| content | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 |
| The region between two loxodromes on a geometric sphere. |
The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as
well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP1.
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.
| ...Archive | Image credit: Karthik Narayanaswami | Read more... |
.png) |
.png) |
| Credit: Inductiveload | |
The exponential function is one of the most important functions in mathematics. It can be written in the form ex, where e is a mathematical constant, sometimes known as Euler's number.
The exponential function exist for any complex number, above to the left is the real part and to the right the imaginary part of various values in the complex plane.
| ...Archive | Read more... |
- ... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
- ...that the primality of a number can be determined using only a single division using Wilson's Theorem?
- ...that the line separating the numerator and denominator of a fraction is called a solidus if written as a diagonal line or a vinculum if written as a horizontal line.
- ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare.
- ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
- ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
- ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
- ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
- ...that a sphere can be cut up and reassembled into two spheres the same size as the original (Banach-Tarski paradox)?
| Showing 9 items out of 21 | More did you know |
The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.
Project pages
Subprojects
Related projects
| General | Foundations | Number theory | Discrete mathematics |
|---|---|---|---|
 |
 |
 |
 |
| Analysis | Algebra | Geometry and topology | Applied mathematics |
 |
 |
 |
 |
| ARTICLE INDEX: | A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0-9 |
| MATHEMATICIANS: | A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
 |
 |
 |
 |
 |
 |
 |
|
| Algebra | Analysis | Category theory |
Computer science |
Cryptography | Discrete mathematics |
Geometry | |
 |
 |
 |
 |
 |
 |
 |
 |
| Logic | Mathematics | Number theory |
Physics | Science | Set theory | Statistics | Topology |
Science: 
History of science 
Philosophy of science 
Scientific method 
Systems science 
Mathematics 
Biology 
Chemistry 
Physics 
Earth sciences 
Technology and applied sciences
