E (mathematical constant)
The number e begins at 2.71828, and it never stops. Like the digits of pi, the digits of e run on forever without settling into a pattern. On the 24th of December 2023, Jordan Ranous pushed a calculation to 35 trillion digits, and still the number refused to repeat or terminate.
A Swiss mathematician named Jacob Bernoulli stumbled onto this constant while puzzling over a humble question about money. He was asking what happens to a savings account when interest is added more and more often. Out of that ordinary problem came one of the most important numbers in mathematics.
Why does a number born from compound interest also describe a losing gambler, a butler who muddles hats, and the spacing of stars in a normal curve? Why did Leonhard Euler attach the letter e to it, and why does e sit beside 0, 1, pi, and i in an equation that mathematicians call beautiful? Those are the threads this documentary follows.
An account starts with $1.00 and pays 100 percent interest a year. If the interest is added once, at year-end, the dollar becomes $2.00. That simple setup is where Jacob Bernoulli began in 1683, and it opened onto something strange.
Credit the interest twice a year instead, and each half earns 50 percent, so the dollar is multiplied by 1.5 twice. Compound it monthly, weekly, daily, and the year-end balance keeps creeping upward. Compounding weekly yields about $2.692596, and compounding daily yields about $2.714567, roughly two cents more.
Bernoulli noticed that this climbing sequence did not run away to infinity. It approached a ceiling, a limit he thought of as the force of interest. With continuous compounding, the account settles at $2.718281828, the value that came to be called e.
The same logic stretches past a single dollar. An account starting at $1 with an annual rate R yields, after t years, a sum governed by e under continuous compounding. Here R is the decimal form of the percentage rate, so 5 percent enters the formula as its decimal equivalent.
A gambler sits at a slot machine that pays out with a probability of one in n, and plays it exactly n times. As n grows, the chance of losing every single bet settles toward roughly 36.79 percent. Even at small n the pattern shows: in one case the losing probability is already about 35.85 percent.
Each pull is a Bernoulli trial, a single event with a fixed chance of success. Playing n times follows the binomial distribution, which ties back to the binomial theorem and to Pascal's triangle. The probability of never winning, taken to the limit as the number of trials grows without bound, lands precisely on a value built from e.
A party offers another disguise for the same number, a puzzle Jacob Bernoulli explored alongside Pierre Remond de Montmort. Called the hat check problem, it imagines n guests handing their hats to a butler who drops them into n labelled boxes at random. De Montmort asked for the probability that no hat lands in its correct box.
That probability approaches a value tied to e as the number of guests increases. The count of arrangements in which not one hat sits in the right box equals a quantity rounded to the nearest integer, for every positive n. A number from a bank account turns out to govern chaos at the coat check.
Exponential growth speeds up as it goes, increasing a quantity faster and faster over time. It happens when the instantaneous rate of change of a quantity is proportional to the quantity itself. Time sits in the exponent, which sets this apart from quadratic growth and other slower kinds.
Flip the sign of the constant of proportionality and growth becomes decay, the quantity shrinking over time instead. The law of exponential change can be written in several mathematically equivalent forms using different bases, and e is a common, convenient choice. In that framing one constant is the growth constant k, and another is the time it takes the quantity to grow by a factor of e.
The standard normal distribution, with zero mean and unit standard deviation, also leans on e through its probability density function. The demand for unit standard deviation shapes the exponent, and the demand for unit total area under the curve sets the leading factor. This bell-shaped curve is symmetric around its center, reaches its peak there, and bends at two inflection points.
The reason e earns its central place in calculus is brutally practical. It lets mathematicians differentiate and integrate exponential functions and logarithms without dragging along an awkward leftover limit. A general exponential function has a derivative defined by a limit, and that limit equals the logarithm of the base to base e.
Set the base equal to e, and that stray logarithm collapses to a value that simplifies everything. The exponential function with base e becomes its own derivative, a property no other base shares so cleanly. Choosing e as the base, rather than some other number, makes calculations with derivatives far simpler.
The logarithm tells the same story from the other side. The base-a logarithm of e equals 1 exactly when a is e itself, which is why the logarithm with this base is called the natural logarithm. It behaves well under differentiation because no undetermined limit lingers in the calculation.
There are two routes to this special base. One sets the derivative of the exponential function to a convenient value and solves for the base. The other sets the derivative of the logarithm to a convenient value and solves. Both roads arrive at the same destination, and that destination is e.
Euler proved that e is irrational by showing that its simple continued fraction expansion never terminates. An irrational number cannot be written as a ratio of integers, no matter how large you allow them to grow. Fourier later gave his own proof that e is irrational.
