Top Ten Applications of Pi

It is pi day!!! It snuck up on me. I was going to make a top ten of cool facts about pi - by stealing them from the internet, of course. But most of those are not math applications, just cool trivia with virtually no applications, e.g., in a Star Trek episode, Spoke once defeated a computer by commanding it to compute the last digit of pi.

Brainstorm. Combine the two into the top ten applications of pi. And also doing so by stealing them from the internet. All wrapped up in one tidy package. By the way, I understand some of these applications fully and some not very well at all. I will try to give a short internet-stolen explanation for these.

1. Circumference of a circle.
2. Area of a circle.
3. Volume of a sphere.
4. Surface area of a sphere.
5. Cosmological constant - Value of the energy density of the vacuum of space. Originally introduced by Albert Einstein in 1917.
6. Heisenberg's uncertainty principle - Precision with which certain pairs of physical properties of a particle (such as momentum and position) can be known. Joke break: Heisenberg is driving down the road and is pulled over by a police officer. Police: "Do you have any idea how fast you were going?" Heisenberg: "No, but I know exactly where I am."
7. Einstein's field equations of general relativity - Describes the fundamental interaction of gravitation as a result of space time being curved by mass and  energy.
8. Coulomb's Law - Force between two charged particles.
9. Period of a pendulum.
10. Buckling formula - Finds the stress an object can handle before "buckling" would occur.

Hidden Figures

This is kind of a review of the movie/book Hidden Figures. Before seeing either one, I was a little concerned that it was going to be preachy. We need to treat African American people / women with respect. Hollywood, you don't need to tell us that. The people that already know that don't need to be told. And the people that don't know that probably aren't going to pay money to see this movie anyway. But it wasn't like that - the movie or the book. It just told their story.

If you want the true story, books are usually a better bet than the movie. The movie seemed quite in line with the book, but there were a few things. The two hour movie obviously had to leave a lot of stuff out that was in the book. There was also a time one of the ladies was at a chalkboard. She was impressing the room with the math she was doing. Some of it was a little unrealistic. Most people can't rattle off sin(23) to ten-thousandths place from memory - stuff like that. But, I'm quibbling. The movie was really good.

I got the book. And to be honest, I did some skimming in parts. I may get around to reading every word at some point, but that will be a ways off. There were a couple of cool things dealing with math applications that I thought was interesting. First of all, the book is Hidden Figures and is written by Margo Lee Shetterly. There. I hope that covers me from violating any copy write issues. Regardless, here we go.

Where do systems of equations take place in real life? Well check this out: "Modeling flight at transonic speeds was a particularly knotty problem, because of the subsonic and supersonic winds that passed over the plane or model simultaneously. Aerodynamic equations describing transonic airflows might contain as many as thirty-five variables. Because each point in the airflow was dependent on the others, an error made in one part of the series would cause an error in all the others. Calculating the pressure distribution over a particular airfoil at a transonic speed could easily take a month to complete for the most experienced of mathematicians." (pages 137, 138)

I once had the father of a student tell me about an older child of his that was in the NASA astronaut program. He said that their astronauts had to have a very good mathematics background in case something happened to the on-board computers. Another part of this book spoke to that issue:  "An astronaut stranded hundreds of thousands of miles from Earth is like a mariner from a previous age, adrift in the most remote part of the ocean. So what do you do when the computers go out? This was precisely the question Katherine [Johnson] and her colleague Al Hamer had asked in the late 1960s, during the most intense preparations for the first Moon landing. And in 1967, Johnson and Hamer coauthored the first of a series of of reports describing a method for using visible stars to navigate a course without a guidance computer and ensure the space vehicle's safe return to earth. This was the method that was available to the stranded astronauts aboard Apollo 13." (page 248)


Speaking of outer space, I give both the book and the movie, four stars.

Book

My book is coming along. My blog is to promote the same idea as my book - applications of mathematics at the high school level. I don't want to be some jerk and use this blog to publicize my book. At least not very often. Let me take this week, though, and tell you where I'm at on it.

I just heard from my peeps at McFarland Publishing. They are pretty much done with what they do. The sent me a PDF of my book - exactly 200 pages. I now have two more jobs.

