Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!

 

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).

Calculus predicts more snow for Boston

Are we there yet? And by “there,” we mean spring and all the lovely weather that comes with it. This winter has been a tough one, and as the New York Times says, “this winter has gotten old.”

snow big[Photo Credit: John Talbot]

Our friends in Boston are feeling the winter blues after seven feet of precipitation over three weeks. But how much is still to come? You may not be the betting kind, but for those with shoveling duty, the probability of more winter weather may give you chills.

For this, we turn to mathematician Oscar Fernandez, professor at Wellesley College. Professor Fernandez uses calculus to predict the probability of Boston getting more snow, and the results may surprise you. In an article for the Huffington Post, he writes:

There are still 12 days left in February, and since we’ve already logged the snowiest month since record-keeping began in 1872 (45.5 inches of snow… so far), every Bostonian is thinking the same thing: how much more snow will we get?

We can answer that question with math, but we need to rephrase it just a bit. Here’s the version we’ll work with: what’s the probability that Boston will get at least s more inches of snow this month?

Check out the full article — including the prediction — over at the Huffington Post.

Math has some pretty cool applications, doesn’t it? Try this one: what is the most effective number of hours of sleep? Or — for those who need to work on the good night’s rest routine — how does hot coffee cool? These and other answers can be found through calculus, and Professor Fernandez shows us how in his book, Everyday Calculus: Discovering the Hidden Math All around Us.

This book was named one of American Association for the Advancement of Science’s “Books for General Audiences and Young Adults” in 2014. See Chapter One for yourself.

For more from Professor Fernandez, head over to his website, Surrounded by Math.

 

Photo Credit: https://www.flickr.com/photos/laserstars/.

Princeton University Press’s best-selling books for the past week

These are the best-selling books for the past week.

1177 BC: The Year Civilization Collapsed by Eric H. Cline
 4-10 Drezner_TheoriesZombies_cvr Theories of International Politics and Zombies by Daniel W. Drezner
Carlson_Tesla jacket
Tesla: Inventor of the Electrical Age by W. Bernard Carlson
DarwinFinches 40 Years of Evolution: Darwin’s Finches on Daphne Major Island by Peter R. Grant and B. Rosemary Grant
OnBullshit On Bullshit by Harry G. Frankfurt
The Banker's New Clothes
The Bankers’ New Clothes: What’s Wrong with Banking and What to Do about It by Anat Admati and Martin Hellwig
RoughCountry Rough Country: How Texas Became America’s Most Powerful Bible-Belt State by Robert Wuthnow
Everyday Calculus, O. Fernandez Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
SocialismCohen Why Not Socialism? by G. A. Cohen
OnWar
On War by Carol von Clausewitz

6 Free to Low-Cost Resources to Teach You Calculus in a Fun and Interactive Way

In his new book, Everyday Calculus: Discovering the Hidden Math All around Us, Oscar E. Fernandez shows that calculus can actually be fun and applicable to our daily lives. Whether you’re trying to regulate your sleep schedule or find the best seat in the movie theater, calculus can help, and Fernandez’s accessible prose conveys complex mathematical concepts in terms understandable even to readers with no prior knowledge of calculus. Fernandez has also provided a list below of his favorite affordable resources for teaching yourself calculus, both on- and offline.

Princeton University Press offers several other books to help you master this most notorious of the mathematics. If you’re already good at calculus, but want to be great at it, check out Adrian Banner’s The Calculus Lifesaver: All the Tools You Need to Excel at Calculus, an informal but comprehensive companion to any single-variable calculus textbook. For high school mathletes and aspiring zombie hunters of all ages, there’s also Colin Adams’s Zombies and Calculus, an interactive reading experience set at a small liberal arts college during a zombie apocalypse. Readers learn as they go, using calculus to defeat the walking dead.

Calculus. There, I said it. If your heart skipped a beat, you might be one of the roughly 1 million students–or the parent of one of these brave souls–that will take the class this coming school year. Math is already tough, you might have been told, and calculus is supposed to be the “make or break” math class that may determine whether you have a future in STEM (science, technology, engineering, or mathematics); no pressure huh?

But you’ve got a little under two months to go. That’s plenty of time to brush up on your precalculus, learn a bit of calculus, and walk in on day one well prepared–assuming you know where to start.

That’s where this article comes in. As a math professor myself I use several free to low-cost resources that help my students prepare for calculus. I’ve grouped these resources below into two categories: Learning Calculus and Interacting with Calculus.

Learning Calculus.

1. Paul’s online math notes–an interactive website (free).

This online site from Paul Dawkins, math professor at Lamar University, is arguably the best (free) online site for learning calculus. In a nutshell, it’s an interactive textbook. There are tons of examples, each followed by a complete solution, and various links that take you to different parts of the course as needed (i.e., instead of saying, for example, “recall in Section 2.1…” the links take you right back to the relevant section). I consider Prof. Dawkins’ site to be just as good, if not better, at teaching calculus than many actual calculus textbooks (and it’s free!). I should also mention that Prof. Dawkins’ site also includes fairly comprehensive precalculus and algebra sections.

