Where did the number e come from?

Question

WHat formula formed the number e and why is it important? Is there any other use for that formula other than finding e?

e=2.7182818284590452353602874713527 and so on. Its an irrational number

Answer 1

The significance of e cannot be understood without knowing some calculus. One way to define it is with the formula mentioned above, as the limit of the sequence (1 + 1/n)^n, i.e., take out your calculator, and plug in n=10, 100, 1000, 10000, 100000, etc. into the above expression -- you will see that the numbers you get out get closer and closer to 2.718281828... It is true that this is the expression involved in computing the interest added to a bank account if it is "compounded continuously."

Another way to find e is to look at the series:

2 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*5) + 1/(2*3*4*5*6) + ..., where what I mean is, add up the first 5 terms of this and see what you get. Then add up the first 10 terms and see what you get. You will find that the more terms you add on, the closer the value gets to 2.718281828..., i.e., to e.

It was named in honor of Leonard Euler (pronounced "Oiler"), a Swiss mathematician who lived in the 1700's and was probably the most prolific mathematician of all time. He published thousands of papers, many of which were seminal results, and he spent the latter years of his life blind, yet still was able to discover tons of new mathematics in his head, and then dictate the results to his servant.

The reason it is important is related to one of the main concepts in calculus, the derivative. Given a function f, i.e., f(x)=x^2, or whatever, you can associate a new function, called its derivative, denoted by f'. Now one may consider all of the so-called exponentical functions f(x)=a^x, where a is some fixed consant. It turns out that the function with a=e=2.718281828... is the nicest of these functions in the setting of calculus, because the associated derivative is itself, i.e., the derivative of e^x is just e^x. One property of this is that whenver you find some quantity in the world that increases, and where the rate of increase of that quantity is proportional to its size, then it will probably be described by a function very close to e^x. For instance, the population of the planet is increasing, and the rate of increase goes up as the number of people on the people go up, because having more people around means that there are more people adding new babies to the population. That is why the population of the Earth is said to exhibit exponential growth.

It turns out the most useful base for a logarithm is log base e, for similar reasons to why e was the most useful base for an exponential function.

Hope this helps.

Answer 2

The number e shows up naturally in processes of continuous growth. If you have a 15% continuous interest rate, the annual growth factor of your capital is not

[a] ... 1 + 0.15 = 1.15

as you would expect if interest was paid at the end of the year, but

[b] ... e^0.15 = 1.16183...

The reason is that some of the interest is paid earlier in the year and starts generating more interest. The value can be approximated by successive corrections to formula [a]:

[c] ... e^0.15 = 1 + 0.15 + 0.15^2/2 + 0.15^3/6 + ...

or in general,

[d] ... e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + ...

The denominators are the factorials, so we can write

[e] ... e^x = SUM x^n/n! { n = 0, 1, ... }

Using x = 1, we find

[f] ... e = e^1 = SUM 1/n!

That is,

[g] ... e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ...

Answer 3

Mathematically, "e" is the solution to the integral of (1/x)dx and thus forms the base of the "natural" logarithms. "e" is most useful when calculating first order logarithmic phenomena, i.e. those things in which the rate of change of something is proportional to the total quantity present. The radioactive decay of substances like Uranium is an example, since the amount lost to decay is always proportional to the amount present.

Answer 4

Be extremely cautious about any health care plan where the authors (those in government) are exempt from mandatory participation. .

Answer 5

Consider the formula y=e^x. Check the slope of the function at any value of x. The slope is 1/x.

Amazing how many people in the Mathematics forum have never heard of "e".

Answer 6

To find e, use a Taylor series:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + .......
e = Σ (1/n!) for n = 0 to infinity
e = 2.718..... (irrational number)

e is important in engineering and was popularized by Euler. "e" usually serves as the base number for a function raised to some power x or as a natural logarithm base....for example: e^x or ln(x).

The numerical value is not all that important, but the concept of using it in natural logarithms and as a base for exponents is. Euler showed that it has very useful properties for relating complex numbers to angles and trig functions. Euler's formula:

e^jθ = cos θ + j sin θ

Part of why it is useful is that its first derivative is itself (d/dx of e^x = e^x). In electronics, the behavior of electronic circuits follows e^-t quite often with the term time constant referring to the time it takes to reach a ratio of 1/e...in general, many things in nature follow a 1/e behavior.

Answer 7

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The first indication of e as a constant was discovered by Jacob Bernoulli

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel after a, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his last initial, since he was a very modest man, and tried to give proper credit to the work of others.
The three most common definitions of e are listed below.

1. The limit
2. The sum of the infinite series
3. The unique real number e > 0 such that
(that is, the number e such that area under the hyperbola f(t) = 1 / t from 1 to e is equal to 1).

Answer 8

I think it had something to do with compound interest. Its the limit as n approaches infinity of (1 + 1/n)^n
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

Answer 9

e is the base of the natural logarithm function and is approximately 2.7182 (rounded to 4 decimal places). It is an "irrational" number---that is it cannot be accurately exactly expressed by any finite number of decimal places, and there is no repeating pattern of the decimal digits.

e is sometimes referred to as Euler's number, after the 18th century Swiss mathematician.

The number e can be calculated in several ways. It can be calculated using the infinite series:

e=1 +1/1! + 1/2! + 1/3! +1/4! + ...
where k! = k x (k-1) X (k-2) x ... x 2 x 1.

It is also be defined in terms of the area under the function f(x)=1/x. (Specifically e is the value such that the area under the curve between 1 and e gives an area of exactly 1.)

Another way to define e is in terms of "continuous compound interest": suppose that $1 is invested at an interest rate of 100%, but instead of calculating the interest only at the end of the year, the interest rate is calculated n times per year (and the annual rate of interest is divided by n). If n increase to infiniity (so that the interest is calculated at every instant), how much money is the $1 worth after one year? The answer turns out to be e.

The number e is extremely important in higher mathematics, and in many mathematical applications. The study of population growth rates and radioactive decay involve formulas using e.
Certain functions in the study of probability and statistics, such as the normal or "bell" curve function, are calculated using e.
The applications of e are too numerous to recount here.

For more info see:http://en.wikipedia.org/wiki/Natural_logarithm
and http://en.wikipedia.org/wiki/Logarithm.

I hope that this helps!

Answer 10

The number e first comes into mathematics in a very minor way. This was in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers. However, that these were logarithms to base e was not recognised since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. We will come back to this point later in this essay. This table in the appendix, although carrying no author's name, was almost certainly written by Oughtred. A few years later, in 1624, again e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work....

...Further work on logarithms followed which still does not see the number e appear as such, but the work does contribute to the development of logarithms. In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x). In this work Mercator uses the term "natural logarithm" for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just round the corner.

Answer 11

e came about when trying to find out the derivative of the function, f(x) = a^x.

The the derivative of this function is
f '(x) = lim((a^(x+h)-a^x)/h = lim(a^x(a^h-1)/h).
.........h-->0.........................h-->0

This simplifies to
f '(x) = a^x*lim((a^h - 1)/h).
................h-->0

This limit is the value of the derivative of f at 0, therefore
lim((a^h-1)/h) = f '(0)
h-->0

As a result, the derivative of the function, f(x) = a^x can be written as f '(x) = f '(0)*a^x. The simplest differentiation formula occurs when f '(0) = 1. Therefore, e is defined as the number such that

lim((e^h-1)/h) = 1.
h-->0

This number turns out to be 2.718281828.....