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
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:
In this equation, e represents the irrational constant 2.71828…, or more specifically , and i represents . 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:
- (Law of exponential)
- (Law and definition of constant e in calculus)
- 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.
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.
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.
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.
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
After the introduction to my website last time, in this blog, I am going to redirect the focus back to the topic of plane tickets prices and the analysis of it through data science approach. Continue reading
As my class has really jumped into our work, everything has really taken off. We have jumped right in to coding and designing simple “game-like” projects. Our first being a simple maze-like machine in order to practice creating objects and effects. Effects are a lot of fun to work with, not much is quite as satisfying as creating an invisible black hole effect that sucks a marble in.
As you can see there was quite a complex system of parts that were both visible and invisible, allowing for effects that seem to appear out of thin air. This lead to quite a fun element of surprise as the marble was rolling through the maze. Though slightly challenging, that project was quite a lot of fun. Continue reading
Hey guys, welcome back! You might remember that in my last blog I mentioned a study-hall sign-in website, which is part of the main focus of my project, and I promised to come back with more details on that, so here I am. Before I dive in though, I just want to provide some quick updates on my quest for answers regarding the manipulated plane ticket prices. Continue reading
This past summer, I flew back and forth between China and the US a lot, which meant I had to book plane tickets a number of times. During this process, I used a Chinese travel website called Ctrip, which my family has always liked and trusted. It is also the largest online travel agency in China. However, this time my experience was not so pleasant. The price for the tickets that I was looking at kept going up every time I returned from looking up similar tickets on other websites, which could be interpreted as normal since that price might go up as the date approached. The part that took me by surprise though was when I tried to log in using a different account and look at the same tickets on the same date, I found that the prices differed. I don’t recall the exact price gap but I just remember that it was enough for me to be upset and intrigued by it at the same time. After a brief search online, I found that there were already news reports accusing this company of manipulating their customers through the use of “big data.” This discovery deeply interested me. I could not help but started to wonder about questions like “How exactly are they using the data they collect to achieve their goal? How are other Internet companies like Netflix and Google using their data? What are the ethical implications of this? What impact does this have on our society as a whole?” Continue reading
Video Games have been a staple of my personal interest for as long as I can remember. At the age of five I clearly remember playing on my friend’s older brother’s PlayStation 2, and I instantly fell in love.
As of recently I have had a growing interest in how video games are made, and I am looking to pursue that interest in college. Thankfully the DigiPen Institute of Technology offers online high school classes. So this week I dove into the world of video game design, a world that I hopefully grow to love. Continue reading
In the past week, I studied Differential Equations through an MIT Online Open Course, which can be found here:
In this blog, I will introduce Differential Equations as well as some of the methods of solving or visualizing them. I will start from the place where most students left out since Calculus II to make it more comprehensible.
Differential Equations, also known as DE, means “an equation involving derivatives of a function of functions” (dictionary.com). Differential Equations have a broad application in subjects like Physics, Engineering, and Biology, which will be discussed in depth in my future blogs. The first several blogs, however, will be focusing on the math behind differential equations including how to solve and visualize the formulas.
One simple example of DE is , in which x is the independent variable and y is the dependent variable. Notice that taking integral is not a way to find a general solution of y; instead, we must employ a method called “Separation of Variables”. Continue reading
For last week’s blog post, I wrote a short tutorial for training a custom object detection model using TensorFlow Object Detection API. Due to the limited space and time constraints, my tutorial was not quite finished. Therefore, in this week’s blog, I will continue my tutorial and include additional steps such as the usage of a tool to test your model’s accuracy.
Have you ever heard of Tesla’s Model S sedan? It is one of the few cars capable of fully autonomous driving. Although U.S. laws currently do not permit this, the Model S can pick you up at your house and drop you off at school, all without you even touching the steering wheel. To create a self-driving vehicle, Tesla engineers had to employ many machine learning techniques, including an object detector that recognizes and classifies objects around the car. For example, the on-board camera is able to recognize pedestrians and instructs the car to stop. Another example is that the object detector recognizes other vehicles on the road, keeping the Tesla from colliding into them.
With the use of the TensorFlow Object Detection API, creating such a model (though probably not as accurate as the one Tesla developed) can now be done with consumer-grade equipment such as a personal computer. As promised in last week’s blog, I will discuss how to create a customized object detector with the TensorFlow API.