2. Khan Academy–short video lectures (free).

A non-profit run by educator Salman Khan, the Khan academy is a popular online site featuring over 6,000 (according to Wikipedia) video mini-lectures–typically lasting about 15 minutes–on everything from art history to mathematics. The link I’ve included here is to the differential calculus set of videos. You can change subjects to integral calculus, or to trigonometry or algebra once you jump onto the site.

3. MIT online lectures–actual course lectures in video format (free).

One of the earliest institutions to do so, MIT records actual courses and puts up the lecture videos and, in some cases, homeworks, class notes, and exams on its Open Courseware site. The link above is to the math section. There you’ll find several calculus courses, in addition to more advanced math courses. Clicking on the videos may take you to iTunes U, Apple’s online library of video lectures. Once there you can also search for “calculus” and you’ll find other universities that have followed in MIT’s footsteps and put their recorded lectures online.

4. How to Ace Calculus: The Streetwise Guide, by Colin Adams, Abigail Thompson, and Joel Hass

If you’re looking for something in print, this book is a great resource. The book will teach you calculus, probably have you laughing throughout due to the authors’ good sense of humor, and also includes content not found in other calculus books, like tips for taking calculus exams and interacting with your instructor. You can read the first few pages on the book’s site.

Interacting with Calculus.

1. Calculus java applets–online interactive demonstrations of calculus topics(free).

There are many sites that include java-based demonstrations that will help you visualize math. Two good ones I’ve come across are David Little’s site and theUniversity of Notre Dame’s site. By dragging a point or function, or changing specific parameters, these applets make important concepts in calculus come alive; they also make it far easier to understand certain things. For example, take this statement: “as the number of sides of a regular polygon inscribed in a circle increases, the area of that polygon better approximates the area of the circle.” Even if you followed that, text is no comparison to this interactive animation.

One technological note: Because these are java applets, some of you will likely run into technology issues (especially if you’re on a Mac). For example, your computer may block these applets because it thinks that they are malicious. Here is a workaround from Java themselves that may help you in these cases.

2. Everyday Calculus, by Oscar E. Fernandez.

Self-promotion aside, calculus teachers often sell students (and parents) on the need to study calculus by telling them about how applicable the subject is. The problem is that the vast majority of the applications usually discussed are to things that many of us will likely never experience, like space shuttle launches and the optimization of company profits. The result: math becomes seen as an abstract subject that, although has applications, only become “real” if you become a scientist or engineer.

In  Everyday Calculus I flip this script and start with ordinary experiences, like taking a shower and driving to work, and showcase the hidden calculus behind these everyday events and things. For example, there’s some neat trigonometry that helps explain why we sometimes wake up feeling groggy, and thinking more carefully about how coffee cools reveals derivatives at work. This sort of approach makes it possible to use the book as an experiential learning tool to discover the calculus hidden all around you.

With so many good resources it’s hard to know where to start and how to use them all effectively. Let me suggest one approach that uses the resources above synergistically.

For starters, the link to Paul’s site takes you to the table of contents of his site. The topic ordering there is roughly the same as what you’d find in a calculus textbook. So, you’d probably want to start with his review of functions. From there, the next steps depend on the sort of learning experience you want.

1. If you’re comfortable learning from Paul’s site you can just stay there, using the other resources to complement your learning along the way.

2. If you learn better from lectures, then use Paul’s topics list and jump on the Khan Academy site and/or the MIT and iTunes U sites to find video lectures on the corresponding topics.

3. If you’re more of a print person, then How to Ace Calculus would be a great way to start. That book’s topics ordering is pretty much the same as Paul’s, so there’d be no need to go back and forth.

Whatever method you decided on, I still recommend that you use Paul’s site, the interactive java applets, and Everyday Calculus. These three resources, used together, will allow you to completely interact with the calculus you’ll be learning. From working through examples and checking your answer (on Paul’s site), to interacting directly with functions, derivatives, and integrals (on the java applet sites), to exploring and experiencing the calculus all around you (Everyday Calculus), you’ll gain an appreciation and understanding of calculus that will no doubt put you miles ahead of your classmates come September.

This article is cross-posted with The Huffington Post with permission of the author.

Recommended Reading:

 Fernandez_Everyday cover Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
The Calculus Lifesaver The Calculus Lifesaver: All the Tools You Need to Excel in Calculus by Adrian Banner
 7-18 Zombies  Zombies & Calculus by Colin Adams

Princeton University Press’s best-selling books for the past week

These are the best-selling books for the past week.