One is to do a final read through for errors. I'm about 3/4 the way through that. It seems pretty good. The main mistake that has come up is where I use a subscript or superscript in an equation and the next symbol mistakenly gets put into that form as well. No big deal. I think that is easily correctable.

They say I can respond by e-mail if there aren't too many errors (under 30) or run off the offending pages with corrections written in. They let me know it isn't time for general editing. This is just to find errors. I'm not sure I always know the difference, but in most cases I think I do.

Job 2 - Create an index. I've never done that before. I thought I would read it for mistakes and plan out the index at the same time. Nope. That turned out to be way more than my head could handle. So by tomorrow I think I'll be through the initial reading. I don't think I'll have to read every word to do the indexing, so perhaps that will go faster.

I've wondered before how they come up with an index. They sent instructions on how it is done, or at least how they want it done. It's pretty interesting. There are some rules, but at the same time a lot is left to the discretion of the author. Rules just on the correct alphabetize an index is a lot more complex than one would think.

I have to do this and send it in by December 31. It goes to the printer then. I'm not exactly sure when it officially comes out, but shouldn't be too long.

I hope this wasn't boring, but I've never done this before and going through the process has been interesting.

Where Are We Going to Use This?

I was browsing through and found one of those site where someone asks a question and others respond with their thoughts. This one asked about where advanced math gets used in real life. I thought, "Hey, that is right up my alley", so I checked it out.

I thought I would include it here in my blog. The responses are quite interesting. I just pasted them in as is, so there might be some grammatical or logical errors. That is just what makes it interesting. The original website was at https://www.physicsforums.com/threads/what-is-advanced-level-mathematics-used-for.500513/

So, here is what people had to say:

I barely understand the bare basics of algebra, my math skills are abysmal. But what applications do advanced mathematics such as stochastic calculus and linear algebra have? Other than in physics, science, and engineering, what other things can advanced-level mathematics be used for? How about in daily life?

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It can enhance your abstract thinking ability. Not directly useful for anything in daily life.

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I fully disagree, how do you think barcodes were invented? A few uses of Linear Algebra: Codabar system Digital image compression Calculating life expectancy Modelling population growth Profit maximization Universal Product Code Lots more. If you want a thorough discussion of exactly how they are used, then just open up some linear algebra books, or do a Google search. Higher math isn't just solving puzzles (In fact, that's not really what math is). People don't just do mathematics to improve their thinking abilities; it certainly helps, but it has many applications. Keep in mind that mathematics need not be applied to anything. Just because you can't use a result of mathematics (at first) for anything practical does not make it useless.

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There is great deal of uses in physics (also era involves some Chemistry) and engineering. In general relativity a lot of linear algebra and calculus is required. In engineering, mostly calculating some basic mechanics problem. Other than these, Economics uses great deal of calculus to model the market which is very important. You can search more on financial mathematics (not accounting~~boring).

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Also, I've heard that matrices can be used to balance out chemical reactions in chemistry. I haven't actually done that myself, but I was happy to know that there was an easier way than what I did in my first year chemistry class!

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Oh sure it's useful in all sorts of professions. But not daily life, and many well paid jobs don't need it either. But the trouble is, when you're 16 you don't quite know what you'll be doing in 10 years time, and by then it's too late to learn so easily. 

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Everything has its applications. Of course, applications of some fields are more obvious than others.

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Mathematics models the natural world. So your question is nearly identical to ...... What use is it to learn English ? Furthermore, Logic, one of the foundations of Mathematics, is the link between The Arts & The Sciences. All art forms (nearly) seek to communicate. How better to make your case than with clear precise easy to understand logic ? Be it painting, screenplay, poem, courtroom summation or a novel. Some of the very the best lawyers were good at Math. That is one of the reasons they excel at the Law. Mathematics underlies nearly everything you see around you. But it will not guarantee a good life. That is the province of religion and moral philosophy.

