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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 .
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
The number e (e = 2.71828...), also known as Euler's number, which occurs widely in mathematical analysis; The number i, the imaginary unit such that = The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
A unique representation of e can be found within the structure of Pascal's Triangle, as discovered by Harlan Brothers. Pascal's Triangle is composed of binomial coefficients, which are traditionally summed to derive polynomial expansions. However, Brothers identified a product-based relationship between these coefficients that links to e.
Define e x as the value of the infinite series = =! = + +! +! +! + (Here n! denotes the factorial of n. One proof that e is irrational uses a special case of this formula.) Inverse of logarithm integral.
The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
The formula defines the energy E of a particle in its rest frame as the product of mass (m) with the speed of light squared (c 2). Because the speed of light is a large number in everyday units (approximately 300 000 km/s or 186 000 mi/s), the formula implies that a small amount of mass corresponds to an enormous amount of energy.
For instance, e x can be defined as (+). Or e x can be defined as f x (1), where f x : R → B is the solution to the differential equation df x / dt (t) = x f x (t), with initial condition f x (0) = 1; it follows that f x (t) = e tx for every t in R.
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