The Genius of René Descartes – Part 1

Descartes-300René Descartes (1596 – 1650), a French philosopher, mathematician and writer, discovered a way to synthesize geometry and algebra that resulted in a revolution in mathematics and science. Without Descartes’s brilliant insight, it would not have been possible to develop differential calculus, integral calculus, and many other branches of mathematics. What was revolutionary to Descartes’s contemporaries, now seems natural and almost intuitively obvious, a part of our culture. (Before Isaac Newton, the concept of gravity was unknown, and now all adults and most children know something about gravity.)

So what was Descartes’s world changing discovery all about? He first invented a right angle based coordinate system in which every point in the Euclidean plane is assigned a unique ordered pair of numbers, which represents the point’s location, denoted by (x, y) where both x and y are real numbers. He then demonstrated how to create algebraic equations or formulas to calculate the distance between two points, midpoint of a line segment, and the slope of a line. With these basics established, he showed how to find an x-y variable equation that describes the relationship between the x-coordinate and y-coordinate for every point on a curve and only those points on the curve. Once the equation of a curve is known, the equation can be algebraically manipulated to reveal important properties of the curve and solve a wide variety of application problems.

The diagrams below illustrates how Descartes’s great discovery is used to calculate the distance between points A and B, and the slope of the line that contains points A and B. The distance calculation is, of course, a direct application of the theorem of Pythagoras. If we let AB equal the distance from point A to point B and let m equal the slope of the line that contains points A and B, then AB = √( 82 + (-6)2 ) = √(100) = 10 units, and m = Δy / Δx = -6/8 = -3/4 or -0.75.

The Genius of RenŽ Descartes Blog

From the definition of a conic section and the theorem of Pythagoras, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on the curve. Study the graph of the circle and its equation. If you listen carefully, you will hear Pythagoras whisper from his grave, “x squared plus y squared equals 4 squared for every point (x, y) on the circle.” The graphs below are the graphs of various conic curves and a line. All equations are special cases of the general conic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are real number constants.

Mathblog13-Edit1The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered as indicated to the right of the graphic. Example: The inequality x2 + 4y2 ≤ 64 is entered as x^2 + 4y^2 ≤ 64. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results.

The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives a quick summary of all the magical mouse control clicks. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

New Resource for Math Teachers

George HeadshotWelcome! I’m glad you’ve joined me for this exciting launch of mathteachersresource.com. My mathematics teaching career has covered over 40 years. I have taught courses ranging from general mathematics through calculus, and I am currently teaching College Algebra and Elementary Statistics at my local junior college. Over the years, I have developed software programs that have helped me do a better job of teaching algebra, trigonometry, pre-calculus, calculus and statistics. It is my core belief that teachers should help students understand math concepts from both an algebraic and geometric point of view, and these programs are designed to do that.

The tools found on my website fall into two major categories. The first includes three main lines of software. The second category includes free teacher-created handouts. The first offering of handouts are those created by me; my future goal is to add to this inventory of handouts and to invite other math teachers to share their handouts through the website. More information on this will be coming in future blogs.

My software offers many unique features that make it easy for teachers to give dynamic presentations of core concepts in mathematics. Please visit my website mathteachersresource.com to see some of the possibilities of my program, Basic Trig Functions.

Thank you for joining me for this launch. Feel free to contact me with feedback, and I hope you’ll join me for more in the weeks to come.

~George Johnson