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e (mathematical constant)

From Wikipedia, the free encyclopedia

Graph of the equation  y = 1 / x . {\displaystyle y=1/x.}  Here, e is the unique number larger than 1 that makes the shaded area equal to 1.
Graph of the equation Here, e is the unique number larger than 1 that makes the shaded area equal to 1.

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series[2]

The constant can be characterized in many different ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0.[3] The function f(x) = ex is called the (natural) exponential function, and is the unique exponential function equal to its own derivative. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor.[4] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[5]

The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in the OEIS).

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Transcription

[James]: We're gonna talk about e! The big, famous constant, e! Okay, it's one of the famous mathematical constants, One of the most important, goes along with pi, and I don't know, golden ratio, and square root of two, Constants in maths that are the most important constants, and e is one of those constants. So e is an irrational number, and it's equal to... 2.718281828, something, something, ... The problem with e, is it's not defined by geometry. Now pi, is a something that is defined by geometry, right, it's the ratio of a circle's circumference and it's diameter. And it's something the ancient Greeks knew about. And a lot of mathematical constants go back to the ancient Greeks, but e is different. e is not based on a shape, it's not based on geometry. It's a mathematical constant that is related to growth, and rate of change, but why is it related to growth and rate of change? So let's look at the original problem where e was first used. So we're going to go back to the seventeenth century, and this is Jacob Bernoulli, and he was interested in compound interest, so, earning interest on your money. So imagine you've got one pound in the bank. And you have a very generous bank and they're gonna offer you 100 percent interest every year. Wow, thanks alot, bank! So, 100 percent interest, so it means after one year, you'll have two pounds. So you've earned one pound interest and you've got your original pound. So, you now have two pounds. What if I offered you instead fifty percent interest, every six months? Now is that better or worse? Well, let's think about it. Ok, you're starting with one pound and then I'm going to offer you fifty percent interest every six months. So after six months, you now have one pound, fifty and then you wait another six months and you're earning fifty percent interest on your total, which is another seventy-five p. and you add that on to what you had so it's two pounds twenty-five Better! It's better. So what happens if I do this more regularly? What if I do it every month? I offer you one-twelfth interest every month Is that better? So, let's think about that. So after the first month, it's gonna be multiplied by this. One plus one-twelfth. So one-twelfth, that's your interest and then you're adding that onto the original pound that you've got. So, you do that, that's your first month, then for your second month you take that and multiply it again by the same value. and your third month you would multiply it again, and again. you actually do that twelve times in a year. So in a year, you'd raise that to a power twelve, and you would get two pounds sixty-one. So it's actually better. In fact, the more frequent your interest is the better the results. Let's start with every week. So if we do it for every week, how much better is that? What I'm saying is you're earning one over fifty two interest every week. And then after the end of the year you got fifty two weeks and you would have two pounds sixty nine. So it's getting better and better and better. In general, you might be able to see a pattern happening here. In general it would look like this: You'd be multiplying by one plus one over n, to the power n. Hopefully you can see that pattern happening. So here n is equal to twelve if you do it every month, fifty two if you do it every week. If you did it every day, it'd be one pound multiplied by [one plus] one over three hundred and sixty five to the power three hundred and sixty five. And that's equal to two pounds, seventy one. Right, and so it would get better if you did it every second, or every nanosecond. What if I could do it continuously? Every instant I'm earning interest. Continuous interest. What does that look like? That means if I take this formula here one plus one over n to the n, I'm gonna n tend to infinity. That would be continuous interest. Now what is that? What is that value? And that's what Bernoulli wanted to know. He didn't work it out. He knew it was between two and three. So fifty years later, Euler worked it out. Euler, he works everything out. [Brady]: Him or Gauss? [James]: It's either Euler or Gauss. Say Euler or Gauss, you're probably going to be right. And the value was 2.718281828459... and so on. [Brady]: We were pretty close when we were doing it daily, weren't we? It was already two seventy one at daily. [James]: You're right, You're right. We were getting closer, weren't we? We were getting close and closer to this value. So already we're quite close to it. If you did it forever though, of course you would have this irrational number. Now Euler called this e. He didn't name it after himself, although it is now known as the Euler constant. [Brady]: Why'd he call it e then? [James]: It was just a letter. He might've used a, b, c, and d already for something else. Right? So you use the next one. [Brady]: Bit of a coincidence! [James]: It's a lovely coincidence! I fully believe that he's not being a jerk here, naming it after himself. But it's a lovely coincidence that it's e for Euler's number. [Brady]: Would you have called it g if you discovered it? [James]: I would not have called it g. No, I would've hoped somebody else would've called it g and then I would have accepted that. Euler proved that this was irrational. He found a formula for e which was a new formula. Not this one here, a different formula. And it showed that it was irrational. I'll quickly show you that. He found that e was equal to two plus one over one plus one over two plus one over one plus one over one plus one over four plus one over one plus one over one plus one over six... and this goes on forever. This is a fraction that goes on forever, continuous fraction. But you can see it goes on forever Because there's a pattern, and that pattern does hold. You got two, one one four, one one six, one one eight. So you can see that pattern goes on forever, and if the fraction goes on forever it means it's an irrational number. If it didn't go on forever, it would terminate, and if you terminate you can write it as a fraction. And he also worked out the value for e. He did it up to eighteen decimal places. To do that, he had a different formula to do that, I'll show you that one. And this time, he worked out e was equal to one plus one over one factorial plus one over two factorial plus one over three factorial plus one over four factorial... and this is something that's going on forever. It's a nice formula, if you're happy with factorials. Factorials means you're multiplying all the numbers up to that value. So if it was four factorial, it'd be four times three times two times one. Okay, why is e a big deal? It's because e is the natural language of growth. And I'll show why. Okay, let's draw a graph y equals e to the x. So we're taking powers of e. So over here at zero, this would cross at one. So if you took a point on this graph, the value at that point is e to the power x. And this is why it's important. The gradient at that point, the gradient of the curve at that point is e to the x. And the area under the curve which means the area under the curve all the way down to minus infinity is e to the x. And it's the only function that has that property. So it has the same value, gradient, and area at every point along the line. So at one, the value is e because it's e to the power one. The value is 2.718, the gradient is 2.718 and the area under the curve is 2.718. The reason this is important then, because it's unique in having this property as well, it becomes the natural language of calculus. And calculus is the maths of rate of change and growth and areas, maths like that. And if you're interested in those things, if you write it in terms of e, then the maths becomes much simpler. Because if you don't write it in terms of e, you get lots of nasty constants and the maths is really messy. If you're trying to deliberately avoid using e, you're making it hard for yourself. It's the natural language of growth. And of course e is famous for bringing together all the famous mathematical constants with this formula, Euler's formula, which is e to the i pi plus one equals zero. So there we have all the big mathematical constants in one formula brought together. We've got e, we've got i, square root of minus one, we've got pi of course, we've got one and zero and they bring them all together in one formula which is often voted as the most beautiful formula in mathematics. I've seen it so often, I'm kinda jaded to it, don't put that in the video. [Brady]: Sometimes here on Numberphile we can make more videos than we'd otherwise be able to thanks to excellent sponsors. And today we'd like to thank "The Great Courses Plus," which is a fabulous service. If you like Numberphile videos, you're really going to like this. They've got an incredible array of lessons, something like over seven thousand videos from world experts, top professors around the world on all sorts of subjects from photography to paleontology to prime numbers. No matter what you're into, you're gonna find something. What I've been looking at just recently on the site is paleontology, and one of my favorite subjects, Egyptology. They've got some great videos on those. Now plans on The Great Courses Plus start from fourteen dollars ninety nine a month, but there's a special offer where you can go and get a whole month, unlimited access to all the videos for free! Go to TheGreatCoursesPlus.com/numberphile and have a free month's access, check it all out. If it's something you like, maybe you'll sign on for more. I think there's a good chance you will. There's the link on the screen and also there's a link in the video description. Why not give it a click? Just to go and have a look. It's a great way of showing your support for Numberphile and showing the people at The Great Courses Plus that you came from here. Our thanks again to them for supporting this video. Okay, I'm gonna go for e, e.

