A peek inside The Calculus of Happiness

What’s the best diet for overall health and weight management? How can we change our finances to retire earlier? How can we maximize our chances of finding our soul mate? In The Calculus of Happiness, Oscar Fernandez shows us that math yields powerful insights into health, wealth, and love. Moreover, the important formulas are linked to a dozen free online interactive calculators on the book’s website, allowing one to personalize the equations. A nutrition, personal finance, and relationship how-to guide all in one, The Calculus of Happiness invites you to discover how empowering mathematics can be. Check out the trailer to learn more:

The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love, Oscar E. Fernandez from Princeton University Press on Vimeo.

FernandezOscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of Everyday Calculus: Discovering the Hidden Math All around Us. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.

Oscar E. Fernandez on The Calculus of Happiness

FernandezIf you think math has little to do with finding a soulmate or any other “real world” preoccupations, Oscar Fernandez says guess again. According to his new book, The Calculus of Happiness, math offers powerful insights into health, wealth, and love, from choosing the best diet, to finding simple “all weather” investment portfolios with great returns. Using only high-school-level math (precalculus with a dash of calculus), Fernandez guides readers through the surprising results. He recently took the time to answer a few questions about the book and how empowering mathematics can be.

The title is intriguing. Can you tell us what calculus has to do with happiness?

Sure. The title is actually a play on words. While there is a sprinkling of calculus in the book (the vast majority of the math is precalculus-level), the title was more meant to convey the main idea of the book: happiness can be calculated, and therefore optimized.

How do you optimize happiness?

Good question. First you have to quantify happiness. We know from a variety of research that good health, healthy finances, and meaningful social relationships are the top contributors to happiness. So, a simplistic “happiness equation” is: health + wealth + love = happiness. This book then does what any good applied mathematician would do (I’m an applied mathematician): quantify each of the “happiness components” on the left-hand side of the equation (health, wealth, and love), and then use math to extract valuable insights and results, like how to optimize each component.

This process sounds very much like the subtitle, how a mathematical approach to life adds up to health, wealth, and love. But just to be sure, can you elaborate on the subtitle?

That’s exactly right. Often we feel like various aspects of our lives are beyond our control. But in fact, many aspects of our lives, including some of the most important ones (like health, wealth, and love), follow mathematical rules. And by studying the equations that emerge from these rules you can quickly learn how to manipulate those equations in your favor. That’s what I do in the book for health, wealth, and love.

Can you give us some examples/applications?

I can actually give you about 30 of them, roughly the number discussed in the book. But let me focus on my three favorite ones. The first is what I called the “rational food choice” function (Chapter 2). It’s a simple formula: divide 100 calories by the weight (say, in grams) of a particular food. This yields a number whose units are calories per gram, the units of “energy density.” Something remarkable then happens when you plot the energy densities of various foods on a graph: the energy densities of nearly all the healthy foods (like fruits and vegetables) are at most about 2 calories per gram. This simple mathematical insight, therefore, helps you instantly make healthier food choices. And following its advice, as I discuss at length in the book, eventually translates to lower risk for developing heart disease and diabetes, weight loss, and even an increase in your life span! The second example comes from Chapter 3; it’s a formula for calculating how many more years you have to work for before you can retire. Among the formula’s many insights is that, in the simplest case, this magic number depends entirely on the ratio of how much you save each year to how much you spend. And the formula, being a formula, tells you exactly how changing that ratio affects your time until retirement. The last example is based on astronomer Frank Drake’s equation for estimating the number of intelligent civilizations in our galaxy (Chapter 5). It turns out that this alien-searching equation can also be used to estimate the number of possible compatible partners that live near you! That sort of equates a good date with an intelligent alien, and I suppose I can see some similarities (like how rare they are to find).

The examples you’ve mentioned have direct relevance to our lives. Is that a feature of the other examples too?

