Global Math Week: The Universal Language

by Oscar Fernandez

FernandezFill in the blank: Some people speak English, some speak French, and some speak ____. I doubt you said “math.” Yet, as I will argue, the thought should have crossed your mind. And moreover, the fact that mathematics being a language likely never has, speaks volumes about how we think of math, and why we should start thinking of it—and teaching it—as a language.

To make my point, consider the following fundamental characteristics shared by most languages:

  •  A set of words or symbols (the language’s vocabulary)
  •  A set of rules for how to use these words or symbols (the language’s rules of grammar)
  •  A set of rules for combining these words or symbols to make statements (the language’s syntax)

Now think back to the math classes you have taken. I bet you will soon remember each of these characteristics present throughout your courses. (For instance, when you learned that 𝑎2 means 𝑎 × 𝑎, you were learning how to combine some of the symbols used in mathematics to make a statement—that the square of a number is the number multiplied by itself.) Indeed, viewed this way, every mathematics lesson can be thought of as a language lesson: new vocabulary, rules of grammar, or syntax is introduced; everyone then practices the new content; and the cycle repeats. By extension, every mathematics course can be thought of as a language course.

Now that I have you thinking of mathematics as a language, let me point out the many benefits of this new viewpoint. For one, this viewpoint helps dispel many myths about the subject. For instance, travel to any country and you will find a diverse set of people speaking that country’s language. Some are smarter than others; some are men and some women; perhaps some are Latino and some Asian. Group them as you wish, they will all share the capacity to speak the same language. The same is true of mathematics. It is not a subject accessible only to people of certain intelligence, sex, or races; we all have the capacity to speak mathematics. And once we start thinking of the subject as a language, we will recognize that learning mathematics is like learning any other language: all you need are good teachers, and lots of practice. And while mastering a language is often the endpoint of the learning process, mastering the language that is mathematics will yield much larger dividends, including the ability to express yourself precisely, and the capacity to understand the Universe. As Alfred Adler put it: “

Mathematics is pure language – the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms.” Galileo—widely regarded the father of modern science—once wrote that Nature is a great book “written in the language of mathematics” (The Assayer, 1623). Centuries later, Einstein, after having discovered the equation for gravity using mathematics, echoed Galileo’s sentiment, writing: “pure mathematics is, in its way, the poetry of logical ideas” (Obituary for Emmy Noether, 1935). Most of us today wouldn’t use words like “language” and “poetry” to describe mathematics. Yet, as I will argue, we should. And moreover, we should start thinking of—and teaching—math as a language.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.

Global Math Week: Counting on Math

by Tim Chartier

The Global Math Project has a goal of sharing the joys of mathematics to 1 million students around the world from October 10th through the 17th. As we watch the ever-increasing number of lives that will share in math’s wonders, let’s talk about counting, which is fundamental to reaching this goal.

Let’s count. Suppose we have five objects, like the plus signs below. We easily enough count five of them.

 

 

 

You could put them in a hat and mix them up.

 

 

 

 

 

 

 

If you take them out, they might be jumbled but you’d still have five.

 

 

 

 

 

 

 

 

 

 

Easy enough! Jumbling can induce subtle complexities, even to something as basic as counting.

Counting to 14 isn’t much more complicated than counting to five. Be careful as it depends what you are counting and how you jumble things! Verify there are 14 of Empire State Buildings in the picture below.

 

 

 

 

 

 

 

If you cut out the image along the straight black lines, you will have three pieces to a puzzle. If you interchange the left and right pieces on the top row, then you get the configuration below. How many buildings do you count now? Look at the puzzle carefully and see if you can determine how your count changed.

 

 

 

 

 

 

 

Can you spot any changes in the buildings in the first versus the second pictures? How we pick up an additional image is more easily seen if we reorder the buildings. So, let’s take the 14 buildings and reorder them as seen below.

 

 

 

 

 

 

 

Swapping the pieces on the top row of the original puzzle has the same effect as shifting the top piece in the picture above. Such a shift creates the picture below. Notice how we pick up that additional building. Further, each image loses 1/14th of its total height.

 

 

 

 

 

 

 

Let’s look at the original puzzle before and after the swap.

 

 

 

 

 

 

 

 

 

 

 

 

 

This type of puzzle is called a Dissection Puzzle. Our eyes can play tricks on us. We know 14 doesn’t equal 15 so something else must be happening when a puzzle indicates that 14 = 15. Mathematics allows us to push through assumptions that can lead to illogical conclusions. Math can also take something that seems quite magical and turn it into something very logical — even something as fundamental as counting to 14.

Want to look at counting through another mathematical lens? A main topic of the Global Math Project will be exploding dots. Use a search engine to find videos of James Tanton introducing exploding dots. James is a main force behind the Global Math Project and quite simply oozes joy of mathematics. You’ll also find resources at the Global Math Project web page. Take the time to look through the Global Math Project resources and watch James explain exploding dots, as the topic can be suitable from elementary to high school levels. You’ll enjoy your time with James. You can count on it!