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I think balancing equations is simply simultaneous equations, I mean for rather complicated equations. Matrices are just simple forms of simultaneous equations, they just save your paper and ink. Of course there are many ways of balancing equations, but many of them might not work for all cases. Also, matrices are very useful in doing statistics, though I haven't learn much of statistics, I heard of something called covariant matrix that is used for complicated systems. And statistics can be applied to many areas. May be you can look more on that.

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How does that even matter? Just because some random CEO doesn't use his knowledge of basket-weaving doesn't make basket-weaving useless. That's a really poor argument. Why does everything have to be immediately useful in daily life, and by daily life, I assume you mean eating, breathing, sleeping, and no more. It seems to me that you think that if you don't use something every day, or can't use it to make lots of money, then it's useless. As said many times in this thread, mathematics is all about logical and abstract thinking; it's basically a form of creativity. Now tell me, how useless are the former?

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When people ask me what the use is for mathematics, I always respond with the following poem by Morris Bishop: There's a tiresome young man in Bay Shore. When his fiancee cried, "I adore the beautiful sea". He replied, "I agree, it's pretty, but what is it for?"

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He never said higher math was useless in the general context, simply useless in daily life. There's nothing wrong with the validity of his statement since we take the meaning of "useless" in every day conversation as "generally useless" rather than "completely useless". However, there are few professions that are useful in daily life, such as cooking, etc. so the statement, although basically true, is misleading.

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Cryptography is a pretty big one.

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I prefer this: A math professor, a native Texan, was asked by one of his students: "What is mathematics good for?" He replied: "This question makes me sick! If you show someone the Grand Canyon for the first time, and he asks you `What's it good for?' What would you do? Well, you kick that guy off the cliff!"

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Yes, I was one of those people who said that. But I think the OP was concerned about direct applications. Like when you would want to sit down with a pen and paper to write an integral or perform a matrix operation. For most people the answer would be never in their life. Us mathy types would think about it all the time when we hear news stories or write on internet forums, but that's not normal people. If you aren't inclined to analyze things for fun, then knowing how to integrate won't make you do it.

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*Science *Physics *Engineering *Computer programming *Genetics and other fields of biology *Chemistry *Business accounting/finance and economics But people other than physicists, scientists, and engineers wouldn't have any real use for any advanced maths. 

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I believe that advanced-level mathematics (especially pure maths) is for fun and for appreciating the beauty. I am currently at high school and love learning math (college-level math), but find the maths at high school very dull and boring. The current education system just turns maths into a very systematic work. For example, when we learn Pythagoras Theorem, after teaching the theorem itself, we are told how to (1) Find the length of the hypotenuse if the sides are given (2) Find the length of one side when the hypotenuse and one of the sides are given while the teacher can just let us find the way of doing it ourselves with the original theorem. Even with the exercises provided, the questions are divided into parts about the first type of problem and second type of problem. It is just plain stupid (sorry for being a bit too rude).

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Finance, cryptography...

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There is a method to this madness. True you can derive these different formula's from the original theorems each and every time you need to use them. But, you will need to apply these things from time to time as you move into higher level work and it's helps if you have a lot of these very basic things like trig formulas memorized (at least somewhat memorized) such that you don]t have to go back and derive them each and every time you need them. Of course there is nothing wrong with learning how to use the theorems and understand their meanings to derive the formulas or, given a formula prove it's validity based on the theorem(s). When i was in high school (and freshman college) many of the more fundamental courses omitted the proofs or simply glossed over them. But I never felt comfortable, I always preferred working through the proofs and, thinking of other approaches I could take to them.

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You must distinguish between these two questions: "What do people often use advanced mathematics for" and "What can advanced mathematics be used for". People who have training in advanced mathematics, and a certain knack for applying it, can apply it to almost any subject, even art and literature. Statistically, the people who know advanced mathematics tend to be engineers, physicists etc. so that's where you most often see advanced mathematics applied. If an artist or literary historian happened to be an expert in differential equations, they might well be able to apply it to their field of study. However, they might not find many of their peers able to understand or appreciate their work.

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Because of it's sometimes mind-boggling complexity and difficulty, it is difficult to wrap your mind around that level of math without having to have an IQ of 140+ Math is despised by most high school and college students. Only the people with high IQ's (a tiny percentage of the population) tend to excel in, and take further interest in it. 

