07Mar 2016 by Vincent Summers No Comments

What are Radians? Where Do They Come From?

Mathematics
Consider a simple equilateral triangle (a triangle that has 3 equal sides and 3 equal angles). Most high-school students know the three angles of such a triangle are 60 degrees (60°) each, for a total of 180°. But degrees is not the only unit used to quantify an angle. Alternately radians can be used. What are radians? Are they just another number? Where do they come from? Degrees Before we get into radians, however, let’s consider where degrees came from, and why it may not be the best choice for the measurement of an angle. If you are facing north and turn to the east, you have turned 90 degrees. Now turn south and you’ve turned another 90 degrees. Turn west, another 90 degrees. Continue the turn so you once…
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29Sep 2015 by Vincent Summers No Comments

Collapsing then Expanding the Equation for a Sphere

Mathematics
[caption id="attachment_8626" align="alignright" width="480"] How simple is a sphere?[/caption] Equation for the Simplest Sphere The equation for a sphere with its center at the origin is: x2 + y2 + z2 = c2 Where c is a positive constant. For simplicity, let's choose a positive constant, k, such that k = c2. Equation for a Circle by Collapsing a Sphere Collapsing it in one dimension generates the equation of one of three circles: x2 + y2 = k x2 + z2 = k y2 + z2 = k Equation for a Point by Collapsing a Circle Collapsing the three circles in one dimension generates two equations representing precisely two points for each of them: For x2 + y2 = k, x²2 = k y2 = k For x2 + z2…
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31Jul 2015 by Vincent Summers No Comments

Generate 2D Math Objects by Collapsing 3D Math Objects

Mathematics
I had some excellent high school mathematics instructors. They both loved their field and took an interest in their students. Since high school, I have had a deep interest in collapsing 3D mathematical equations to derive equations for 2D mathematical objects or modifying 2D objects into other 2D, or 2D objects into 1D. A 3D sphere becomes a 2D circle. A 2D parabola becomes a 1D line. The 2D hyperbola shown, if collapsed along the x-axis, becomes two 1D line segments stretching at one end to infinity. The same hyperbola collapsed along the y-axis becomes a complete line. A 2D circle becomes a single 1D line segment of a length equal to the diameter. An Example of Collapsing 3D into 2D What can be obtained by collapsing 3D math objects…
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15Jun 2015 by Vincent Summers 8 Comments

Is a Circle a Polygon or Not? Implications for Calculus

Mathematics
[caption id="attachment_16285" align="alignright" width="380"] A 12-sided dodecagon.[/caption] What do you think? Is the circle a polygon, or not? As a result of watching a child’s video, I previously wrote a brief piece about the “corners” of a circle. The video was designed to teach children the various shapes—how many sides does a shape have, and how many corners? The video maintained a circle has no corners. I called that into question. I still do. And yet, I do not. I now think it’s all in how you look at it. Or, you might say, it’s all in the mathematical perspective. Consider. The Circle by Definition One can define the two-dimensional circle as the complete collection or “set” of points equidistant from a set point, not part of the circle. In…
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15Nov 2014 by Vincent Summers No Comments

Analytic Geometry: The Hyperbola

Mathematics
[caption id="attachment_6399" align="alignright" width="380"] Simple hyperbola with asymptotes. Image by author.[/caption] The parabola, ellipse, circle, and hyperbola are all termed conic sections. This means that a plane that cuts into a cone in just the right way will generate one of these figures. We will consider the basic equation of a hyperbola and graph one. Equation of an Hyperbola It may be recalled the equation for an ellipse centered at the origin is, x2/a2 + y2/b2 = 1 where 2a is the length of the ellipse and 2b is its height. The equation for an hyperbola centered at the origin is very similar, x2/a2 - y2/b2 = 1 The graph of this function is completely different from that of an ellipse. Let's look at a very basic one, for which…
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28Oct 2014 by Vincent Summers No Comments

Analytic Geometry: The Ellipse and the Circle

Mathematics
The circle is really a special type of ellipse. In analytic geometry, an ellipse is a mathematical equation that, when graphed, resembles an egg. An ellipse has two focal points. The distance apart between the two points is one way of describing a particular ellipse. If the two points come together the ellipses become a circle with the point at its center. The equation for an ellipse is, x2/a2 + y2/b2 = 1 In this equation, "a" and "b" are constants that determine the shape of the ellipse, whereas x and y are variables, i.e., they can take on a host of values. When the value for x is known, the value for y is determined. Or, if it is y that is known, then x is determined. If a…
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