ChartierTim Chartier is associate professor of mathematics at Davidson College. He is the coauthor of Numerical Methods and the author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing.

Global Math Week: Around the World from Unsolved to Solved

by Craig Bauer

BauerWhat hope do we have of solving ciphers that go back decades, centuries, or even all the way back to the ancient world? Well, we have a lot more hope than we did in the days before the Internet. Today’s mathematicians form a global community that poses a much greater threat to unsolved problems, of every imaginable sort, than they have every faced before.

In my Princeton University Press book, Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies, I collected scores of the most intriguing unsolved ciphers. It’s a big book, in proper proportion to its title, and I believe many of the ciphers in it will fall to the onslaught the book welcomes from the world’s codebreakers, both professionals and amateurs. Why am I making this prediction with such confidence? Well, I gave a few lectures based on material from the book, while I was still writing it, and the results bode well for the ciphers falling.

Here’s what happened.

Early in the writing process, I was invited to give a lecture on unsolved ciphers at the United States Naval Academy. I was surprised, when I got there, by the presence of a video camera. I was asked if I was okay with the lecture being filmed and placed on YouTube. I said yes, but inside I was cursing myself for not having gotten a much needed haircut before the talk. Oh well. Despite my rough appearance, the lecture went well.[1] I surveyed some of the unsolved ciphers that I was aware of at the time, including one that had been put forth by a German colleague and friend of mine, Klaus Schmeh. It was a double transposition cipher that he had created himself to show how difficult it is to solve such ciphers. He had placed it in a book he had written on unsolved ciphers, a book which is unfortunately only available in German.[2] But to make the cipher as accessible as possible, he assured everyone that that particular bit of writing was in English.

 

VESINTNVONMWSFEWNOEALWRNRNCFITEEICRHCODEEA

HEACAEOHMYTONTDFIFMDANGTDRVAONRRTORMTDHE

OUALTHNFHHWHLESLIIAOETOUTOSCDNRITYEELSOANGP

VSHLRMUGTNUITASETNENASNNANRTTRHGUODAAARAO

EGHEESAODWIDEHUNNTFMUSISCDLEDTRNARTMOOIREEY

EIMINFELORWETDANEUTHEEEENENTHEOOEAUEAEAHUHI

CNCGDTUROUTNAEYLOEINRDHEENMEIAHREEDOLNNIRAR

PNVEAHEOAATGEFITWMYSOTHTHAANIUPTADLRSRSDNOT

GEOSRLAAAURPEETARMFEHIREAQEEOILSEHERAHAOTNT

RDEDRSDOOEGAEFPUOBENADRNLEIAFRHSASHSNAMRLT

UNNTPHIOERNESRHAMHIGTAETOHSENGFTRUANIPARTAOR

SIHOOAEUTRMERETIDALSDIRUAIEFHRHADRESEDNDOION

ITDRSTIEIRHARARRSETOIHOKETHRSRUAODTSCTTAFSTHCA

HTSYAOLONDNDWORIWHLENTHHMHTLCVROSTXVDRESDR

Figure 1. Klaus Schmeh’s double transposition cipher challenge.

When the YouTube video went online, it was seen by an Israeli computer scientist, George Lasry, who became obsessed with it. He was not employed at the time, so he was able to devote a massive amount of time to seeking the solution to this cipher. As is natural for George, he attacked it with computer programs of his own design. He eventually found himself doing almost nothing other than working on the cipher. His persistence paid off and he found himself reading the solution.

I ended up being among the very first to see George’s solution, not because I’m the one who introduced him to the challenge via the YouTube video, but because I’m the editor-in-chief of the international journal (it’s owned by the British company Taylor and Francis) Cryptologia. This journal covers everything having to do with codes and ciphers, from cutting edge cryptosystems and attacks on them, to history, pedagogy, and more. Most of the papers that appear in it are written by men and women who live somewhere other than America and it was to this journal that George submitted a paper describing how he obtained his solution to Klaus’s challenge.

George’s solution looked great to me, but I sent it to Klaus to review, just to be sure. As expected, he was impressed by the paper and I queued it up to see print. The solution generated some media attention for George, which led to him being noticed by people at Google (an American company, of course). They approached him and, after he cleared the interviewing hurdles, offered him a position, which he accepted. I was very happy that George found the solution, but of course that left me with one less unsolved cipher to write about in my forthcoming book. Not a problem. As it turns out there are far more intriguing unsolved ciphers than can be fit in a single volume. One less won’t make any difference.

Later on, but still before the book saw print, I delivered a similar lecture at the Charlotte International Cryptologic Symposium held in Charlotte, North Carolina. This time, unlike at the Naval Academy, Klaus Schmeh was in the audience.

