11May 2017 by Vincent Summers No Comments

Sphere Reciprocal? Not Inside Out, But Equation Inverse?

Education, Mathematics
One way of mathematically representing a very simple sphere in 3D space is, r2 = x2 + y2 + z2 where r equals a radius of the sphere. Solving in terms of x, y, and z, we get, x = √(r2 — y2 — z2) y = √(r2 — x2 — z2) z = √(r2 — x2 — y2) A Sphere Reciprocal Now a sphere may be the most aesthetically pleasing of the simple geometric curves. So it is natural to wonder, concerning a sphere, what if…? So what if we convert the equation into an equation for a sphere reciprocal? No, not turn the sphere inside out. Rather an inverse of the equation of a sphere? What is the graph of, r2 = 1/( x2 + y2 + z2)…
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08Aug 2016 by Vincent Summers 1 Comment

Constants and Variables: A Simple Introduction to Algebra

Logic, Mathematics
Please bear with me on this article. You see, I am a chemist, not a mathematician. Yet, as an individual who struggled with the concepts behind algebra (yet I grasped it soon enough to ace it), I can understand how others – intelligent individuals – can find algebra disconcerting. What are constants and variables? Two Basic Participants - Constants and Variables There are two primary participants in algebra – variables (which change) and constants (which do not change). Constants are specific numbers that never change. 27 is always 27. 43-1/4 is always 43-1/4. It never changes; it is constant. So let’s consider your age. Your age changes! This year you may be 16. Next year, you will be 17. Age is variable. Let’s write an equation. Your First Algebra Equation…
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05Aug 2016 by Vincent Summers 2 Comments

A Negative Times a Negative Makes a Positive Number

Logic, Mathematics
In high school we were introduced to negative numbers. Why high school? Why not earlier? Because we cannot picture in our minds what a negative number is. We know what positive numbers are. For instance, if we have three apples and someone gives us four more apples, we know we now have seven apples. And as to multiplication, if we have three groups of four three apples each, we know we have 12 apples. But can you visualize what a negative apple might be? How can you demonstrate negative times negative makes positive? Pure Numbers Forget units for the moment. We will concentrate on pure numbers. In the above example, the four groups of three apples becomes simply 4(3) = 12. Suppose, instead of 4(3) we make on of the…
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02Aug 2016 by Vincent Summers 2 Comments

XY-Coordinate System Symmetry with Examples

Logic, Mathematics
[caption id="attachment_28582" align="alignright" width="480"] Image Department of Energy[/caption]In high school mathematics, the topic of symmetry is bound to arise. Especially is this so in analytic geometry. For curve C, what is its XY coordinate system symmetry? How is it symmetric about the y-axis? The x-axis? The origin? The line y = x? The line y = -x? Symmetric about some point other than the origin? Symmetry About the Y-Axis Symmetry about the y-axis means that if there is a curve that lies to the right of the y-axis, there is an identical copy of it to the left of the y-axis. That is, it is symmetrical if each x value can be replaced with –x. Thus, the parabola y = 1/2x2 is symmetric with regard to the y-axis. For every…
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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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07Mar 2014 by Vincent Summers 2 Comments

More High School Math

Mathematics
[caption id="attachment_5627" align="alignright" width="400"] Calculations[/caption] The most practical math for people to understand is undoubtedly high school math, rather than college math. After all, how much calculus is used when you go grocery shopping, get your plumbing fixed, or you go skiing on the weekend? High School Math You've got to love it. Here's the first high school math problem. Problem 1: Simplify the mathematical expression: (x-2y3)4 (x-3y4)-2 Simplifying the first parenthetical expression, we get (x-8y12) It is the powers we multiply when powers are raised to powers. Doing similarly with the second parenthetical expression, we get for that (x6y-8) The equation now reads, (x-8y12) (x6y-8) When we multiply numbers, we add and subtract powers. This gives, (x-2y4) [Answer] ------------------------- Problem 2: 2/10 divided by n equals 3-1/2. What does…
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04Mar 2014 by Vincent Summers 2 Comments

Sample High School Math Problems with Answers

Mathematics
[caption id="attachment_5578" align="alignright" width="440"] Fractal - CCA Share Alike 3.0 Unported by Wolfgang Beyer[/caption] Want some sample high school math problems with answers? Well then here you go! High School Math Problems Problem 1:     A change purse has 100 nickels and dimes. The total value of the coins is $7. How many coins of each type does the purse contain? If the number of nickels is N and the number of dimes is D, then 5N + 10D = 700 (the 5, 10 and 700 representing the number of cents) However, N + D = 100 (the number of nickels plus the number of dimes equals 100) So, solving for N for both equations, we get as the result N = – 2D + 140 and N = 100 – D…
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21Feb 2014 by Vincent Summers 1 Comment

Determining the Equation for a Line from Two Points

Mathematics
Lines can be drawn in three dimensions, but most analytical geometry courses stick to lines in two dimensions, generally using the Cartesian or XY coordinate system. The generic equation for a line may follow the form: y = mx + b where m is the slope (measure of tilt or steep-ness) of the line, while b is its intercept or intersection with the y-axis. Equation for a Line from Two Points A line can be determined and an equation derived from two points. In the Cartesian system, for instance, take two points, (2 , 3) and (– 1 , 5). The first number in each pair represents the x-value of a point and the second number in each pair represents the y-value. Writing these points into the general equation y…
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