1177 BC: The Year Civilization Collapsed by Eric H. Cline
 BankersClothes The Bankers’ New Clothes: What’s Wrong with Banking and What to Do about It by Anat Admati and Martin Hellwig
shtetl
The Golden Age Shtetl: A New History of Jewish Life in East Europe by Yohanan Petrovsky-Shtern
Fernandez_Everyday cover Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
Carlson_Tesla jacket Tesla: Inventor of the Electrical Age by W. Bernard Carlson
FriendlyFire
Friendly Fire: The Accidental Shootdown of U.S. Black Hawks over Northern Iraq by Scott A. Snook
OnBullshit On Bullshit by Harry G. Frankfurt
SouloftheWorld The Soul of the World by Roger Scruton
The Five Elements of Effective Thinking The 5 Elements of Effective Thinking by Edward B. Burger and Michael Starbird
OnWar
On War by Carol von Clausewitz

Princeton University Press’s best-selling books for the past week

 

These are the best-selling books for the past week.

1177 BC: The Year Civilization Collapsed by Eric H. Cline
 Carlson_Tesla jacket Tesla: Inventor of the Electrical Age by W. Bernard Carlson
DarwinFinches
40 Years of Evolution: Darwin’s Finches on Daphne Major Island by Peter R. Grant and B. Rosemary Grant
Fernandez_Everyday cover Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
Fawcett_Liberalism_S14 Liberalism: The Life of an Idea by Edmund Fawcett
The Five Elements of Effective Thinking
The 5 Elements of Effective Thinking by Edward B. Burger and Michael Starbird
I Ching The I Ching or Book of Changes, edited by Hellmut Wilhelm, translated by Cary F. Baynes
thebox The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger by Marc Levinson
OnBullshit On Bullshit by Harry G. Frankfurt
shtetl
The Golden Age Shtetl: A New History of Jewish Life in East Europe by Yohanan Petrovsky-Shtern

 

 

Paying It Forward, Using Math: Oscar Fernandez’s ‘Everyday Calculus’ Donated to Libraries in Franklin County, PA

Everyday Calculus, O. FernandezWhat a week!

It was recently announced that one of our books, Everyday Calculus by Oscar Fernandez, is to be donated by the United Way of Franklin County, in partnership with the Franklin County Library System, to public libraries all throughout Franklin County. The decision recognizes the 2013 Campaign Chair, Jim Zeger, who has demonstrated a dedication to service and a “willingness to teach others” during the course of his four-year tenure on the board of directors.

But the choice of text was far from random; Everyday Calculus was selected “because of the need for materials that support financial and mathematical literacy within our library systems,” says Mr. Zeger. He’s one to know; before coming to United Way, Zeger studied math at Juniata College and taught mathematics at the Maryland Correctional Institute. He also served for a number of years as part of the Tuscarora School District school board, and “is very supportive and understanding of the value of relating and connecting applied math to students.”

Bernice Crouse, executive director of the Franklin County Library System, accepted the books and has found them a place in each County library, including the bookmobile, in order to make them more accessible to readers. According to Crouse, this book fits perfectly with Pennsylvania Library Association’s PA Forward initiative, which “highlights Financial Literacy as a key to economic vitality in Pennsylvania.”

Mr. Fernandez is reportedly “delighted” and “honored” by the decision, and looks forward to further collaborating with United Way.

Princeton University Press’s best-selling books for the past week

These are the best-selling books for the past week.

1177 BC: The Year Civilization Collapsed by Eric H. Cline
shtetl The Golden Age Shtetl: A New History of Jewish Life in East Europe by Yohanan Petrovsky-Shtern
Fawcett_Liberalism_S14
Liberalism: The Life of an Idea by Edmund Fawcett
Fernandez_Everyday cover Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
Carlson_Tesla jacket Tesla: Inventor of the Electrical Age by W. Bernard Carlson
thebox
The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger by Marc Levinson
OnBullshit On Bullshit by Harry G. Frankfurt
The Five Elements of Effective Thinking The 5 Elements of Effective Thinking by Edward B. Burger and Michael Starbird
I Ching The I Ching or Book of Changes, edited by Hellmut Wilhelm, translated by Cary F. Baynes
SouloftheWorld
The Soul of the World by Roger Scruton

 

Princeton University Press’s best-selling books for the past week

 

Princeton University Press’s best selling books for the past week

These are the best-selling books for the past week.

1177 BC: The Year Civilization Collapsed by Eric H. Cline
 7-17 Government Why Government Fails So Often: And How It Can Do Better by Peter H. Schuck
CountLikeanEgyptian
Count Like an Egyptian: A Hands-On Introduction to Ancient Mathematics by David Reimer
Carlson_Tesla jacket Tesla: Inventor of the Electrical Age by W. Bernard Carlson
Fernandez_Everyday cover Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez
Evans_Beetles
Beetles of Eastern North America by Arthur V. Evans
SouloftheWorld The Soul of the World by Roger Scruton
FaustI&II Faust I & II by Johann Wolfgang von Goethe
OnBullshit On Bullshit by Harry G. Frankfurt
Osterhammel_Transformation The Transformation of the World: A Global History of the Nineteenth Century by Jürgen Osterhammel (trans. Patrick Camiller)