One of the ciphers that I shared was fairly new to me. I had not spoken about it publicly prior to this event. It appeared on a tombstone in Ohio and seemed to be a Masonic cipher. It didn’t look to be sophisticated, but it was very short and shorter ciphers are harder to break. Brent Morris, a 33rd degree Mason with whom I had discussed it, thought that it might be a listing of initials of offices, such as PM, PHP, PIM (Past Master, Past High Priest, Past Illustrious Master), that the deceased had held. This cipher was new to Klaus and he made note of it and later blogged about it. Some of his followers collaborated in an attempt to solve it and succeeded. Because I hadn’t even devoted a full page to this cipher in my book, I left it in as a challenge for readers, but also added a link to the solution for those who want to see the solution right away.

Bauer

Figure 2. A once mysterious tombstone just south of Metamora, Ohio.

So, what was my role in all of this? Getting the ball rolling, that’s all. The work was done by Germans and an Israeli, but America and England benefited as well, as Google gained yet another highly intelligent and creative employee and a British owned journal received another great paper.

I look forward to hearing from other people from around the globe, as they dive into the challenges I’ve brought forth. The puzzles of the past don’t stand a chance against the globally networked geniuses of today!

Craig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

 

[1] It was split into two parts for the YouTube channel. You can see them at https://www.youtube.com/watch?v=qe0JhEajfj8 (Part 1) and https://www.youtube.com/watch?v=5L12gjgMOMw (Part 2). A few years later, I got cleaned up and delivered an updated version of the talk at the International Spy Museum. That talk, aimed at a wider audience, may be seen at https://www.youtube.com/watch?v=rsdUDdkjdQg.

[2] Schmeh, Klaus, Nicht zu Knacken, Carl Hanser Verlag, Munich, 2012.

Vickie Kearn kicks off Global Math Week

October 10 – 17 marks the first ever Global Math Week. This is exciting for many reasons and if you go to the official website, you’ll find that there are already 736,546—and counting— reasons there. One more: PUP will be celebrating with a series of posts from some of our most fascinating math authors, so check this space tomorrow for the first, on ciphers, by Craig Bauer. Global Math Week provides a purposeful opportunity to have a global math conversation with your friends, colleagues, students, and family.

Mathematics is for everyone, as evidenced in the launch of Exploding Dots, which James Tanton brilliantly demonstrates at the link above. It is a mathematical story that looks at math in a new way, from grade school arithmetic, all the way to infinite sums and on to unsolved problems that are still stumping our brightest mathematicians. Best of all, you can ace this and no longer say “math is hard”, “math is boring”, or “I hate math”.

Vickie Kearn visits the Great Wall during her trip to our new office in Beijing

I personally started celebrating early as I traveled to Beijing in August to attend the Beijing International Book Fair. I met with the mathematics editors at a dozen different publishers to discuss Chinese editions of our math books. Although we did not speak the same language, we had no trouble communicating. We all knew what a differential equation is and a picture in a book of a driverless car caused lots of hand clapping. I was thrilled to be presented with the first Chinese editions of two books written by Elias Stein (Real Analysis and Complex Analysis) from the editor at China Machine Press. Although I love getting announcements from our rights department that one of our math books is being translated into Chinese, Japanese, German, French, etc., there is nothing like the thrill I had of meeting the people who love math as much as I do and who actually make our books come to life for people all over the world.

Because Princeton University Press now has offices in Oxford and Beijing, in addition to Princeton, and because I go to many conferences each year, I am fortunate to travel internationally and experience global math firsthand. No matter where you live, it is possible to share experiences through doing math. I urge you to visit the Global Math Project website and learn how to do math(s) in a global way.

Check back tomorrow for the start of our PUP blog series on what doing math globally means to our authors. Find someone who says they don’t like math and tell them your global math story.

Craig Bauer on unsolved ciphers

In 1953, a man was found dead from cyanide poisoning near the Philadelphia airport with a picture of a Nazi aircraft in his wallet. Taped to his abdomen was an enciphered message. In 1912, a book dealer named Wilfrid Voynich came into possession of an illuminated cipher manuscript once belonging to Emperor Rudolf II, who was obsessed with alchemy and the occult. Wartime codebreakers tried—and failed—to unlock the book’s secrets, and it remains an enigma to this day. In Unsolved, Craig Bauer examines these and other vexing ciphers yet to be cracked. Recently he took the time to answer some questions about his new book.

Why focus on unsolved ciphers?

They’re much more intriguing because they could be concealing anything. Some might reveal the identities of serial killers. Others could unmask spies, rewrite history, expose secret societies, or even give the location of buried treasure worth millions. This sense of mystery is very appealing to me.

Did you try to solve the ciphers yourself first?

