## Mathematics Intrinsic Value, or Simply a Tool?

It is a sunny day; you want to take a walk. We here define walking as the taking of repeated steps in any given direction. Our walk will take us into the field of mathematics. An Unusual Walk Imagine a stretch of sidewalk that is precisely 6 feet long. An initial 3 foot step is taken. Every step after that must be 1/2 the distance of the previous step. So our second step is 1/2 the length of the first step. The third step is 1/2 the length of the second step, and so on. Questions: 1. Theoretically, how long would it take you to reach the end of the 6′stretch? 2. How far would you have traveled, after each step? Answers: 1. You would, again theoretically, never make it…

## Simplifying Mathematics: Introducing Vectors and Vector Addition

[caption id="attachment_23153" align="alignright" width="480"] Magnetic vector force field[/caption] We all know how to add, subtract, multiply and divide ordinary numbers, even if basic units are attached to them such as gallons, apples, feet, tons, and so forth. 4+3 = 7 5 lbs x 3.2 = 16 lbs These two examples illustrate pure numbers in the first instance, and simple quantities in the second instance. But What About Direction? What if we toss in direction? Imagine a huge square, 5 miles on a side. We have to travel along the perimeter to travel from Point A to Point B, and then on to our destination, Point C (see the image). We thus travel 10 miles to reach Point C. If we could travel "as the crow flies", we would only have…

## How Far Can Perfect Eyes See? An Ideal Earthly Scenario

[caption id="attachment_26540" align="alignright" width="480"] Image by Meg Learner[/caption]I once met a fellow in Virginia who said, 'The human eye is an amazing thing. Why, if there were no mountains in-between, we could see California!' Of course, he was referring to perfect eyes... Of course, that is just plain nuts. Nevertheless, it raised the question, "Just how far away could a person see an object if nothing interfered? Let's consider the answer to that question. Conditions First, we need to set conditions or ground rules. Earth is sufficiently round to call it a sphere, so we treat it as such. In fact, we assume it is perfectly smooth even to an ant. Further, we assume the atmosphere is perfectly clear, and the observer has perfect eyes. When we look out toward…

## Deriving Basic (Circular) Trigonometric Functions

[caption id="attachment_18674" align="alignright" width="401"] Figure 1[/caption] Draw two intersecting lines in space, illustrated below. The mathematician will not want to leave this simple drawing without completely pointing out its features and labeling those features. We do so to begin our understanding of basic (circular) trigonometric functions. [caption id="attachment_18675" align="alignleft" width="387"] Figure 2[/caption] We label the point of intersection of course – P will do. But the intersection produces what looks like slices in a pie. The size of those slices of pie were determined by how the two lines intersected, how "wide apart" the lines are. We label these as angles α (alpha) and β (beta). Superimposing a Circle [caption id="attachment_18676" align="alignright" width="387"] Figure 3[/caption] The title of this paper is understanding basic circular trigonometric functions. So at this stage,…

## Simple Algebra II Graph Symmetries Discussion and Examples

[caption id="attachment_17854" align="alignright" width="480"] Typical functions in two variables.[/caption] College preparatory classes in high school often include Algebra and Algebra II. Perhaps the most memorable aspect of Algebra II is the two-dimensional (2D) graphing of mathematical functions in two variables. This is typically introduced beginning with the Cartesian coordinate system. The generic function is written y = f (x). This reads y equals a function of x. See the illustration for some examples of functions. Cartesian Coordinate System In the Cartesian system, two variables, often x and y, are assigned their own line, one horizontal (x), one vertical (y). The intersection between the two axes is called the origin, and is assigned the value (0, 0). The value of x is the value written on the left in the brackets;…

## Sphere Reciprocal? Not Inside Out, But Equation Inverse?

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)…

## Interior Angles of Polygons Starting with a Triangle

Regular polygons are closed curves with a set number of sides and interior angles, each of identical value. How can one calculate the interior angles of polygons? An equilateral triangle has three equal sides and three equal interior angles. Each angle is 60o or π/3 radians, adding up to 180o or π radians total. A square has four equal sides and four equal interior angles. Each interior angle is 90o or π/2 radians, adding up to 360o or 2π radians total. Interior Angles of Polygons A short chart of additional polygons with their interior angles provides the following data: Pentagon    5 interior angles of 108o (540o or 3π radians total) Hexagon    6 interior angles of 120o (720o or 4π radians total) Heptagon    7 interior angles of about 128.57o (900o or…

## Memorizing Long Numbers – Two Quick Memory Aids

[caption id="attachment_15687" align="alignright" width="440"] Dice - Image Pixabay[/caption] Memorizing long numbers? How can I do that? I recall being told the average American can repeat quickly only numbers with five or fewer digits. For example, hearing several numbers, say 17, 38294, 584, and 127532, most can only say back the 17, 38294, and 584 – not the 127532. How can such a person improve in memorizing long numbers so he can recall 6, 7, and even more digits? There are two ways. The first involves a kind of 'device'. One definition of mnemonic device is “a memory technique to help your brain better encode and recall important information”. Memory Aid – Grouping Numbers Almost anyone can repeat a string of three. Jill speaks a three-digit number. Bob repeats it back…

## Factorials? What are They? A Simple Kind of Mathematics Shorthand

What are factorials? A variable is a symbol, often written as a letter of the alphabet that stands for a number that can vary in value. For example, take your age. That varies every year, doesn’t it? This year your age may be, say 21. If so, in 365 days your age will be 22. Another 365 days after that and your age will be 23. Thus age is a function of time. For you, we can write right now: A = 21 If the number of years that pass equals n, then for next year, n = 1 and An = 21 + n So, A₀ = 21 + 0 = 21 A₁ = 21 + 1 = 22 A₂ = 21 + 2 = 23 A₃ = 21…

## The Algebra Distributive Property – A Simple Introduction

The algebra distributive property lets you multiply a sum by multiplying each part separately and then adding those amounts together. These words are bound to confuse the reader, so let’s consider an example that will demonstrate what we mean. The Example We want to multiply 4x7. Let’s write it as (4)(7). Then, (4)(7) = 28 Now let’s replace 4 with its equivalent, 3+1. And let’s replace 7 with its equivalent, 5+2. Then, (3+1)(5+2) = 28 This seems to be a pretty strange way to write 4x7, doesn’t it? Yet in mathematics – in algebra-style notation – it is just as correct as 4x7. In this form, we can hopefully explain in an understandable way, how the algebra distributive property works. Refer to the diagram to see how we can do…