Contents

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[5] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[7][8] who attempted to find the value of the following expression (which is in fact e):

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 introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731.[9][10] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[11] and the first appearance of e in a publication was in Euler's Mechanica (1736).[12] While in the subsequent years some researchers used the letter c, the letter e was more common and eventually became standard.[citation needed]

The constant has been historically typeset as "e", in italics, although the ISO 80000-2:2009 standard recommends typesetting constants in an upright style.

Applications

Compound interest

The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies
The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies

Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:[5]

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.4414..., and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00×(1 + 1/n)n.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818...

More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.)

Bernoulli trials

Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 - P  vs n ; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.
Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 - P  vs n ; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.

The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, for large n (such as a million) the probability that the gambler will lose every bet is approximately 1/e. For n = 20 it is already approximately 1/2.79.

This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. The probability of winning k times out of a million trials is:

In particular, the probability of winning zero times (k = 0) is

This is very close to the limit

Derangements

Another application of e, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem:[13] n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into n boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. The answer is:

As the number n of guests tends to infinity, pn approaches 1/e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is n!/e rounded to the nearest integer, for every positive n.[14]

Optimal planning problems

A stick of length L is broken into n equal parts. The value of n that maximizes the product of the lengths is then either[15]

The stated result follows because the maximum value of occurs at (Steiner's problem, discussed below). The quantity is a measure of information gleaned from an event occurring with probability , so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

Asymptotics

The number e occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

As a consequence,

Standard normal distribution

The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution, given by the probability density function

The constraint of unit variance (and thus also unit standard deviation) results in the 1/2 in the exponent, and the constraint of unit total area under the curve ϕ(x) results in the factor .[proof] This function is symmetric around x = 0, where it attains its maximum value , and has inflection points at x = ±1.