There are so many unsolved ciphers that I realized I would never finish writing about them if I kept stopping to try to solve them. There’s one that I’m confident I could solve, but instead of doing so, I simply presented the approach I think will work and am leaving it for a reader to pursue. I expect that several of them will be solved by readers and I look forward to seeing their results!

Does someone who wants to attack these mysteries need to know a lot of mathematics or have computer programming skills?

No. Many of the ciphers were created by people with very little knowledge in either area. Also, past solvers of important ciphers have included amateurs. One of the Zodiac killer’s ciphers was solved by a high school history teacher. Some of the ciphers might be solved in a manner that completely bypasses mathematics. A reader may find a solution through papers the cipher’s creator left behind, perhaps in some library’s archives, in government storage, or in a relative’s possession. I think some may be solved by pursuing a paper trail or some other non-mathematical avenue. Of course, there are mathematical challenges as well, for those who have the skills to take them on. The puzzles span thousands of years, from ancient Egypt to today’s online community. Twentieth century challenges come from people as diverse as Richard Feynman (a world-class physicist) and Ricky McCormick (thought to have been illiterate).

Are all of the unsolved ciphers covered in the book?

No, far from it. There are enough unsolved ciphers to fill many volumes. I limited myself to only the most interesting examples, and still there were too many! I originally set out to write a book about half the size of what was ultimately published. The problem was that there was so much fascinating material that I had to go to 600 pages or experience the agony of omitting something fabulous. Also, unsolved ciphers from various eras are constantly coming to light, and new ones are created every year. I will likely return to the topic with a sequel covering the best of these.

Which cipher is your favorite?

I’m the most excited about the Paul Rubin case. It involves a cipher found taped to the abdomen of a teenage whiz-kid who was found dead in a ditch by the Philadelphia airport, way back in 1953. While I like well-known unsolved ciphers like the Voynich Manuscript and Kryptos, I have higher hopes for this one being solved because it hasn’t attracted any attention since the 1950s. The codebreakers have made a lot of progress since then, so it’s time to take another look and see what can be learned about this young man’s death. I felt it was very important to include cases that will be new even to those who have read a great deal about cryptology already and this is one such case.

Should the potential reader have some prior knowledge of the subject?

If he or she does, there will still be much that is new, but for those with no previous exposure to cryptology, everything is explained from the ground up. As a teenager I loved books at the popular level on a wide range of topics. In particular, the nonfiction of Isaac Asimov instilled in me a love for many subjects. He always started at the beginning, assuming his readers were smart, but new to the topic he was covering. This is the approach that I have taken. I hope that the book finds a wide readership among the young and inspires them in the same way Asimov inspired me.

Is there anything that especially qualifies you to write on this topic?

Early work on this book was supported by the National Security Agency through their Scholar-in-Residence program at the Center for Cryptologic History. They wanted me in this role because, while I have a PhD in mathematics and have carried out mathematical research in cryptology, I also have a passion for history and other disciplines. In fact, both of my books have the word “history” in their titles. The journal Cryptologia, for which I serve as the editor-in-chief, is devoted to all aspects of cryptology, mathematical, historical, pedagogical, etc. My love of diverse fields allows me to write with enthusiasm about ciphers in music, art, criminal cases, ancient history, and other areas. The broad approach to the subject is more entertaining and ensures that there’s something in the book for nearly every reader.

BauerCraig Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

Craig Bauer: The Ongoing Mystery of Unsolved Ciphers (and new hope)

When a civilization first develops writing and few people are literate, simply putting a message down on paper can be all that is required to keep an enemy from understanding it. As literacy spreads, a more sophisticated method is needed, which is why codes and ciphers, a.k.a. “secret writing,” always follow closely on the heels of the discovery of writing. Over the millennia, ciphers have become extremely sophisticated, but so too have the techniques used by those attempting to break them.

In recent decades, everyone from mathematicians and computer scientists to artists and authors have created ciphers as challenges to specialists or the general public, to see if anyone is clever enough to unravel the secrets. Some, like the first three parts of James Sanborn’s sculpture Kryptos and the ciphers appearing in the television show Gravity Falls, have been solved, while others remain mysteries. The highly secretive online society known as Cicada 3301 has repeatedly issued such challenges as a means of talent scouting, though for what purpose such talented individuals are sought remains unknown. One unsolved cipher was laid down as a challenge by former British army intelligence officer Alexander d’Agapeyeff in his book Codes & Ciphers (1939). Sadly, when frustrated letters of enquiry reached the author, he admitted that he had forgotten how to solve it! Another was made by the famous composer Edward Elgar in 1897 as a riddle for a young lady friend of his. She, along with various experts, all failed to ferret out the meaning and Elgar himself refused to reveal it.

