René Descartes AB = 1. DE is parallel to AC. Check the result using a scale drawing for the following values FG = 1

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MEI Conference 2016 René Descartes Multiplication and division AB = 1 DE is parallel to AC BC and BD are given values (lengths) Show that BE is the product of BC and BD Check the result using a scale

1 MEI Conference 2016 René Descartes Multiplication and division AB = 1 DE is parallel to AC BC and BD are given values (lengths) Show that BE is the product of BC and BD Check the result using a scale drawing for the following values a) BC = 3 and BD = 4 b) BC = 2.5 and BD = 6 c) BE = 10 and BD = 5 Finding a square root FG = 1 GH is the given value A semi-circular is constructed with centre K such that FK = KH Show that IG is the square root of GH Using a ruler and compass construction, check the result using a scale drawing for the following values a) GH = 9 b) GH = 16 c) GH = 20

2 Descartes rule of signs Descartes s Rule of Signs is a method for finding the number and sign of real roots of a polynomial equation written in decreasing powers of the variable. The number of true (positive real) roots of a polynomial equation P(x) = 0, with real coefficients, is equal to the number of sign changes (from positive to negative or vice versa) between the coefficients of the terms of P(x), or is less than this number by a multiple of two. The number of false (negative real) roots of such a polynomial equation is equal to the number of sign changes between the coefficients of the terms of P(-x), or is less than this number by a multiple of two. Example Determine the possible number of real roots of x 4 2x 3 + 9x 2 8x 12 = 0. Solution Count the number of sign changes of P(x) = 0. P(x) = x 4 2x 3 + 9x 2 8x 12 From + to - is one sign change. There are three sign changes. The number of positive real roots of P(x) = 0 is three or one. Count the number of sign changes of P(-x) = 0. P(-x) = ( x) 4 2( x) 3 + 9( x) 2 8( x) 12 = x 4 + 2x 3 + 9x 2 + 8x 12. There is one sign change. The number of negative real roots of P(x) = 0 is one. Problems Determine the possible number of positive and negative real roots of each polynomial equation. a) x 4 + 4x 3 19x 2 106x 120 = 0 b) x 4 x 3 x 2 + x 1 = 0 c) x 4 3x 2 6x + 1 = 0 d) x 3 + x 2 x + 6 = 0 e) x 2 6x + 5 = 0 f) x = 0 g) x 2 + bx + c = 0 2

3 Transforming Polynomials Descartes used the technique of translating the polynomial equation in order to remove one of the terms. Consider the polynomial of degree 3 (or dimension 3 as Descartes referred to it) x 3 6x 2 + 5x + 12 = 0 From Descartes rule of signs we can see this has one negative and either 2 or 0 positive roots. In order to remove the 6x 2 term, let y = x 2, (i.e. 6/3). Therefore x = y + 2 ; x 2 = (y + 2) 2 = y 2 + 4y + 4; and x 3 = (y + 2) 3 = y 3 + 6y y + 8. Substituting into the original equation gives y 3 + 6y y + 8 6(y 2 + 4y + 4) +5(y + 2) +12 y 3 + 0y 2 7y + 6 = 0 or y 3 7y + 6 = 0 This is the original equation transformed by 2 units in the negative x direction. In general for a polynomial of degree n, where a is the coefficient of x n 1, this term can be removed by substituting x = y a. n Transform the following equations using the above method. h) x 4 + 4x 3 19x 2 106x 120 = 0 i) x 2 6x + 5 = 0 j) x x + 20 = 0 k) x 2 + bx + c = 0 For each of b) c) and d) the transformed equation can be used to find the roots of the original quadratic equation. 3

4 Folium of Descartes x 3 + y 3 = 3axy a) Show that the folium equation is given by the parametric equations x = 3at 3at2 and y = 3 b) Find the coordinates of the extreme point of the loop, A. c) Find the equation of the tangent to the curve at A. d) Find the maximum point e) Find the area of the loop f) Find the equation of the tangent for a general point (x,y). 1+t 1+t 3 4

6 René Descartes Kevin Lord MEI

7 Me and René Tours, France Summer 1990

8 Early Life Born 31 March 1596 in La Haye, France Mother died when he was 1 years old Started Jesuit College in La Fleche aged 11 Given permission to stay in bed until 11am due to ill health Obtained a law degree Joined the Military School in Breda Met Dutch mathematician Isaac Beeckman Left the Netherlands to join the Bavarian Army