In calculus

The graphs of the functions x ↦ ax are shown for a = 2 (dotted), a = e (blue), and a = 4 (dashed). They all pass through the point (0,1), but the red line (which has slope 1) is tangent to only ex there.
The graphs of the functions xax are shown for a = 2 (dotted), a = e (blue), and a = 4 (dashed). They all pass through the point (0,1), but the red line (which has slope 1) is tangent to only ex there.
The value of the natural log function for argument e, i.e. ln(e), equals 1.
The value of the natural log function for argument e, i.e. ln(e), equals 1.

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[16] A general exponential function y = ax has a derivative, given by a limit:

The parenthesized limit on the right is independent of the variable x: it depends only on the base a. When the base is set to e, this limit is equal to 1, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the derivative of the base-a logarithm,[17] i.e., of loga x for x > 0:

where the substitution u = h/x was made. The a-logarithm of e is 1, if a equals e. So symbolically,

The logarithm with this special base is called the natural logarithm and is denoted as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

There are thus two ways in which to select such special numbers a. One way is to set the derivative of the exponential function ax equal to ax, and solve for a. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for a are actually the same, the number e.

Alternative characterizations

The five shaded regions are of equal area, and define units of hyperbolic angle along the hyperbola  x y = 1 {\displaystyle xy=1} .
The five shaded regions are of equal area, and define units of hyperbolic angle along the hyperbola .

Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced:

  1. The number e is the unique positive real number such that .
  2. The number e is the unique positive real number such that .

The following four characterizations can be proven equivalent:

  1. The number e is the limit
    Similarly:
  2. The number e is the sum of the infinite series
    where n! is the factorial of n.
  3. The number e is the unique positive real number such that
  4. If f(t) is an exponential function, then the quantity is a constant, sometimes called the time constant (it is the reciprocal of the exponential growth constant or decay constant). The time constant is the time it takes for the exponential function to increase by a factor of e: .

Properties

Calculus

As in the motivation, the exponential function ex is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative

and therefore its own antiderivative as well:

Inequalities

Exponential functions and intersect the graph of , respectively, at and . The number is the unique base such that intersects only at . We may infer that lies between 2 and 4.

The number e is the unique real number such that

for all positive x.[18]

Also, we have the inequality

for all real x, with equality if and only if x = 0. Furthermore, e is the unique base of the exponential for which the inequality axx + 1 holds for all x.[19] This is a limiting case of Bernoulli's inequality.

Exponential-like functions

The global maximum of  x x {\displaystyle {\sqrt[{x}]{x}}}  occurs at x = e.
The global maximum of occurs at x = e.

Steiner's problem asks to find the global maximum for the function

This maximum occurs precisely at x = e. For proof, the inequality , from above, evaluated at and simplifying gives . So for all positive x.[20]

Similarly, x = 1/e is where the global minimum occurs for the function

defined for positive x. More generally, for the function

the global maximum for positive x occurs at x = 1/e for any n < 0; and the global minimum occurs at x = e−1/n for any n > 0.

The infinite tetration

or

converges if and only if eexe1/e (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.[21]

Number theory

The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite.[22] (See also Fourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.

It is conjectured that e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

Complex numbers

The exponential function ex may be written as a Taylor series

Because this series keeps many important properties for ex even when x is complex, it is commonly used to extend the definition of ex to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

which holds for all x. The special case with x = π is Euler's identity:

from which it follows that, in the principal branch of the logarithm,

Furthermore, using the laws for exponentiation,

which is de Moivre's formula.

The expression

is sometimes referred to as cis(x).

Differential equations

The general function

is the solution to the differential equation:

Representations

The number e can be represented as a real number in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. The chief among these representations, particularly in introductory calculus courses is the limit

given above, as well as the series

given by evaluating the above power series for ex at x = 1.

Less common is the continued fraction (sequence A003417 in the OEIS).

[23]

which written out looks like

This continued fraction for e converges three times as quickly:[citation needed]

Many other series, sequence, continued fraction, and infinite product representations of e have been developed.

