An Eraser-Full of Mistakes

This is my pencil. I've used it throughout the entire spring semester and it was the only pencil I used this semester. When I began in January, the pencil was brand new and the eraser hadn't been used. It turns out that I've had to use this eraser in every single one of my classes at least once. Now, the eraser is completely gone; I've made an eraser full of mistakes throughout this semester. Some have been bigger than others, some I'll continue to make, and others I'll never make again. But with all my mistakes, I've learned a lot along the way. I'll continue to make mistakes, maybe even some really big mistakes, but hopefully, I'll start to make less mistakes and learn more things as I go on in life.

A Few Mathematical Concepts That Blow My Mind (Part Two)

Euler's Identity

This post is the true sequel to the first of the Mathematical concepts that blow my mind series. In this post, I'll talk about Euler's (I recently learned this is actually pronounced roughly how American's would say Oiler's) Identity. Without further adieu, Euler's identity is as follows:

The first time I saw this equation, I was drawn to it; It’s a pretty elegant equation from a mathematical perspective. I mean you have the number e, which is the base of the natural logarithm which is prevalent everywhere in nature. the next thing you see is an exponential operation and in that there’s the number i (-1) which is the basis for all imaginary numbers, then you have a multiplication operation attaching i with the number which is extremely important when dealing with circles and is the basis of measurement in radians. This whole term is being added to the number 1, which is then equated to the number 0. I mean wow, it’s amazing how you can take all of these numbers and principals which are vastly important in their own field of mathematics and join them into one, simple looking equation, which is no more than an inch or two long. For this reason, many people regard this equation as beautiful; some even venture to call it the most beautiful equation ever. When I first came across this equation, I had never heard of the notion of mathematical beauty, but when I saw this, there was just something about it that give it more value than just how one can use it in mathematics. I kind of see it as a rite of passage for the human race, being able to show to the universe that we understand these individual concepts to the point where we feel comfortable combining them all and using them for our gain.

Now, Let’s get down to what this equation actually means from a mathematical point of view. Well this equation is actually a special case of Euler’s formula relating to the field of complex analysis. the general formula states:

note that since x is being used here, we’re assuming radians for the trigonometric portion of this equation. So if we take x to be pi, we’ll have:

and since the cosine of pi is -1 and the sin of pi is 0, this gives us:

after some cleaning up of the equation, this will eventually give us Euler’s famous equation. So, now that we have this equation. What does it actually do? What can we use this for? Well, the first thing that comes to mind is that it can be used to simplify things in differential equations. This is important because differential equations in my opinion are kind of like a cornerstone of physics. without our understanding of differential equations, our knowledge of physics would be severely lacking and no where near as accurate. Some other uses I’ve heard of but don’t really know too much about are it’s ability to allow us traverse circles easily, as well as help explain electronic signals that vary overtime in the realm of electrical engineering since the formula combines sine and cosine.

Alrighty, that’s all for now. I’ll probably post another one of these in the near future because I just keep running into cool mathematical principles, but I don’t exactly want to saturate this blog with math. The next post in this series will most likely be one the continuum hypothesis. Thanks for reading,

-Carlos