9 Descartes and the fly

10 Early Life continued Obtained a law degree Joined the Military School in Breda Met Dutch mathematician Isaac Beeckman Left the Netherlands to join the Bavarian Army

11 Descartes Dreams November 1619 whilst in the Bavarian Army First - never to accept anything for true which I did not clearly know to be such. Second - to divide each of the difficulties under examination into as many parts as possible and as might be necessary for its adequate solution. Third - to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and as it were, step by step, to the knowledge of the more complex. Last - in every case to make enumerations so complete and reviews so general that I might be assured that nothing was omitted.

12 Travelled through Europe Returned to France Began correspondence with Mersenne Moved to Holland to work on Le Monde, ou Traité de la Lumière Withheld from publication after hearing news of Galileo s house arrest

13 Descartes place in history Oresme ( ) Cardano ( ) Harriot ( ) Galileo ( ) Mersenne ( ) Desargues ( ) Fermat ( ) Pascal ( ) Newton ( ) Leibniz ( ) English Monarchy Elizabeth I ( ) James I ( ) Charles I ( ) Europe 30 years war ( )

14 Coordinates Egyptian surveyors laid out towns with a grid system Latitude and Longitude Hipparchus (c.140 BC) located points on Earth s surface Romans arranged the streets on a rectangular coordinate system Oresme (c.1360) made use of a coordinate system arrangements of points This idea was later used by Kepler to show the course of a planet

15 Coordinates

16 Algebra and Geometry Arabic mathematicians (c. 9 th century) make links between geometric figures and algebra Fibonacci (1220) uses algebra in solving geometric problems relating to triangles

17 Analytic Geometry Descartes or Fermat Discours de la Methode Published in 1637

18 je pense, donc je suis Discourse de la Methode, 1637

19 La Geométrie In three sectons which included Arithmetic operations related to geometry Geometric problems posed by Pappus Conic sections and loci Nature of curves Roots of polynomials Algebraic methods for solving geometric problems

20 Arithmetic and Geometry Multiplication and division x AB = 1 DE is parallel to AC y 1 unit BC.BD = BE = xy

21 Arithmetic and Geometry Finding the square root FG = 1 GH is the unknown FH is diameter of a circle 1 unit x IG = GH = x

22 Notation Wrote algebraic equations using notation similar to that used today, using a, b, c for known values and x, y and z for unknowns For equals sign, Descartes used æ for aequalis Refers to imaginary roots of polynomials

23 Descartes Rule of Signs Distinguished between true roots (positive), that could be solutions to geometric problems, and false roots (negative). Observed that the order or dimension of a polynomial indicated the number of possible roots. The number of changes of sign indicate the maximum number of true roots of the polynomial

24 Transforming Polynomials

25 Solving polynomials Finding roots by eliminating terms x 2 + 4x 3 = 0 Let y = x + 2, therefore x = y 2 Equation becomes y 2 4y y 2 3 = 0 y 2 7 = 0

26 Solving polynomials x 2 + 4x 3 = 0 y 2 7 = 0

27 Folium of Descartes x 3 + y 3 = 3axy Find the coordinates of the extreme point of the loop Find the equation of the tangent to the curve Find the area of the loop Find the maximum point Parametric equations are x = 3at 3at2 1+t3 and y = 1+t 3

28 Example

29 Later work and life Francine, his daughter, born 1635, died 1640 Meditations on First Philosophy, published 1641 Principles of Philosophy, published 1644 Passions of the Soul, published in 1649 Winter 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm Descartes died of pneumonia, 11 February 1650

30 Legacy Van Schooten's 1649 Latin translation of and commentary on La Géométrie was responsible for the spread of analytic geometry to the world. Van Schooten extended ideas to 3 dimensions and introduced the x and y axes. Work of Newton and Leibniz in developing the Calculus drew on methods and ideas of Descartes.

31 La Geométrie I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.

32 A final word from Descartes It is not enough to have a good mind. The main thing is to use it well.

34 Bibliography The Geometry of Rene Descartes Downloaded pdf from Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace L Mlodinow MacTutor History of Mathematics archive University of St. Andrews Men of Mathematics E.T. Bell

35 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with industry to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

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