 

Elgar's cipher

Elgar’s cipher

 

Many unsolved ciphers appear in much more serious contexts. The serial killer who referred to himself as “The Zodiac” was responsible for at least five murders, as well as the creation of several ciphers sent to San Francisco newspapers. While the first of these ciphers was solved, others remain unbroken. Could a solution to one of these lead to an identification of the killer? Although many have speculated on his identity, it has never been firmly established. The Zodiac is not the only murderer to have left us such mysterious communiques, he is just the best known. Other killers’ secrets have persisted through relative obscurity. How many readers have heard of Henry Debsonys? In 1883, a jury sentenced him to death for the murder of his wife, after deliberating for only nine minutes. But this unfortunate woman was Henry’s third wife and the first two died under strange circumstances. Had Henry killed all of them? Will the ciphers he left behind confirm this? I think his ciphers will be among the first to fall this year, thanks to a major clue I provide in my book, Unsolved: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. There are many more such criminal ciphers. One deranged individual even sent threatening letters containing ciphers to John Walsh of America’s Most Wanted fame! The FBI’s codebreakers maintain a list of their top unsolved ciphers. At present, only two of these are known to the public, but many others that didn’t make the top 10 are available for anyone to try to crack.

How do codebreakers, whether amateur or professional, meet the challenges they face? Statistics and other areas of mathematics often help, as do computers, but two of the codebreakers’ most powerful tools are context and intuition. This is why ciphers have often been broken by amateurs with no programming skills and little knowledge of mathematics. Enter Donald Harden, a high school history teacher, who with assistance from his wife Bettye, broke one of the Zodiac killer’s ciphers by guessing that the egotistical killer’s message would begin with “I” and contain the word “KILL.” Context allows the attacker to guess words, sometimes entire phrases, that might appear in the message. These are known as cribs. During World War II, the German word eins (meaning one) appeared in so many Nazi messages that a process known as “einsing” was developed, searching the cipher for the appearance of this word in every possible position. In today’s ciphers, the word President appears frequently.

Of course, time and again cribs and intuition can lead in the wrong direction. Indeed, the single most important attribute for a codebreaker is patience. A good codebreaker will have the ability to work on a cipher for months, for that is sometimes what it takes to reach a solution, ignoring the body’s normal demands for food and sleep; during World War I, the French codebreaker Georges Painvin lost 33 pounds over three months while sitting at a desk breaking the German ADFGX and ADFGVX ciphers.

Fig 2

Fig 3Is it possible that some of the earliest known ciphers, dating from the ancient world, have survived unread by anyone other than those they were created for? I believe this is the case and that they’ve been hiding in plain sight, like the purloined letter in Poe’s classic tale. Those studying ancient cultures have long been aware of so-called “nonsense inscriptions.” These appear on Egyptian sarcophagi, Greek vases, runestones, and elsewhere. They are typically dismissed as the work of illiterates imitating writing, merely because the experts cannot read them. But all of these cultures are known to have made use of ciphers and some of the contexts of the inscriptions are so solemn (e.g. sarcophagi) that it’s hard to believe they could be meaningless. I’d like to see a closer examination of these important objects. I expect some of the messages will be read in the near future, if cryptologists can form collaborations with linguists. These two groups have worked together successfully in military contexts for many decades. It is time that they also join forces for historical studies.

With a very large number of unsolved ciphers, spanning millennia, having been composed by a diverse group of individuals, it seems likely that it will take a diverse group of attackers, with skills ranging over many disciplines, to solve them. Some mysterious texts may reveal themselves to clever computer programmers or linguists, others to those taking the psychological approach, getting into the creator’s head and guessing phrases he or she used in the cipher, and some may be broken by readers who manage to discover related material in government archives or private hands that provides just enough extra information to make the break. I look forward to seeing the results!

BauerCraig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

Oscar Fernandez: A Healthier You is Just a Few Equations Away

This post appears concurrently on the Wellesley College Summer blog.

How many calories should you eat each day? What proportion should come from carbohydrates, or protein? How can we improve our health through diets based on research findings?

You might be surprised to find that we can answer all of these questions using math.  Indeed, mathematics is at the heart of nutrition and health research. Scientists in these fields often use math to analyze the results from their experiments and clinical trials.  Based on decades of research (and yes, math), scientists have developed a handful of formulas that have been proven to improve your health (and even help you lose weight!).

So, back to our first question: How many calories should we eat each day?  Let’s find out…

Each of us has a “total daily energy expenditure” (TDEE), the total number of calories your body burns each day. Theoretically, if you consume more calories than your TDEE, you will gain weight. If you consume less, you will lose weight. Eat exactly your TDEE in calories and you won’t gain or lose weight.

“Great! So how do I calculate my TDEE?” I hear you saying. Good question. Here’s a preliminary answer:

TDEE = RMR + CBE + DIT         (1)                                                                                                                                                                  

Here’s what the acronyms on the right-hand side of the equation mean.