Absolutely. And it’s more than just relevance—the examples and applications I chose are all meant to highlight how empowering mathematics can be. Indeed, the entire book is designed to empower the reader—via math—with concrete, math-backed and science-backed strategies for improving their health, wealth, and love life. This is a sampling of the broader principle embodied in the subtitle: taking a mathematical approach to life can help you optimize nearly every aspect of your life.

Will I need to know calculus to enjoy the book?

Not at all. Most of the math discussed is precalculus-level. Therefore, I expect that nearly every reader will have studied the math used in the book at some point in their K-12 education. Nonetheless, I guide the reader through the math as each chapter progresses. And once we get to an important equation, you’ll see a little computer icon next to it in the margin. These indicate that there are online interactive demonstrations and calculators I created that go along with the formula. The online calculators make it possible to customize the most important formulas in the book, so even if the math leading up to them gets tough, you can still use the online resources to help you optimize various aspects of health, wealth, and love.

Finally, you mention a few other features of the book in the preface. Can you tell us about some of those?

Sure, I’ll mention two particular important ones. Firstly, at least 1/3 of the book is dedicated to personal finance. I wrote that part of the book to explicitly address the low financial literacy in this country. You’ll find understandable discussions of everything from taxes to investing to retirement (in addition to the various formulas derived that will help you optimize those aspects of your financial life). Finally, I organized the book to follow the sequence of math topics covered in a typical precalculus textbook. So if you’re a precalculus student, or giving this book to someone who is, this book will complement their course well. (I also included the mathematical derivations of the equations presented in the chapter appendixes.) This way the youngest readers among us can read about how empowering and applicable mathematics can be. It’s my hope that this will encourage them to continue studying math beyond high school.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of Everyday Calculus: Discovering the Hidden Math All around Us and The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love.

This Halloween, a few books that won’t (shouldn’t!) die

If Halloween has you looking for a way to combine your love (or terror) of zombies and academic books, you’re in luck: Princeton University Press has quite a distinguished publishing history when it comes to the undead.

 

As you noticed if you follow us on Instagram, a few of our favorites have come back to haunt us this October morning. What is this motley crew of titles doing in a pile of withered leaves? Well, The Origins of Monsters offers a peek at the reasons behind the spread of monstrous imagery in ancient empires; Zombies and Calculus  features a veritable course on how to use higher math skills to survive the zombie apocalypse, and International Politics and Zombies invites you to ponder how well-known theories from international relations might be applied to a war with zombies. Is neuroscience your thing? Do Zombies Dream of Undead Sheep? shows how zombism can be understood in terms of current knowledge regarding how the brain works. Or of course, you can take a trip to the graveyard of economic ideology with Zombie Economics, which was probably off marauding when this photo was snapped.

If you’re feeling more ascetic, Black: The History of a Color tells the social history of the color black—archetypal color of darkness and death—but also, Michel Pastoureau tells us, monastic virtue. A strikingly designed choice:

In the beginning was black, Michel Pastoureau tells us in Black: A History of a Color

A post shared by Princeton University Press (@princetonupress) on

 

Happy Halloween, bookworms.

Raffi Grinberg: Survival Techniques for Proof-Based Math

GrinbergReal analysis is difficult. In addition to learning new material about real numbers, topology, and sequences, most students are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver by Raffi Grinberg is an innovative guide that helps students through their first real analysis course while giving them a solid foundation for further study. Below, Grinberg offers an introduction to proof-based math:

 

 

 

Raffi Grinberg is an entrepreneur and former management consultant. He graduated with honors from Princeton University with a degree in mathematics in 2012. He is the author of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs.

Why Calculus Will Save You from the Zombie Apocalypse

To survive a zombie apocalypse, one will need more than instinct and short term solutions – one will need logic and, most importantly, math. A thought-out plan comprised of sophisticated calculus equations will ensure long-term safety objectives. Thankfully, Zombies and Calculus by Colin Adams colorfully illustrates the critical implementation of calculus components when going head-to-head with zombies. Adams demonstrates how a professor and his students successfully exercise calculus to survive the attacks of zombies who not only disrupt their calculus class (the horror!), but are also out for human flesh.

