So I was thinking about trigonometry the other day after I had been tutoring someone about to take a mathematics placement course focusing on the wide net of mathematical animals known a “pre-calculus.” Basically it covers everything from “Explain how many fingers you think you have, and don’t worry, there are no wrong answers, including leaving it blank.” to “Find a polynomial time algorithm for the traveling salesman problem and have it submitted for peer review for the past five years.”
One of my favorite mathematical topics to explain to people is geometry. I suspect this is because I am a very visual person and I have almost no ability to draw. Good thing I hardly ever help people with their zoology placement tests. “OK, let me draw you two slightly different bird species and explain how different evolutionary patters in their lower beak have allowed them to both cooperate and thrive together for thousands of years.”
Right triangles are one of the most talked about objects in geometry. This, of course, explains why trapizoids are so bitter and jealous. Take the following triangle. (But remember I “borrowed” it from the wikipedia website, so put it back when you are finished.)
So questions often arise here such as: How do you “know” that the long side of the triangle has the length of the square root of two? Why not make it something easier like 1 1/2? And why does it matter anyways? When am I ever going to need a right triangle at a job interview?
Suppose you have a right triangle which has two sides of length 1 and you want to find the length of the unknown side:
My favorite way to prove this is to start by finding the area of this triangle. (And yes, there are more ways to prove this than there are incorrect proofs about squaring the circle.) Singe the area is 1/2*b*h we know the area of this triangle is 1/2. Now imagine we have four of these triangles:
These triangles together have an area of 2. Now suppose the triangles get rearranged as follows:
So now you have a square with an area of 2. This means that each side of the square must have the length of the square root of two. I like this approach because it uses the least number of tools to get the job done. Also, this is the philosophy I use to build my kinetics crafts, but with mixed results.
Then I started thinking of a different approach using a concept called limits. Suppose we started building a staircase along the unknown length of the triangle. As we use smaller and smaller steps it starts to look more like a straight line. We can use limits to see what this would look like as we approach an infinite number of smaller and smaller steps.
Each time the steps get smaller, but the total length of the blue line is always two. Now the big question is: What happens when we use a limit to see what happens as we approach an infinite number of steps? I’m warning you– this is where some weird shit is going to go down. If you are standing up, I suggest sitting down. If you are on public transportation, please activate the emergency stop mechanism. If you are sitting on the toilet, I think you should be OK.
So as we approach the limit of this exercise, the length stays the same at 2, but all the points of the staircase line up exactly with the diagonal line. But at the beginning I told everyone the length was the square root of two, which is somewhere in the neighborhood of 1.4. So where did the extra 0.6 go? Rounding error? Did the dog run off with it?
Honestly, I’m not sure. First of all, I’ve been a UPS driver for the past 10 years. My number skills aren’t quite what they used to be. Eigthly, I hope this goes on to be one of the most discussed mathematical oddities of this generation– somewhere between the “Let’s Make a Deal” dilemma (people have literally written entire books on the subject) and understanding how Leonard is dating Penny on “The Big Bang Theory.”