  • RMR: Your resting metabolic rate, roughly defined as the number of calories your body burns while awake and at rest
  • CBE: The calories you burned during the day exercising (including walking)
  • DIT: Your diet’s diet-induced thermogenesis, which quantifies what percentage of calories from dietary fat, protein, and carbohydrates are left over for your body to use after you ingest those calories

So, in order to calculate TDEE, we need to calculate each of these three components. This requires very precise knowledge of your daily activities, for example: what exercises you did, how many minutes you spent doing them, what foods you ate, and how much protein, carbohydrates, and dietary fat these foods contained. Luckily, nutrition scientists have developed a simpler formula that takes all of these factors into account:

    TDEE = RMR(Activity Factor) + 0.1C.         (2)

Here C is how many calories you eat each day, and the “Activity Factor” (below) estimates the calories you burn through exercise:

 

Level of Activity Activity Factor
Little to no physical activity 1.2
Light-intensity exercise 1-3 days/week 1.4
Moderate-intensity exercise 3-5 days/week 1.5
Moderate- to vigorous-intensity exercise 6-7 days/week 1.7
Vigorous daily training 1.9

 

As an example, picture a tall young man named Alberto. Suppose his RMR is 2,000 calories, that he eats 2,100 calories a day, and that his Activity Factor is 1.2. Alberto’s TDEE estimate from (2) would then be

TDEE = 2,000(1.2) + 0.1(2,100) = 2,610.

Since Alberto’s caloric intake (2,100) is lower than his TDEE, in theory, Alberto would lose weight if he kept eating and exercising as he is currently doing.

Formula (2) is certainly more user-friendly than formula (1). But in either case we still need to know the RMR number. Luckily, RMR is one of the most studied components of TDEE, and there are several fairly accurate equations for it that only require your weight, height, age, and sex as inputs. I’ve created a free online RMR calculator to make the calculation easier: Resting Metabolic Heart Rate. In addition, I’ve also created a TDEE calculator (based on equation (2)) to help you estimate your TDEE: Total Daily Energy Expenditure.

I hope this short tour of nutrition science has helped you see that mathematics can be empowering, life-changing, and personally relevant. I encourage you to continue exploring the subject and discovering the hidden math all around you.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College. He is 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. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.

 

Joshua Holden: Quantum cryptography is unbreakable. So is human ingenuity

Two basic types of encryption schemes are used on the internet today. One, known as symmetric-key cryptography, follows the same pattern that people have been using to send secret messages for thousands of years. If Alice wants to send Bob a secret message, they start by getting together somewhere they can’t be overheard and agree on a secret key; later, when they are separated, they can use this key to send messages that Eve the eavesdropper can’t understand even if she overhears them. This is the sort of encryption used when you set up an online account with your neighbourhood bank; you and your bank already know private information about each other, and use that information to set up a secret password to protect your messages.

The second scheme is called public-key cryptography, and it was invented only in the 1970s. As the name suggests, these are systems where Alice and Bob agree on their key, or part of it, by exchanging only public information. This is incredibly useful in modern electronic commerce: if you want to send your credit card number safely over the internet to Amazon, for instance, you don’t want to have to drive to their headquarters to have a secret meeting first. Public-key systems rely on the fact that some mathematical processes seem to be easy to do, but difficult to undo. For example, for Alice to take two large whole numbers and multiply them is relatively easy; for Eve to take the result and recover the original numbers seems much harder.

Public-key cryptography was invented by researchers at the Government Communications Headquarters (GCHQ) – the British equivalent (more or less) of the US National Security Agency (NSA) – who wanted to protect communications between a large number of people in a security organisation. Their work was classified, and the British government neither used it nor allowed it to be released to the public. The idea of electronic commerce apparently never occurred to them. A few years later, academic researchers at Stanford and MIT rediscovered public-key systems. This time they were thinking about the benefits that widespread cryptography could bring to everyday people, not least the ability to do business over computers.

Now cryptographers think that a new kind of computer based on quantum physics could make public-key cryptography insecure. Bits in a normal computer are either 0 or 1. Quantum physics allows bits to be in a superposition of 0 and 1, in the same way that Schrödinger’s cat can be in a superposition of alive and dead states. This sometimes lets quantum computers explore possibilities more quickly than normal computers. While no one has yet built a quantum computer capable of solving problems of nontrivial size (unless they kept it secret), over the past 20 years, researchers have started figuring out how to write programs for such computers and predict that, once built, quantum computers will quickly solve ‘hidden subgroup problems’. Since all public-key systems currently rely on variations of these problems, they could, in theory, be broken by a quantum computer.

Cryptographers aren’t just giving up, however. They’re exploring replacements for the current systems, in two principal ways. One deploys quantum-resistant ciphers, which are ways to encrypt messages using current computers but without involving hidden subgroup problems. Thus they seem to be safe against code-breakers using quantum computers. The other idea is to make truly quantum ciphers. These would ‘fight quantum with quantum’, using the same quantum physics that could allow us to build quantum computers to protect against quantum-computational attacks. Progress is being made in both areas, but both require more research, which is currently being done at universities and other institutions around the world.

