To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Generality of algebra

In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange,[1] particularly in manipulating infinite series. According to Koetsier,[2] the generality of algebra principle assumed, roughly, that the algebraic rules that hold for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions.

In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis.

• 1/3
Views:
146 672
263 054
5 229
• Abstract vector spaces | Essence of linear algebra, chapter 11
• Futurama and Keeler's Theorem: Original Edit
• Tensor Calculus Lecture 10a: The Covariant Surface Derivative in Its Full Generality

Example

An example[2] is Euler's derivation of the series

${\displaystyle {\frac {\pi -x}{2}}=\sin x+{\frac {1}{2}}\sin 2x+{\frac {1}{3}}\sin 3x+\cdots }$

(1)

for ${\displaystyle 0. He first evaluated the identity

${\displaystyle {\frac {1-r\cos x}{1-2r\cos x+r^{2}}}=1+r\cos x+r^{2}\cos 2x+r^{3}\cos 3x+\cdots }$

(2)

at ${\displaystyle r=1}$ to obtain

${\displaystyle 0={\frac {1}{2}}+\cos x+\cos 2x+\cos 3x+\cdots .}$

(3)

The infinite series on the right hand side of (3) diverges for all real ${\displaystyle x}$. But nevertheless integrating this term-by-term gives (1), an identity which is known to be true by modern methods.[example  needed]