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# Cartesian coordinates

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René Descartes (1596—1650) French philosopher, mathematician

Gottfried Wilhelm Leibniz (1646—1716) German rationalist philosopher, mathematician, and logician

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The usual system for identifying the location of a point in two, or more, dimensions. The position of a point P in a plane can be represented by a pair of numbers (x, y), relative to two axes Ox and Oy which are straight lines meeting at the origin O represented by (0, 0). In the usual case of rectangular (or perpendicular, or orthogonal) axes, the abscissa x is the perpendicular distance of P from Oy and the ordinate y is the perpendicular distance of P from Ox. The value of x is positive if P lies in the half-plane to the right of O, and is negative in the left-hand half-plane. Similarly, the value of y is positive in the upper half-plane and negative in the lower half-plane. The plane is therefore divided into fourquadrants corresponding to the four combinations of coordinate signs. The units of distance in the directions Ox and Oy may be different and are usually indicated by numbers on the axes.

The term ‘Cartesian’ is derived from Descartes who first introduced coordinates. The word ‘coordinate’ was introduced by Leibniz in about 1693. The phrase ‘Cartesian coordinates’ was used in 1844.

The idea of coordinates can be generalized to three-dimensional space. The process can be reversed by considering any ordered set of three numbers (x, y, z) as a point in three-dimensional space. This can be further generalized by considering the row vector (x1x2xn) to be the coordinates of a point in an n-dimensional space. This space is usually denoted by ℝn.

Cartesian coordinates. The axes meet at the origin O. The coordinates of a point are (x, y), where x and y are the corresponding signed distances along the two axes.

Subjects: Probability and Statistics.

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