Yet some government agencies still want to restrict or control research into cryptographic security. They argue that if everyone in the world has strong cryptography, then terrorists, kidnappers and child pornographers will be able to make plans that law enforcement and national security personnel can’t penetrate.

But that’s not really true. What is true is that pretty much anyone can get hold of software that, when used properly, is secure against any publicly known attacks. The key here is ‘when used properly’. In reality, hardly any system is always used properly. And when terrorists or criminals use a system incorrectly even once, that can allow an experienced codebreaker working for the government to read all the messages sent with that system. Law enforcement and national security personnel can put those messages together with information gathered in other ways – surveillance, confidential informants, analysis of metadata and transmission characteristics, etc – and still have a potent tool against wrongdoers.

In his essay ‘A Few Words on Secret Writing’ (1841), Edgar Allan Poe wrote: ‘[I]t may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve.’ In theory, he has been proven wrong: when executed properly under the proper conditions, techniques such as quantum cryptography are secure against any possible attack by Eve. In real-life situations, however, Poe was undoubtedly right. Every time an ‘unbreakable’ system has been put into actual use, some sort of unexpected mischance eventually has given Eve an opportunity to break it. Conversely, whenever it has seemed that Eve has irretrievably gained the upper hand, Alice and Bob have found a clever way to get back in the game. I am convinced of one thing: if society does not give ‘human ingenuity’ as much room to flourish as we can manage, we will all be poorer for it.Aeon counter – do not remove

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology and the author of The Mathematics of Secrets.

This article was originally published at Aeon and has been republished under Creative Commons.

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.

Keith Devlin: Fibonacci introduced modern arithmetic —then disappeared

More than a decade ago, Keith Devlin, a math expositor, set out to research the life and legacy of the medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, whose book Liber abbaci has quite literally affected the lives of everyone alive today. Although he is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent—Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death. Finding Fibonacci is a compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin recently answered some questions about his new book for the PUP blog:

You’ve written 33 math books, including many for general readers. What is different about this one?

KD: This is my third book about the history of mathematics, which already makes it different from most of my books where the focus was on abstract concepts and ideas, not how they were discovered. What makes it truly unique is that it’s the first book I have written that I have been in! It is a first-person account, based on a diary I kept during a research project spread over a decade.

If you had to convey the book’s flavor in a few sentences, what would you say?

KD: Finding Fibonacci is a first-person account of a ten-year quest to uncover and tell the story of one of the most influential figures in human history. It started out as a diary, a simple record of events. It turned into a story when it became clear that it was far more than a record of dates, sources consulted, places visited, and facts checked. Like any good story, it has false starts and disappointments, tragedies and unexpected turns, more than a few hilarious episodes, and several lucky breaks. Along the way, I encountered some amazing individuals who, each for their own reasons, became fascinated by Fibonacci: a Yale professor who traced modern finance back to Fibonacci, an Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci’s astonishing story, an American math professor who fought against cancer to complete the world’s first (and only) modern language translation of Liber abbaci, and the widow who took over and brought his efforts to fruition after he lost that battle. And behind it all, the man who was the focus of my quest. Fibonacci played a major role in creating the modern commercial world. Yet he vanished from the pages of history for five hundred years, made “obsolete,” and in consequence all but forgotten forever, by a new technology.

What made you decide to write this book?

KD: There were really two key decisions that led to this book. One was deciding, back in the year 2000, to keep a diary of my experiences writing The Man of Numbers. My first history book was The Unfinished Game. For that, all I had to do was consult a number of reference works. It was not intended to be original research. Basic Books asked me to write a short, readable account of a single mathematical document that changed the course of human history, to form part of a series they were bringing out. I chose the letter Pierre De Fermat wrote to his colleague Blaise Pascal in 1654, which most experts agree established modern probability theory, in particular how it can be used to predict the future.

In The Man of Numbers, in contrast, I set out to tell a story that no one had told before; indeed, the consensus among the historians was that it could not be told—there simply was not enough information available. So writing that book would require engaging in a lot of original historical research. I had never done that. I would be stepping well outside my comfort zone. That was in part why I decided to keep a diary. The other reason for keeping a record was to ensure I had enough anecdotes to use when the time came to promote the book—assuming I was able to complete it, that is. (I had written enough popular mathematics books to appreciate the need for author promotional activities!)

The second decision, to turn my diary into a book (which only at the end found the title, Finding Fibonacci), came after The Man of Numbers was published in 2011. The ten-year process of researching and writing that book had turned out to be so rich, and so full of unexpected twists and turns, including several strokes of immense luck, that it was clear there was a good story to be told. What was not clear was whether I would be able to write such a book. All my other books are third-person accounts, where I am simply the messenger. In Finding Fibonacci, I would of necessity be a central character. Once again, I would be stepping outside my comfort zone. In particular, I would be laying out on the printed page, part of my inner self. It took five years and a lot of help from my agent Ted Weinstein and then my Princeton University Press editor Vickie Kearn to find the right voice and make it work.

