compounded continuously. How much money will be in the account after 4 years?**

Solution:

As we know, A = Pe^{rt}

Here,

P = $850

r = 4% = 0.04

t = 4 years

Now, A = 850 ⋅ e^{0.04 ⋅ 4} = 850 ⋅ e^{0.16} = 997.484

Thus, $997.484 will be in the account after four years.