Calculus and Abbott and Costello

Although n + 30 > n for all n, it’s also true that

\displaystyle \lim_{n \to \infty} \frac{n+30}{n} = 1.

That’s the subtle mathematical premise behind this classic comedy routine from Abbott and Costello. (This routine was the basis of a recent article in The College Mathematics Journal.)

A Visual Proof of a Remarkable Trig Identity

Strange but true (try it on a calculator):

\displaystyle \cos \left( \frac{\pi}{9} \right) \cos \left( \frac{2\pi}{9} \right) \cos \left( \frac{4\pi}{9} \right) = \displaystyle \frac{1}{8}.

Richard Feynman learned this from a friend when he was young, and it stuck with him his whole life.

Recently, the American Mathematical Monthly published a visual proof of this identity using a regular 9-gon:

Feynman identity


This same argument would work for any 2^n+1-gon. For example, a regular pentagon can be used to show that

\displaystyle \cos \left( \frac{\pi}{5} \right)  \cos \left( \frac{2\pi}{5} \right) = \displaystyle \frac{1}{4},

and a regular 17-gon can be used to show that

\displaystyle \cos \left( \frac{\pi}{17} \right) \cos \left( \frac{2\pi}{17} \right) \cos \left( \frac{4\pi}{17} \right) \cos \left( \frac{8\pi}{17} \right) = \displaystyle \frac{1}{16}.

Folding a New Tomorrow: Origami Meets Math and Science

From the YouTube description:

Origami, the art of paper folding, has been practiced in Japan and all over the world for centuries. The past decade, however, has witnessed a surge of interest in using origami for science. Applications in robotics, airbag design, deployment of space structures, and even medicine and bioengineering are appearing in the popular science press. Videos of origami robots folding themselves up and walking away or performing tasks have gone viral in recent years. But if the art of paper folding is so old, why has there been an increase in origami applications now? One answer is because of mathematics. Advances in our understanding of how folding processes work has arisen due to success in modeling origami mathematically. In this presentation we will explore why origami lends itself to mathematical study and see some of the math that has allowed applications to become so fruitful.

Larger or smaller?

Suppose I write down two different numbers on two slips of paper. You have no idea what the two numbers are. They could be really large or really small, positive or negative, rational or irrational. All you know is that the two numbers are different.Your job is to pick the larger number.

Is there a way for you to guess the larger number with a probability greater than 50%?

The surprising answer is yes.

Mathematics that Swings: The Math Behind Golf

From the YouTube description:

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

Math autobiography

I recently read a very interesting opinion piece: asking students to write a math autobiography as the first assignment of the semester. I may try this out in a future semester. From the opinion piece:

Want to know one of my favorite assignments that I have ever given my students? Want to know learn a lot of useful information about your students in a short amount of time?

I know it sounds too good to be true, but this one simple assignment could change how you teach your classes and how well you know your audience…

Math Autobiography

Purpose of the Assignment 
As your instructor, I want to get to know you as a person and as a student of mathematics. This will help me better meet your needs. It also helps our department as we work to improve our services to students.

Your autobiography should address the four sections listed below. I’ve listed some questions to help guide you, but please don’t just go through and answer each question separately. The questions are just to help get you thinking. Remember the purpose of the paper. Write about the things that will give me a picture of you. The key to writing a good piece is to give lots of detail…

Section 1: Introduction

  • How would you describe yourself?
  • Where are you from? How did you decide to attend Fort Lewis?
  • What is your educational background? Did you just graduate from high school? Have you been out of school for a few years? If so, what have you been doing since then?
  • General interests: favorite subjects in school, favorite activities or hobbies.

Section 2: Experience with Math

  • What math classes have you taken and when?
  • What have your experiences in math classes been like?
  • How do you feel about math?
  • In what ways have you used math outside of school?

Section 3: Learning Styles and Habits (specifically for math)

  •  Do you learn best from reading, listening or doing?
  • Do you prefer to work alone or in groups?
  • What do you do when you get “stuck”?
  • Do you ask for help? From whom?
  • Describe some of your study habits. For example: Do you take notes? Are they helpful? Are you organized? Do you procrastinate? Do you read the text?

Section 4: The Future

  • What are your expectations for this course?
  • What are your responsibilities as a student in this course? What do you expect from your instructor?
  • What are your educational and life goals?
  • How does this course fit into your educational goals?

The author’s conclusions:

It was fantastic! Students took it way more seriously than I could have imagined. Some wrote pages and all wrote enough to get to know them. It made me realize that we don’t give our students opportunities to share their math baggage/backgrounds/etc. with us often enough. Students shared everything from horror stories about being shamed in math courses to their excitement about math. Some let you know what they have heard about your class and even fears they may have such as a fear of presenting or working with others.

Preparation for Industrial Careers in the Mathematical Sciences: Building a Better Filter

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the third pair of videos describing how mathematics is used for certain problems in materials science. From the YouTube descriptions:

Dr. Sumanth Swaminathan of W. L. Gore & Associates talks about his career path and the research questions about filtration that he considers. He works to understand the different waste capture mechanisms of filtration devices and to mathematically optimize the microstructure to create better filters.

Prof. Louis Rossi of the Department of Mathematical Sciences of the University of Delaware presents two introductory mathematical models that one can use to understand and characterize filters and the filtration processes.


Useless Numerology for 2016: Part 1

The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.

2016 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3

2016 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^5

2016 = 1+2+3 + \dots + 62 + 63

2016 = \displaystyle \sum_{n=0}^{63} (-1)^{n+1} n^2

2016 = (1+2+...+8+9)^2 - (1+2)^2

(2 + 0 + 1)! = 6

2016 = 2^{11} - 2^5

2016 = 2016 \times 1

The last tongue-in-check equation is my favorite.

In this series, I’ll explain why these different expressions for 2016 have to be equal to each other. I’ll begin with tomorrow’s post.

The Wrong Door, Or Why Math Gets a Bad Rap

I really enjoyed this news article on how to motivate young students to enjoy mathematics. Surprise, surprise: it isn’t by doing a whole bunch of rote arithmetic or algebra problems.

Deciphering recommendation engines

From the video’s description: “Data scientist Cathy O’Neil provides a glimpse of the methods that Netflix, Google, and others apply to recommend or offer to users selections based on their apparent interests.” This is a non-intuitive but real application of linear algebra.