Name: Rene Descartes Born: 31 March 1596, La Haye en Touraine, France Died: 11 February 1650, Stockholm, Sweden Famous for: Cartesian Coordinates, Algebraic Notation, Analytic Geometry, Rationalist Philosophy

## Biography of Rene Descartes

Rene Descartes was a French philosopher, mathematician and scientist. He is most famous for inventing the Cartesian coordinate system (named after him). He also introduced important conventions in notation: such as using x, y, z to denote unknowns and a, b, c for knowns, and index notation for exponents, x2.

## Rene Descartes' Life

Rene Descartes was born on 31 March 1596 in France (in a town now named after him). He studied law at University but joined the Dutch States Army where he would have learned a lot of mathematics when studying military engineering. After leaving the army, he spent the next twenty years at Universities in the Netherlands - a period of his life where he published his major philosophical and mathematical works. Queen Christina of Sweden invited him to her Court to organise a new scientific academy and to tutor her. It was in Sweden that he died on 11 February 1650.

## Cartesian Coordinates

Cartesian coordinates are used to describe the position of a point on a graph. The image below shows a graph. A point is plotted on the graph (a blue cross) with its Cartesian coordinates written beside it: (2, 4).

## Lesson Slides

The slider below discusses the Cartesian coordinate system.

## Rene Descartes' Notation

Rene Descartes introduced the use of letters to stand for numbers. Consider the general form equation of a straight line:
• x and y are variables. They can taken any number.

• a, b and c are constants. On any particular line, they can each only be one number.
So an example of an equation for a line is: Descartes also introduced index notation for exponents. So a squared number has a raised 2 written after it:

## "I Think, Therefore I Am

Rene Descartes was a great philosopher, as well as making advances in mathematics. He was very sceptical of what we can learn about the world through our experience of it. We can see and hear imaginary things in our dreams, so how can we trust our senses in waking life? Does the world we live in really exist? Do we really exist? The fact that there is a human mind asking these questions proves beyond doubt that the person who possesses that mind exists. Thinking proves that the thinker exists. In Descartes words: "I think, therefore I am" In Latin: "Cogito ergo sum" In French: "Je pense, donc je suis" Rene Descartes was a rationalist. He thought everything we know about the world has to be reasoned out by thought alone. This is opposed to empiricism, which learns about the world by experiencing it.

## Algebra

Descartes made algebra a fundamental part of mathematics. Before, all algebra had been derived from geometry. This meant that mathematicians were suspicious of any algebraic equation that could not be linked to simple shapes. Descartes made algebra a more foundational subject, that could be used to reason about things that can't easily be pictured. It freed the subject and allowed it to be developed by future mathematicians into calculus and other forms of analysis.

## Analytic Geometry

Before Descartes, all geometry had been derived from Euclid's work. Euclid's geometry defined lines, angles and circles and set to use them to find rules of geometry. But his shapes only existed relative to other shapes, and didn't exist anywhere particularly in space. With Descartes work, shapes now existed at particular points in space and could be described using numbers and equations, not just words expressing other shapes.

## Influences

Descartes' work was a big influence on Isaac Newton. His work paved the way for Newton and Leibniz to develop differential calculus, a branch of mathematics that is very useful at describing the physical world.