Ren Descartes: His mathematical legacy

Ren Descartes:His mathematical legacyBy Nicolas Synnott

Ren Descartes(1596-1650)French philosopher, mathematician, scientist and writerWesterner/European (lived most of his adulthood in the Dutch Republic)17th century (Early Modern period of Europe, Dutch Golden Age, Baroque cultural movement, French Grand Sicle, Scientific revolution, The General Crisis)Revolutionized philosophy and mathematics (Father of Modern Philosophy and first of the Modern School of Mathematics)Scientific method: foundation of all his works

Summarized BiographyBorn in conservative France - 1596Finished his studies in lawEngaged as a soldier in diverse armyMet Isaac Beeckman (Dutch philosopher and scientist; sparked his interest in mathematics and the New Physics)After several dreams, he quitted army career for the study of philosophy and mathematicsDied in liberal Netherlands - 1650

The MathematicianBefore Descartes, the different mathematical fields were considered as separated entities. Descartes had a pioneering approach: he merged geometry and algebra.

Le Discours de la MthodeExplain everything in terms of matter and motionLa Gomtrie (one of three appendices)Presented analytic geometryThree main contributionsNotationCartesian coordinate systemAnalytic or Cartesian geometry (most important legacy)

Father of Standard NotationLetters at the beginning of the alphabet: data or known quantities (a, b, c)Letters at the end of the alphabet: variables or unknown quantities (x, y, z)The number of distinct roots of an equation is equal to the degree of the equation (ax+bx^2+cx^3=y)Use of superscripts to show powers or exponents (4x4x4x4x4=45)

Cartesian Coordinate SystemDescartes defined analytic geometry as a way of visualizing algebraic formulas. To do so, he first had to develop the coordinates system: (a device to locate points on a plane).

CompositionTwo perpendicular lines: the vertical (y) and horizontal (x) axisThe intersection of the two axis: the origin (0,0)Axis are both positive and negative4 quadrants: (x,y), (-x,y), (-x,-y), (x,-y)

Location of a point on the Cartesian coordinates:Indicated by its two coordinates (x,y)The coordinated are determined by how far perpendicularly is the point from each axis in term of units