Mathematical constants | Real transcendental numbers | E (mathematical constant)
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828, which can be characterized in many ways: * It is the base of the natural logarithms * It is the limit of as n approaches infinity, an expression that arises in the study of compound interest * It can also be calculated as the sum of the infinite series It is also the unique positive number a, such that the graph of the function has a slope of 1 at . The (natural) exponential function is the unique function f that equals its own derivative and satisfies the equation ; hence, one can also define e as . The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the area under the curve between and , in which case e is the value of k for which this area equals one (see image). There are various . The number e is sometimes called Euler's number (not to be confused with Euler's constant )—after the Swiss mathematician Leonhard Euler—or Napier's constant—after John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity and play important and recurring roles across mathematics. Like the constant π, e is 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). To 50 decimal places, the value of e is: 2.71828182845904523536028747135266249775724709369995.... (Wikipedia).
What is the mathematical constant e???
In this video, I talk about two ways of "deriving" e and the important of the number! "Why do we study e?" is often over-shadowed by procedural grind, so here we explicitly talk about where it comes from and how it is used. It is important to note that I explained e *without* direct use of
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Infinite Sum for e -- without Calculus (just a few limits)
The number e can be expressed as an infinite sum of factorial recipriocals. You usually see this for the first time in Calculus I when studying Taylor Series. In this video, we derive that sum using only a few limits, starting with the limit definition of the logarithm.
From playlist e
Calculating e^A for a matrix A, explaining what this has to do with diagonalization, and solving systems of differential equations Check out my Eigenvalues playlist: https://www.youtube.com/watch?v=H-NxPABQlxI&list=PLJb1qAQIrmmC72x-amTHgG-H_5S19jOSf Subscribe to my channel: https://www.y
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Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is the number “e”. e=2.718218284590..., “e” is called the natural number” because in nature things tend to increase and decrease according to that number. Some of the examples of that inc
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Free trial at The Great Courses Plus: http://ow.ly/tKWt306Gg7a Dr James Grime discusses "e" - the famed Euler's Number. More links & stuff in full description below ↓↓↓ A bit extra from this video: https://youtu.be/uawO3-tjP1c More James Grime videos from Numberphile: http://bit.ly/grimev
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