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Encyclopedia > Analytic geometry

Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics. Table of Geometry, from the 1728 Cyclopaedia. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Fig. ... An equation is a mathematical statement, in symbols, that two things are the same. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...


RenĂ© Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native language (French), and its philosophical principles, provided the foundation for calculus in Europe. René Descartes (March 31, 1596 – February 11, 1650), also known as Cartesius, was a noted French philosopher, mathematician, and scientist. ... Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ... The Discourse on Method is a philosophical and mathematical treatise published by René Descartes in 1637. ... For other uses of Calculus, see Calculus (disambiguation) Calculus is an important branch of mathematics. ...


The fact that the results obtained by analytic geometry and by Euclidean geometry must be consistent tends to be assumed tacitly. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...


Important themes of analytical geometry

Many of these problems involve linear algebra In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ... Distance is a numerical description of how far apart things lie. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...


Example

Here is an example of a problem from the USAMTS that can be solved via analytic geometry: The United States of America Mathematical Talent Search (USAMTS) is a mathematics competition open to all United States high school students, sponsored by the National Security Agency. ...


Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in that order. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DE, respectively. Let X be the midpoint of segment FH, and Y be the midpoint of segment GI. The length of segment XY is an integer. Find all possible values for the length of side AE.


Solution: Let A, B, C, D, and E be located at A(0,0), B(a,0), C(b,e), D(c,f), and E(d,g).


Using the midpoint formula, the points F, G, H, I, X, and Y are located at In mathematics, a line segment is a part of a line that is bounded by two end points. ...

Fleft(frac{a}{2},0right), Gleft(frac{a+b}{2},frac{e}{2}right), Hleft(frac{b+c}{2},frac{e+f}{2}right), Ileft(frac{c+d}{2},frac{f+g}{2}right), Xleft(frac{a+b+c}{4},frac{e+f}{4}right), and Yleft(frac{a+b+c+d}{4},frac{e+f+g}{4}right).

Using the distance formula, Distance is a numerical description of how far apart things lie. ...

AE=sqrt{d^2+g^2}

and

XY=sqrt{frac{d^2}{16}+frac{g^2}{16}}=frac{sqrt{d^2+g^2}}{4}.

Since XY has to be an integer, The integers are commonly denoted by the above symbol. ...

AEequiv 0pmod{4}

(see modular arithmetic) so AE = 4. Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...


Other uses

Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones). It is closely linked to algebraic geometry, especially through the work of Jean-Pierre Serre in GAGA. It is strictly a larger area than algebraic geometry, but studied by similar methods. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. ...


  Results from FactBites:
 
Geometry - MSN Encarta (1664 words)
Analytic geometry was of great value in the development of mathematics because it unified the concepts of analysis (number relationships) and geometry (space relationships).
The techniques of analytic geometry, which made possible the representation of numbers and of algebraic expressions in geometric terms, have cast new light on calculus, the theory of functions, and other problems in higher mathematics.
Analytical methods may also be used to investigate regular geometrical figures in four or more dimensions and to compare them with similar figures in three or fewer dimensions.
  More results at FactBites »

 
 

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