Category Archives: Math and Technology

The Fourier Series: An Introduction – Baiting

In the past few weeks and Thanksgiving break, I finished Unit II and moved on to Unit III. In this unit, I will primarily focus on the Fourier Series, Laplace Transformation, their connections with Differential Equations, and their applications in reality. In this blog, I will introduce the Fourier Series of periodic functions, including the trigonometry functions, the sin and cos. Continue reading

What Makes a Game Great – Dylan

This week we took a break from the technical practice and stepped in to a new realm. What truly makes a game fun to play? In class this week we talked about game design, less about creating a game, and more about designing one. In class we sat and discussed, what made our favorite games special, and what did we dislike about them?

Most people chose series of many games, or a single game that has been updated over several years, which allowed us to discuss our likes and dislikes over time.  People mostly chose well enjoyed series like Animal Crossing or Fallout. I personally chose the Pokemon DS games as well as World of Warcraft (WoW), two very different games. Continue reading

The Spring Model Continued – Baiting

ΦΘΙΝΟΥΣΕΣ ΤΑΛΑΝΤΩΣΕΙΣ(6)

In the past two weeks, I worked on Second Order Differential Equations with constant coefficients and learned more about the Spring-Damper Model. In this blog, I will provide a brief recap of the basic knowledge, and then provide further analysis of the same model. If you are interested in my last blog, please visit here. Continue reading

A Failed Attempt – Alina

This past week marks the end of the data collection period of my project. After I figured out how to scrape data generally on websites with simple structures in the last blog post, I had been experimenting with pulling data down from the Expedia website which was way more complex. However, as I tried to do this, I encountered some difficulties. At first, I decided to start experimenting with data that should be easily pulled to see if the code would indeed even work for this site. Therefore, I picked the date of the flight shown on the website. It had the tag class=”title-date-rtv“. I put this value into the code. Continue reading

The Trigonometry of Blasters! – Dylan

Blasters are such a staple to video games. Think back to space invaders, that game is almost completely just a simple blaster that moves side to side. That works great if you only want to shoot in a single direction, but what happens when you would like to aim in a full 360 degrees? Continue reading

Data Collection Continued – Alina

Updated Table

If you remembered from my last blog, my focus in this project has recently been on figuring out a way to scrape data off of the travel websites using code instead of doing it manually since it is indeed a tedious job. Of course, while working on the code, I have also kept with the primitive collecting method since data collection is the objective of this month’s work in my project. So here’s an updated version of my data table: Continue reading

Power-ups! – Dylan

What is the best part of a game to you? The sweet end music? The awesome rewards? Well, for me, it was always power-ups, there is nothing more staple in video games than a power up. I mean, who doesn’t know what a Mario mushroom looks like by 2018? That’s what I have been working on lately, making simple power-ups that make a game feel more alive. Continue reading

Euler’s Formula – Baiting

As mentioned in my last post, I continued to develop my algebraic skills for solving differential equations during the past two weeks. In this blog, I want to introduce Euler’s Formula, the most impressive piece of math work I have ever seen. In specific, I will dive deep into the mathematical proof of this formula and explain its broad application.

Euler’s Formula consists of a simple line:

0011.png

In this equation, e represents the irrational constant 2.71828…, or more specifically WeChat Screenshot_20181009230402 , and i represents WeChat Screenshot_20181009230659.png. With an irrational number and an imaginary number involved, Euler’s Formula is not easy to visualize in the regular coordinate system. In order to make this blog more comprehensive to everyone, I will offer an algebraic proof, so that we don’t have to learn the complex Cartesian and Polar coordinate system from the beginning.

To prove Euler’s Formula, we have to prove it satisfies these three fundamental properties:

  1. WeChat Screenshot_20181007153513.png(Law of exponential)
  2. WeChat Screenshot_20181007153856 (Law and definition of constant e in calculus)
  3. Taylor’s series of the left should equal to Taylor’s series on the right as the number of terms approaches infinite.

So let’s start from the first one! Remember we are now assuming Euler’s Formula is correct and testing if it [actually] follows all of the properties.

WeChat Screenshot_20181007160416.png

After proving the Euler’s Formula follows exponential identity, we can move on to proving the identity of e.

The concept of constant e was first brought up by Euler himself,  and is sometimes referred to as “Euler’s Number.” The core definition of e is the rate of change always equal to its self. If you want to know more about this mysterious number, please visit here. Due to its ingenious definition, e is used in multiple areas including charging/discharging a capacitor or calculating quantities related to half-life.

Now here is a simple and straightforward proof for Part II.

WeChat Screenshot_20181007161023

In the last part, we are proving the polynomials of e^iθ equals the polynomials of cos(x)+i*sin(x) from Taylor’s series. If you remember some relevant knowledge from Calculus II, this would also be straightforward.

Now, we have proved the Euler’s Formula. And I will briefly and broadly cover its applications. If you are interested in knowing more about any of these topics, please tell me and I will discuss them in the future!

Trigonometry, Fourier transformation, Taylor’s series, …. the span of the impact of Euler’s Formula goes on and on. With Euler’s Formula, we can easily model rotations in complex coordinate system or even explain the spiral movements of the starts; with Euler’s Formula, the trigonometry relationships become easier than ever; and most importantly, with Euler’s Formula, we can, for the first time in history, cross the wall between transcendental numbers and algebraic numbers.

Euler’s Formula is a piece of math theorem, but also a piece of art. Like Mona Lisa’s Smile by Leonardo da Vinci, Euler’s Formula contains much more than it seems. When rewritten as e^(iπ)+1=0, the simple line involves plus/minus, multiplication/division, exponential, trigonometry, complex number, the concept of zero, and transcendental numbers. Even within itself, there is a sea of knowledge waiting for me to explore.

To me, Euler’s Formula is just a beginning. On my journey of learning, more fascinating math topics will be closely studied and researched. What I will not forget, however, is the excitement after seeing this beautiful math work. Like a line of a poem that eulogizes our life, Euler’s Formula thoroughly depicts the beauty of math.

 

 

References

Fourier Transforms [Illustration]. (2010, Fall). Retrieved from https://slideplayer.com/slide/10994153/

Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

 

Images:

Leonhard Euler [Photograph]. (n.d.). Retrieved from https://www.usna.edu/Users/math/meh/euler.html

O’Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved October 8, 2018, from MacTutor History of Mathematics archive website: http://www-history.mcs.st-and.ac.uk/HistTopics/e.html