My Book

I might have mentioned before that I was writing a book. It is meant for high school math teachers. The best way to describe it that it might be the reply to students that ask, "Where are we ever going to use this?"

It isn't really out yet. It's supposedly being released in the fall. It is called "Math for Real Life" and is published by McFarland Publishing. The company has been great and know what their doing. It is been an interesting education for me as to how publishing works. In the contract they made clear what they would do and what I would do. Also, they let me know what things I might have input on, but that they would be making the final decisions.

For example, I wanted a little different title. They apparently didn't care for my title. I also could have input on the cover, I didn't really have any ideas on that, and they came up with something cooler than I would have ever come up with.

After I signed a contract with them I sent them my manuscript. They said there would be several months in which I wouldn't hear much from them as they prepared the book and not to get anxious and bug them. It was a little disconcerting, but I remained patient. Supposedly I'll get it back in a bit and make any final corrections and then I'm required to come up with an index.

Anyway, they've been very professional and great to work with. Its been a fun process. When its actually available, which I think will be in a couple months, I'll mention that in the blog.

Math Occupations

I copied this from the University of Northern British Columbia website. I'm claiming ignorance as to copy write laws, but I'm giving them credit, so i think I'm fine. This seems like a good list. I edited this down just a bit. The full page is at http://www.unbc.ca/math-statistics/careers-mathematics

Careers with a Mathematics Degree

Aside from being useful to becoming a High School teacher, a degree in Mathematics is quite useful in the following job areas (where people with a Mathematics degree have been hired).

1. It prepares you for an MBA (Masters of Business Administration) degree and become, for example, an Accounting Manager or a Certified Public Accountant.
2. Can work as a computer scientist at the National Institute of Standards and Technology.
3. Can work as a mathematician in places like Rockwell International Corporation.
4. With a degree in mathematics you can go to Law School.
5. Can work as a project manager in a company like Hewlett-Packard.
6. You can go on to an engineering school and become an industrial engineer or work in companies like Westinghouse Wireless Solutions.
7. Informations Systems Consultant.
8. Manager at a place like Advanced Research Computing Services.
9. Can work as an actuary for a life insurance company.
10. Senior software engineer (e.g. at Harris Scientific Calculations).
11. With a MSc in Mathematics one can work as an aerospace mathematician in space centers like the NASA Goddard Space Flight Center.
12. Can be a mathematics editor for a publisher (e.g. Simon & Schuster, Inc.).
13. A computer systems specialist for a chemical company.
14. A professional relations representative for a health care company.
15. Educational markets manager for companies like Texas Instruments.
16. Operations research analyst (e.g. for FedEx or any courier company).
17. Director for inventory control.
18. Can work in companies like Exxon Production as a research specialist.
19. A statistician at a health research laboratory.
20. Can work at a Boeing company as instructor and/or consultant on quality control.
21. An environmental mathematician for an engineering company.
22. With a double major with economics you can become an economist for an oceanographic company.
23. Some national laboratories hire mathematicians to analyze problems numerically (Numerical analysis).
24. A statistician in a census bureau.
25. Telephone companies are known to hire mathematicians, even ones with only a BSc degree.
26. Financial analyst.

Math Videos

Many math teachers lament the dearth of good math videos. How many Friday afternoons have you thought how great it would be to be a history teacher and just pop in one of thousands of video they have? From some one who has shown "Donald Duck in Mathemagicland" maybe a millions times, the value of a good math video cannot be overstated. I was somewhat surprised to see how many good math related things there are on the PBS website. I literally stumbled on to (via http://www.stumbleupon.com/) a video entitled "The Great Math Mystery". It can be found at http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html. I think it's good. It perhaps tries to do a little too much, but it is a quality video. It shows many good applications in mathematics. It doesn't get into the nitty-gritty of the math, making it showable to pretty much any level of student.  And at 53 minutes, you could probably milk it for two days worth.

Along with other videos, there are also a number of well-written articles. For example one called "Describing Nature with Math" is at http://www.pbs.org/wgbh/nova/physics/describing-nature-math.html. There are quite a few more you can find in their "physics & math" section.