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The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
The number e (e = 2.718...), also known as Euler's number, which occurs widely in mathematical analysis The number i , the imaginary unit such that i 2 = − 1 {\displaystyle i^{2}=-1} The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth. As an example, suppose that a slot machine with a one in n probability of winning is played n times, then for large n (e.g., one million), the probability that nothing will be won will tend to 1/e as n tends to infinity.
The idea is to then analyze the scaled-up difference (here denoted x) between the series representation of e and its strictly smaller b-th partial sum, which approximates the limiting value e. By choosing the scale factor to be the factorial of b , the fraction a / b and the b -th partial sum are turned into integers , hence x must be a ...
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
The area of the blue region converges to Euler's constant. Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
It has been shown that both e + π and π/e do not satisfy any polynomial equation of degree and integer coefficients of average size 10 9. [47] [48] At least one of the numbers e e and e e 2 is transcendental. [49] Schanuel's conjecture would imply that all of the above numbers are transcendental and algebraically independent. [50]
The mathematical notation for using the common logarithm is log(x), [4] log 10 (x), [5] or sometimes Log(x) with a capital L; [a] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log".