Being transcendental goes further than being irrational. By the Lindemann-Weierstrass theorem, e is not a solution to any non-zero polynomial equation with rational coefficients. Charles Hermite delivered the proof in 1873, and e holds a particular distinction. It was the first number proved transcendental that had not been specially constructed for the purpose.
Some of the deepest questions about e remain unanswered. Whether e and pi are algebraically independent is unresolved, and would follow from the unproven Schanuel's conjecture. Mathematicians conjecture that e is normal, meaning its digits in any base are uniformly distributed, and that, unlike pi, e is not a period.
The exponential function can be written as a Taylor series that converges for every complex value of its input. That convergence lets mathematicians extend the function from real numbers to complex ones. Combined with the series for sine and cosine, it yields Euler's formula, which holds for every complex input.
One special case of Euler's formula stands above the rest. It binds together the most fundamental numbers in mathematics, and is held up as an exemplar of mathematical beauty for the profound connection it reveals. This is Euler's identity, the formulation in which 0, 1, pi, i, and e all appear at once.
The reach of this identity runs deep into other proofs. It is used directly in a proof that pi is transcendental, which in turn settles the impossibility of squaring the circle. From the same machinery flows de Moivre's formula, true for any integer power.
The first traces of this constant appeared in 1618, in an appendix to John Napier's work on logarithms, a table thought to have been written by William Oughtred. That table did not contain e itself, only a list of logarithms. In 1661 Christiaan Huygens calculated what we now recognize as the base-10 logarithm of e, yet never saw e as a quantity worth naming.
The symbols came before the standard one stuck. Gottfried Leibniz used the letter b in letters to Huygens in 1690 and 1691. Leonhard Euler began using e in 1727 or 1728, in an unpublished paper on the explosive force in cannons, then in a letter to Christian Goldbach on the 25th of November 1731. Its first printed appearance was in Euler's Mechanica in 1736, and why he chose e remains unknown.
The race to know more of its digits stretches across centuries. Jacob Bernoulli knew 1 digit in 1690; Roger Cotes reached 13 in 1714; William Shanks pushed to 205 by 1871. John von Neumann computed 2,010 digits on the ENIAC in 1949, and Steve Wozniak found 116,000 on the Apple II in 1981.
The number even slipped into the culture of computing. Donald Knuth numbered versions of his Metafont program to approach e, releasing 2, 2.7, 2.71, 2.718, and onward. In 2004 Google filed to raise 2,718,281,828 dollars, which is e billion dollars rounded to the nearest dollar. A Google billboard in Silicon Valley read simply, the first 10-digit prime found in consecutive digits of e, which is 7427466391, beginning at the 99th digit, and solving it led toward a hidden invitation to submit a resume.
Common questions
What is the mathematical constant e and what is its value?
The number e is a mathematical constant approximately equal to 2.71828, and it is the base of the natural logarithm and the exponential function. It is irrational, meaning it cannot be written as a ratio of integers, and transcendental, meaning it is not a root of any non-zero polynomial with rational coefficients.
Who discovered the constant e?
The Swiss mathematician Jacob Bernoulli discovered the constant e in 1683 while studying the problem of continuous compounding of interest. He introduced it as the limit a climbing sequence of compound-interest values approached.
Why is e called Euler's number?
The number e is called Euler's number after the Swiss mathematician Leonhard Euler, who began using the letter e for it in 1727 or 1728. It can also be called Napier's constant after John Napier. The name Euler's number can invite confusion with Euler numbers or with Euler's constant, which are different.
How does e relate to compound interest?
The number e arises when interest is compounded continuously. An account starting at $1.00 with 100 percent annual interest settles at $2.718281828 under continuous compounding, which is the value of e. Compounding weekly yields about $2.692596 and compounding daily about $2.714567.
Why is e important in calculus?
The number e is important in calculus because the exponential function with base e is its own derivative, which makes differential and integral calculus with exponentials and logarithms much simpler. The logarithm with base e is the natural logarithm, and it behaves well under differentiation because no undetermined limit lingers.
When was e proved to be transcendental?
Charles Hermite proved that e is transcendental in 1873. It was the first number proved transcendental that had not been specifically constructed for that purpose, and the result follows from the Lindemann-Weierstrass theorem.
How many digits of e have been calculated?
On the 24th of December 2023, Jordan Ranous set a record by computing e to 35 trillion digits. Earlier milestones include John von Neumann reaching 2,010 digits on the ENIAC in 1949 and Steve Wozniak reaching 116,000 digits on the Apple II in 1981.
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