e (mathematical constant)

[math]\displaystyle{ \boldsymbol e }[/math] is a number. It is the base of the natural logarithm and is about 2.71828. It is an important mathematical constant. The number [math]\displaystyle{ e }[/math] is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is equally important in mathematics as [math]\displaystyle{ \pi }[/math] and [math]\displaystyle{ i }[/math]. [math]\displaystyle{ e }[/math] is an irrational number, and Euler himself gave the first 23 digits of [math]\displaystyle{ e }[/math].

The number [math]\displaystyle{ e }[/math] has great importance in mathematics, as do 0, 1, [math]\displaystyle{ \pi }[/math], and [math]\displaystyle{ i }[/math]. All five of these numbers are important and occur again and again in mathematics. The five constants appear in one formulation of Euler's identity. Like the constant [math]\displaystyle{ \pi }[/math], [math]\displaystyle{ e }[/math] is also irrational (it cannot be represented as a ratio of Integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients).

The number [math]\displaystyle{ e }[/math] is very important for exponential functions. For example, the exponential function applied to the number one, has a value of [math]\displaystyle{ e }[/math].

[math]\displaystyle{ e }[/math] was discovered in 1683 by the Swiss mathematician Jacob Bernoulli, while he was studying compound interest. The numerical value of [math]\displaystyle{ e }[/math] truncated to 20 places is:

[math]\displaystyle{ e\approx2.71828182845904523536 }[/math]

Alternate definitions of e

The area shown in blue (under the graph of the equation [math]\displaystyle{ y=1/x }[/math]) stretching from 1 to [math]\displaystyle{ e }[/math] is exactly 1.

There are many different ways to define [math]\displaystyle{ e }[/math]. Jacob Bernoulli, who discovered [math]\displaystyle{ e }[/math], was trying to solve the problem:

[math]\displaystyle{ \lim_{n\to\infty}(1+\frac1n)^n. }[/math]

In other words, there is a number that the expression [math]\displaystyle{ (1+\frac1n)^n }[/math] approaches as [math]\displaystyle{ n }[/math] becomes larger. This number is [math]\displaystyle{ e }[/math].

Another definition is to find the solution of the following formula:

[math]\displaystyle{ 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{\ddots\,}}}}} }[/math]

E (mathematical Constant) Media

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  The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies. The limiting curve on top is the graph y=1000e^{0.2t}, where y is in dollars, t in years, and 0.2 = 20%.

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  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.

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  The graphs of the functions x ↦ a^x 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 e^x there.

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  The value of the natural log function for argument e, i.e. ln e, equals 1.

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  The five colored regions are of equal area, and define units of hyperbolic angle along the {{{1}}}

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  Exponential functions y = 2x and y = 4x intersect the graph of y = x + 1, respectively, at x = 1 and x = -1/2. The number e is the unique base such that y = e'x intersects only at x = 0. We may infer that e lies between 2 and 4.

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  The global maximum of √x occurs at x = e.

Related pages

• Imaginary unit
• Pi
• Tau (mathematical constant)