Here are a few need-to-knows:

Zombies travel approximately at one yard per second – a constant derivative.

A derivative of a function is its rate of change. If a function is changing quickly, its derivate will be high, while if a function is changing slowly, its derivate will be low. Adams explains that we can measure the function’s rate of change through the steepness of the tangent line. zombies and calculus rate of change
Since speed is defined as distance divided by time, one can calculate the speed required to get from Point A to Point B in a specific time, while being able to evade any unwanted visitors (zombies). Keep in mind — speed tends to vary (not for zombies, remember, they travel in a constant derivative!), so the derivate of the function has the potential to increase or decrease. Using these simple formulas, one is able to plan out the distance, time, and speed needed to outrun these deadly predators.

It’s hard to crack a zombie’s skull. It’s easier to knock a zombie unconscious.

As detailed in Zombies and Calculus, the amount of force necessary to crack a human skull is 10,000 newtons (a newton is a measurement for force that equals 1 kilogram meter per second squared). Adams offers an example: if a baseball is going 90 miles per hour (40.2 meters/second), weighs 5 ounces (0.145 kilograms), and comes into contact with a head for .007 seconds, its force can be calculated through:Screen Shot 2015-10-29 at 4.44.31 PMSo since a baseball, with said specifications, can only create approximately 800 newtons, imagine how much force is needed to produce 10,000 newtons! When attacking a zombie with force, do not try to go for the easy kill — rather play strategically by knocking the zombie unconscious with a sudden sharp blow to the head. This will create a dramatic head jerk, causing the brain to get knocked around in the cranial cavity, thus causing a short circuit. The benefit of knocking a zombie unconscious, of course, is additional planning and escape time!

Zombies pursue in a radiodrome path.

Like a dog pursues a rabbit, a zombie pursues its human prey. A zombie will follow its prey’s path at the prey’s given location at that specific instant. In a scene from Zombies and Calculus, (pause to imagine it), a Dean is running towards the safe haven of an academic building in a straight line. However, a zombie is present and begins to pursue the Dean, always having its tangent vector pointing at the Dean. The zombie is going to travel to wherever the Dean is in that current moment. Screen Shot 2015-10-29 at 4.39.23 PM

Since zombies are incapable of developing an efficient plan, the zombie does not run at a diagonal towards the academic building, which would cut-off the dean’s path. Instead of recognizing the Dean’s travel pattern or destination, the zombie is chasing the dean like a dog chasing a bunny’s tail to the rabbit hole. If only the dog knew that its radiodrome procedure was flawed, the dog would be able (with a speed higher than the rabbit) to cut-off the rabbit at its hole and claim victory. If dogs were to catch on, there would probably be fewer bunnies hopping around.

Cold-blooded creatures are unable to regulate their body heat.

Like other cold-blooded creatures, zombies hibernate. A zombie’s body temperature will decrease according to the differential equation that guides the temperature change of an object placed in a space with a different temperature (so for instance, if a zombie with a temperature of 60 degrees is placed a room of 30 degrees.) According to Newton’s Law of Cooling (remember Newton from discussing the measurement ‘newton’ for force?), the temperature of a body’s rate of change is proportional to the difference between the present temperature of that body and the ambient temperature (basically, the temperature of its surroundings). Given as a function of time, the zombie’s temperature (where Tg is the specific location):Screen Shot 2015-10-29 at 4.42.31 PMThe larger the contrast of temperatures, the faster the body temperature will drop. As the characters in the book discover, if there is a zombie apocalypse, it might be time to consider a move to our friendly neighbor to the north, Canada.

 

Zombies and CalculusTo discover more lifesaving tips, fun and entertaining mathematical applications, and learn the fate of the brave calculus professor and his students, read Colin Adam’s  Zombies and Calculus. Just in case the zombie apocalypse does occurs (maybe tomorrow?) it should be comforting to know there’s a mathematical guide to survival on your bookshelf.

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/.

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
 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