E (mathematics): Difference between revisions
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'''e''' is a [[constant]] [[real number]] equal to 2.71828 18284 59045 23536... | '''e''' is a [[constant]] [[real number]] equal to 2.71828 18284 59045 23536.... [[Irrational number|Irrational]] and [[transcendental number|transcendental]], ''e'' is the base of the [[natural logarithm]]s. Its inverse, the [[exponential function]] | ||
:<math>f(x) = e^x \,</math> | |||
is equal to its [[derivative]], i.e. <math> f^'(x) = f(x) \,</math>. More generally, for any differentiable function <math>u </math>, we have | |||
:<math> \frac{d}{dx}(K e^u) = K e^u \frac{du}{dx}</math> | |||
for K constant and | |||
:<math> \int K e^u du = K e^u + C</math> | |||
for K and C constants. For this reason, the exponential function plays a central role in [[analysis]]. | |||
''e'' is sometimes called "Euler's number" in honor of the Swiss mathematician [[Leonhard Euler]] who studied it and has shown its mathematical importance. Equally, it is sometimes called "Napier's constant" in honor of the Scottish mathematician [[John Napier]] who introduced [[logarithm]]s. | ''e'' is sometimes called "Euler's number" in honor of the Swiss mathematician [[Leonhard Euler]] who studied it and has shown its mathematical importance. Equally, it is sometimes called "Napier's constant" in honor of the Scottish mathematician [[John Napier]] who introduced [[logarithm]]s. | ||
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In 1737, [[Leonhard Euler]] proved that ''e'' is an [[irrational number]]<ref name="maor_37">Eli Maor, ''e: The Story of a Number'', Princeton University Press, 1994, p.37. ISBN 0-691-05854-7.</ref>, i.e. it cannot be expressed as a [[fraction]], only as an infinite [[continued fraction]]. In 1873, [[Charles Hermite]] proved it was a [[transcendental number]]<ref name="maor_37"/>, i.e. it is not solution of any [[polynomial]] having a finite number of [[rational number|rational]] coefficients. | In 1737, [[Leonhard Euler]] proved that ''e'' is an [[irrational number]]<ref name="maor_37">Eli Maor, ''e: The Story of a Number'', Princeton University Press, 1994, p.37. ISBN 0-691-05854-7.</ref>, i.e. it cannot be expressed as a [[fraction]], only as an infinite [[continued fraction]]. In 1873, [[Charles Hermite]] proved it was a [[transcendental number]]<ref name="maor_37"/>, i.e. it is not solution of any [[polynomial]] having a finite number of [[rational number|rational]] coefficients. | ||
''e'' is the base of the [[natural logarithm]]s. | ''e'' is the base of the [[natural logarithm]]s. The [[exponential function]] | ||
:<math>f(x) = K e^x \,</math> | |||
for K constant, is equal to its [[derivative]], i.e. <math> f^'(x) = f(x) \,</math>. For any differentiable function <math>u </math>, we have | |||
:<math> \frac{d}{dx}(K e^u) = K e^u \frac{du}{dx}</math> | |||
for K constant and | |||
:<math> \int K e^u du = K e^u + C</math> | |||
for K and C constants. The solutions of many [[differential equation]]s are based on those properties. | |||
== History == | == History == | ||
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There are many ways to define ''e''. The most common are probably | There are many ways to define ''e''. The most common are probably | ||
<math> e = \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n</math> | ::<math> e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n</math> | ||
and | and | ||
<math> e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots</math><ref>This equation is a special case of the [[exponential function]] : | ::<math> e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots</math><ref>This equation is a special case of the [[exponential function]] : | ||
<math> e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots</math> | ::<math> e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots</math> | ||
with ''x'' set to 1.</ref> | with ''x'' set to 1.</ref> | ||
== References == | == References == | ||
<References/> | <References/>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 9 August 2024
e is a constant real number equal to 2.71828 18284 59045 23536.... Irrational and transcendental, e is the base of the natural logarithms. Its inverse, the exponential function
is equal to its derivative, i.e. . More generally, for any differentiable function , we have
for K constant and
for K and C constants. For this reason, the exponential function plays a central role in analysis.
e is sometimes called "Euler's number" in honor of the Swiss mathematician Leonhard Euler who studied it and has shown its mathematical importance. Equally, it is sometimes called "Napier's constant" in honor of the Scottish mathematician John Napier who introduced logarithms.
Properties
In 1737, Leonhard Euler proved that e is an irrational number[1], i.e. it cannot be expressed as a fraction, only as an infinite continued fraction. In 1873, Charles Hermite proved it was a transcendental number[1], i.e. it is not solution of any polynomial having a finite number of rational coefficients.
e is the base of the natural logarithms. The exponential function
for K constant, is equal to its derivative, i.e. . For any differentiable function , we have
for K constant and
for K and C constants. The solutions of many differential equations are based on those properties.
History
There is no precise date for the discovery of this number[2]. In 1624, Henry Briggs, one of the first to publish a logarithm table, gives its logarithm, but does not formally identify e. In 1661, Christiaan Huygens remarks the match between the area under the hyperbola and logarithmic functions. In 1683, Jakob Bernoulli studies the limit of , but nobody links that limit to natural logarithms. Finally, in a letter sent to Huyghens, Gottfried Leibniz sets e as the base of natural logarithm (even if he names it b).[3]
Definitions
There are many ways to define e. The most common are probably
and
References
- ↑ 1.0 1.1 Eli Maor, e: The Story of a Number, Princeton University Press, 1994, p.37. ISBN 0-691-05854-7.
- ↑ John J. O'Connor et Edmund F. Robertson, The number e, MacTutor History of Mathematics archive. Consulted 2008-01-10.
- ↑ Eli Maor, e: The Story of a Number, Princeton University Press, 1994. ISBN 0-691-05854-7.
- ↑ This equation is a special case of the exponential function :
