Thoughts with Richard Bleil
Back in high school, my academic career was in serious jeopardy. I started off with a far less than stellar beginning. I wonder how many people would be surprised by that. In my junior year, something really clicked in Mrs. Beemer’s algebra 2 class. It just suddenly made sense, and I was off and running. Math, and science, suddenly became very easy for me. I realized that, in math, I can do anything I want on one side of the equation provided I do the same to the other, and all of those math problems just fell into place.
Following that class came trigonometry. Here we learned trigonometric functions that would become so critical in my work as a theoretical chemist. Angles and distances between atoms became critical to calculate potential energy and fell into my mathematical modeling in critical ways. Among those trigonometric functions are things like sine, which is the ratio of the opposite axis divided by the hypotenuse, but at ninety degrees the hypotenuse falls into the hypotenuse (the adjacent axis is zero). Because of this, anybody who knows trigonometry knows that the sine of the right angle is equal exactly to one. The reason, if you understand what sine really is, falls into place, and of course it equals exactly one.
One of my classmates, a true knucklehead if ever I’d seen one, whipped out his calculator (a relatively new product in those days in the early eighties) punches ninety into his calculator and comes up with 0.99999999 as the answer. He raised his hand and proceeded to explain to the teacher why the answer is not one after all, and that he will choose to believe his calculator rather than, well, frankly his own mind.
Trigonometric functions (along with many others such as logs) are actually based on algorithms. The calculators, and modern computers, do not have built in tables of these functions. As a result, they’re really only estimates to these problems. Older calculators, as a result, would sometimes give answers such as this because they didn’t round. I wonder if Brian the knucklehead ever thinks about this, and repeats the experiment, and if so, how does he reconcile his old calculator results with the new ones?
The true point that I want to make here has really nothing to do with the sine of half pi radians, but rather, the overreliance on technology. Here’s Brian, with the definition of sine, and the perfectly good, understandable and reasonable answer provided by both the teacher and the textbook as to why the answer is exactly one. Not only did he refuse to recognize and understand why sine is one at this point, but he also failed to understand the technology not only uses an algorithmic estimate of the value, or why it should have been rounded up. In fact, had he done the opposite experiment and taken the inverse sine of one he would have come up with ninety degrees.
Today, people still have an overreliance on technology, and the flawed humans behind it. One of my favorite long-term problems came about because a well-renowned biochemist believed a program that suggested the answer to his question was a proverbial “bifurcated hydrogen bond”, a concept I didn’t believe, but the program flagged it. The program that he was using flagged this because the position and angle of the hydrogen fell within the ranges for a hydrogen bond, not because of any advanced calculations. In other words, it was an estimate, but because the computer said it was a hydrogen bond, and because this biochemist didn’t understand how the program worked, he believed it.
We have an important midterm election literally days away, and many of the people who will be helping to decide the direction of our nation still believe in simply ridiculous conspiracy theories because, after all, they saw these theories on their computer, and computers can’t be wrong, right? Logic, on the other hand, clearly dictate the flaws in logic of a vast majority of these conspiracy theories. All one has to do is think about them, and they fall apart, but people don’t like thinking for themselves. Bill Gates is putting microchips into every Covid vaccine, for example, so the government can track us. Never mind the enormous expense it would take to develop such technology, to build the chips, the excessive power to transmit the data which would be impossible in such a small device, or even the incredible database that would be required to store the data and the manpower to monitor it. Any one of these arguments explains why this conspiracy theory is nonsense, but even without the technical arguments, why would anybody believe that they are so significant that the government would want to track them anyway? But people would rather believe what they read on their computer than the logic of their own critical thinking. These kinds of people really should elect Brian the Knucklehead as their leader in anarchy. And, yes, people who do think critically will realize the humor in that last sentence.