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Algebraic Geometry

  This area of mathematics originated with René Descartes (1596 - 1650), after whom the Cartesian co-ordinate system is named. It has developed considerably since, but is still fundamentally concerned with co-ordinate systems, which have shown their importance in three main areas.

First, algebraic geometry demonstrated an intimate connection between two branches of mathematics which had previously seemed to be far apart, algebra and geometry. Equations from algebra define curves in the co-ordinate systems of the appropriate dimension. For example, the linear equation y=3x + 2 defines the curve in which every point has a co-ordinate (x,3x + 2). This turns out to be a straight line (which is why such equations are called linear); more complicated equations give rise to more complicated curves; the equation x2 + y2 = 1 produces the circle of radius 1 centred at the origin. This connection led to the formulation of the idea of a function, which (in its original form) was an algebraic connection between x and y in the co-ordinates of the points in a particular figure.

Second, the Cartesian co-ordinate system was one in which the theorems of Euclidean geometry could be shown to hold by algebraic methods (which the original Greek methods of proof would not have allowed); they therefore showed that Euclidean geometry is consistent, that is, allowed of no contradiction, assuming that arithmetic is in its turn consistent. This is not, of course the same as showing that Euclidean geometry is true; there are co-ordinate systems which have non-Euclidean geometries (which, because they are curved, today are thought to give a truer reflection of the universe, which is also curved). For example, the system of latitude and longitude on the surface of the Earth is a co-ordinate system (except that the poles have no well-defined co-ordinates), but the theorems of Euclidean geometry are not all true in this system; for example, there are triangles which have the sum of their angles greater than two right angles (putting one corner at the north pole and the other two on the equator makes each angle a right angle). It is also the case that using algebraic co-ordinates measuring the length of lines, it is easy to do things which Galois theory later proved could not be done with Euclidean geometry.

The chief importance of algebraic geometry is in the way it clearly demonstrates the fact that clarity of notation is all important to mathematical development. SMcL

See also symbolism.



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