Stochastic representations

In addition to exact analytical expressions for representation of e, there are stochastic techniques for estimating e. One such approach begins with an infinite sequence of independent random variables X1, X2..., drawn from the uniform distribution on [0, 1]. Let V be the least number n such that the sum of the first n observations exceeds 1:

Then the expected value of V is e: E(V) = e.[24][25]

Known digits

The number of known digits of e has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.[26][27]

Number of known decimal digits of e
Date Decimal digits Computation performed by
1690 1 Jacob Bernoulli[7]
1714 13 Roger Cotes[28]
1748 23 Leonhard Euler[29]
1853 137 William Shanks[30]
1871 205 William Shanks[31]
1884 346 J. Marcus Boorman[32]
1949 2,010 John von Neumann (on the ENIAC)
1961 100,265 Daniel Shanks and John Wrench[33]
1978 116,000 Steve Wozniak on the Apple II[34]

Since that time, the proliferation of modern high-speed desktop computers has made it possible for amateurs, with the right hardware, to compute trillions of digits of e.[35]

In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number e.

For instance, in the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is e billion dollars rounded to the nearest dollar. Google was also responsible for a billboard[36] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of e}.com". Solving this problem and visiting the advertised (now defunct) web site led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a résumé.[37] The first 10-digit prime in e is 7427466391, which starts at the 99th digit.[38]

In another instance, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth.[39]

Notes

  1. ^ Oxford English Dictionary, 2nd ed.: natural logarithm
  2. ^ Encyclopedic Dictionary of Mathematics 142.D
  3. ^ Jerrold E. Marsden, Alan Weinstein (1985). Calculus. Springer. ISBN 978-0387909745.
  4. ^ Sondow, Jonathan. "e". Wolfram Mathworld. Wolfram Research. Retrieved 10 May 2011.
  5. ^ a b c O'Connor, J J; Robertson, E F. "The number e". MacTutor History of Mathematics.
  6. ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston. ISBN 978-0030295584.
  7. ^ a b Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for e. See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–23. On page 222, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si a=b, debebitur plu quam 2½a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) If a=b, the geometric series reduces to the series for a × e, so 2.5 < e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)
  8. ^ Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd ed.). Wiley. p. 419.
  9. ^ Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60, see especially p. 58. From p. 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )
  10. ^ Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. p. 136. ISBN 978-0387971957
  11. ^ Euler, Meditatio in experimenta explosione tormentorum nuper instituta.
  12. ^ Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. From page 68: Erit enim seu ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be or , where e denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)
  13. ^ Grinstead, C.M. and Snell, J.L.Introduction to probability theory (published online under the GFDL), p. 85.
  14. ^ Knuth (1997) The Art of Computer Programming Volume I, Addison-Wesley, p. 183 ISBN 0201038013.
  15. ^ Steven Finch (2003). Mathematical constants. Cambridge University Press. p. 14.
  16. ^ Kline, M. (1998) Calculus: An intuitive and physical approach, section 12.3 "The Derived Functions of Logarithmic Functions.", pp. 337 ff, Courier Dover Publications, 1998, ISBN 0486404536
  17. ^ This is the approach taken by Kline (1998).
  18. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. pp. 44–48.
  19. ^ A standard calculus exercise using the mean value theorem; see for example Apostol (1967) Calculus, §6.17.41.
  20. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 359.
  21. ^ Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–69, 1921. (facsimile)
  22. ^ Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Archived from the original (PDF) on 2014-02-23. Retrieved 2010-06-18.
  23. ^ Hofstadter, D.R., "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books (1995) ISBN 0713991550
  24. ^ Russell, K.G. (1991) Estimating the Value of e by Simulation The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66–68.
  25. ^ Dinov, ID (2007) Estimating e using SOCR simulation, SOCR Hands-on Activities (retrieved December 26, 2007).
  26. ^ Sebah, P. and Gourdon, X.; The constant e and its computation
  27. ^ Gourdon, X.; Reported large computations with PiFast
  28. ^ Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5–45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )
  29. ^ Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.
  30. ^ William Shanks, Contributions to Mathematics, … (London, England: G. Bell, 1853), page 89.
  31. ^ William Shanks (1871) "On the numerical values of e, loge 2, loge 3, loge 5, and loge 10, also on the numerical value of M the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27–29.
  32. ^ J. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204–05.
  33. ^ Daniel Shanks and John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99 (78). doi:10.2307/2003813. JSTOR 2003813. We have computed e on a 7090 to 100,265D by the obvious program
  34. ^ Wozniak, Steve (June 1981). "The Impossible Dream: Computing e to 116,000 Places with a Personal Computer". BYTE. p. 392. Retrieved 18 October 2013.
  35. ^ Alexander Yee. "e".
  36. ^ First 10-digit prime found in consecutive digits of e}. Brain Tags. Retrieved on 2012-02-24.
  37. ^ Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle". NPR. Retrieved 2007-06-09.
  38. ^ Kazmierczak, Marcus (2004-07-29). "Google Billboard". mkaz.com. Retrieved 2007-06-09.
  39. ^ Knuth, Donald (1990-10-03). "The Future of TeX and Metafont" (PDF). TeX Mag. 5 (1): 145. Retrieved 2017-02-17.

Further reading

External links

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