Euler's Constant
(Redirected from Euler's Number)
Jump to navigation
Jump to search
A Euler's constant is a transcendental irrational number based on the limit of (1 + 1/n)n as n approaches infinity
- AKA: [math]\displaystyle{ e }[/math].
- Context:
- It is used by an Natural Exponential Function, a Natural Logarithm Function, ....
- Example(s):
- 2.71828 1828459045 23536….
- …
- Counter-Example(s):
- See: Mathematical Constant, Natural Logarithm, Limit of a Sequence, Compound Interest, Integral.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/e_(mathematical_constant) Retrieved:2015-11-9.
- The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, [1] and is the limit of (1 + 1/n)n as 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] : [math]\displaystyle{ e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots }[/math] The constant can be defined in many ways. For example, can be defined as the unique positive number such that the graph of the function y = ax has unit slope at x = 0. The function f(x) = ex is called the exponential function, and its inverse is the natural logarithm, or logarithm to base . The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case is the number whose natural logarithm is 1. There are alternative characterizations. Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number is also known as Napier's constant, but Euler's choice of the symbol is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The number is of eminent importance in mathematics, alongside 0, 1, pi and imaginary unit. 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 , is irrational: it is not a ratio of integers. Also like , is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of truncated to 50 decimal places is
:.
- The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, [1] and is the limit of (1 + 1/n)n as 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] : [math]\displaystyle{ e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots }[/math] The constant can be defined in many ways. For example, can be defined as the unique positive number such that the graph of the function y = ax has unit slope at x = 0. The function f(x) = ex is called the exponential function, and its inverse is the natural logarithm, or logarithm to base . The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case is the number whose natural logarithm is 1. There are alternative characterizations. Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number is also known as Napier's constant, but Euler's choice of the symbol is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The number is of eminent importance in mathematics, alongside 0, 1, pi and imaginary unit. 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 , is irrational: it is not a ratio of integers. Also like , is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of truncated to 50 decimal places is