Who do you expect will enjoy reading this book?

KD: I have a solid readership around the world. I am sure they will all read it. In particular, everyone who read The Man of Numbers will likely end up taking a look. Not least because, in addition to providing a window into the process of writing that earlier book, I also put in some details of that story that I did not fully appreciate until after the book had been published. But I hope, and in fact expect, that Finding Fibonacci will appeal to a whole new group of readers. Whereas the star of all my previous books was a discipline, mathematics, this is a book about people, for the most part people alive today. It’s a human story. It has a number of stars, all people, connected by having embarked on a quest to try to tell parts of the story of one of the most influential figures in human history: Leonardo of Pisa, popularly known as Fibonacci.

Now that the book is out, in one sentence if you can, how would you summarize writing it?

KD: Leaving my author’s comfort zone. Without a doubt. I’ve never been less certain how a book would be received.

DevlinKeith Devlin is a mathematician at Stanford University and cofounder and president of BrainQuake, an educational technology company that creates mathematics learning video games. His many books include The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern and The Man of Numbers: Fibonacci’s Arithmetic Revolution. He is the author of Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.

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.

PUP math editor Vickie Kearn: How real mathematicians celebrate Pi Day

Who doesn’t love Pi (aka Pie) Day? Residents here in Princeton, NJ love it so much that we spend four days celebrating. Now, to be honest, we’re also celebrating Einstein’s birthday, so we do need the full four days. I know what I will be doing on 3.14159265 but I wondered what some of my friends will be doing. Not surprisingly, a lot will either be making or eating pie. These include Oscar Fernandez (Wellesley), Ron Graham (UCSD), and Art Benjamin (who will be performing his mathemagics show later in the week). Anna Pierrehumbert (who teaches in NYC) will be working with upper school students on a pi recitation and middle school students on making pi-day buttons. Brent Ferguson (The Lawrenceville School) has celebrated at The National Museum of Mathematics in NYC, Ireland, Greece, and this year Princeton. Here he is celebrating in Alaska:

Pi

The Princeton University Math Club will be celebrating with a party in Fine Hall. In addition to eating pie and playing games, they will have a digit reciting contest. Tim Chartier (Davidson College) will be spending his time demonstrating how to estimate pi with chocolate chips while also fielding interview requests for his expert opinion on March Madness (a lot going on this month for mathematicians). Dave Richeson (Dickinson College) goes to the local elementary school each year and talks with the fifth graders about pi and its history and then eats creatively rendered pi themed pie provided by the parents.

You might be wondering why we celebrate a mathematical constant every year. How did it get to be so important? Again I went back to my pi experts and asked them to tell me the most important uses of pi. This question is open to debate by mathematicians but many think that the most important is Euler’s Identity, e(i*pi) + 1 = 0. As Jenny Kaufmann (President of the Princeton University Math Club) puts it, “Besides elegantly encoding the way that multiplication by i results in a rotation in the complex plane, this identity unites what one might consider the five most important numbers in a single equation. That’s pretty impressive!” My most practical friend is Oscar and here is what he told me: “There are so many uses for pi, but given my interest in everyday explanations of math, here’s one I like: If you drive to work every day, you take many, many pi’s with you. That’s because the circumference of your car’s tires is pi multiplied by the tires’ diameter. The most common car tire has a diameter of about 29 inches, so one full revolution covers a distance of about 29 times pi (about 7.5 feet). Many, many revolutions of your tires later you arrive at work, with lots and lots of pi’s!” Anna is also practical in that she will be using pi to calculate the area of the circular pastry she will be eating, but she also likes the infinite series for pi (pi/4 = 1 – 1/3 + 1/5 – 1/7 etc.). Avner Ash (Boston College) sums it up nicely, “ We can’t live without pi—how would we have circles, normal distributions, etc.?”

One of the most important questions one asks on Pi Day is how many digits can you recite? The largest number I got was 300 from the Princeton Math Club. However, there are quite a few impressive numbers from others, as well as some creative answers and ways to remember the digits. For example, Oscar can remember 3/14/15 at 9:26:53 because it was an epic Day and Pi Time for him. Art Benjamin can recite 100 digits from a phonetic code and 5 silly sentences. Ron Graham can recite all of the digits of pi, even thousands, as long as they don’t have to be in order. Dave Richeson also knows all of the digits of pi which are 0,1,2,3,4,5,6,7,8,and 9.

No matter how you celebrate, remember math, especially pi(e) is useful, fun, and delicious.

Vickie Kearn is Executive Editor of Mathematics at Princeton University Press.