Math Drives Careers: Paul Nahin on Electrical Engineering and √-1

Paul Nahin is the author of many books we’ve proudly published over the years, including An Imaginary Tale, Dr. Euler’s Fabulous Formula, and Number Crunching. For today’s installment in our Math Awareness Month series, he writes about his first encounter with √-1.

Electrical Engineering and √-1

It won’t come as a surprise to very many to learn that mathematics is central to electrical engineering. Probably more surprising is that the cornerstone of that mathematical foundation is the mysterious (some even think mystical) square-root of minus one. Every electrical engineer almost surely has a story to tell about their first encounter with √-1, and in this essay I’ll tell you mine.

Lots of different kinds of mathematics have been important in my personal career at different times; in particular, Boolean algebra (when I worked as a digital logic designer), and probability theory (when I wore the label of radar system engineer). But it’s the mathematics of √-1 that has been the most important. My introduction to √-1 came when I was still in high school. In my freshman year (1954) my father gave me the gift of a subscription to a new magazine called Popular Electronics. From it I learned how to read electrical schematics from the projects that appeared in each issue, but my most important lesson came when I opened the April 1955 issue.

It had an article in it about something called contra-polar power: a desk lamp plugged into a contra-polar outlet plug would emit not a cone of light, but a cone of darkness! There was even a photograph of this, and my eyes bugged-out when I saw that: What wondrous science was at work here?, I gasped to myself —I really was a naive 14-year old kid! It was, of course, all a huge editorial joke, along with some nifty photo-retouching, but the lead sentence had me hooked: “One of the reasons why atomic energy has not yet become popular among home experimenters is that an understanding of its production requires knowledge of very advanced mathematics.” Just algebra, however, was all that was required to understand contra-polar power.

contra power scan

Contra-polar power ‘worked’ by simply using the negative square root (instead of the positive root) in calculating the resonant frequency in a circuit containing both inductance and capacitance. The idea of negative frequency was intriguing to me (and electrical engineers have actually made sense of it when combined with √-1, but then the editors played a few more clever math tricks and came up with negative resistance. Now, there really is such a thing as negative resistance, and it has long been known by electrical engineers to occur in the operation of electric arcs. Such arcs were used, in the very early, pre-electronic days of radio, to build powerful AM transmitters that could broadcast music and human speech, and not just the on-off telegraph code signals that were all the Marconi transmitters could send. I eventually came to appreciate that the operation of AM/FM radio is impossible to understand, at a deep, theoretical level, without √-1.

When, in my high school algebra classes, I was introduced to complex numbers as the solutions to certain quadratic equations, I knew (unlike my mostly perplexed classmates) that they were not just part of a sterile intellectual game, but that √-1 was important to electrical engineers, and to their ability to construct truly amazing devices. That early, teenage fascination with mathematics in general, and √-1 in particular, was the start of my entire professional life. I wish my dad was still alive, so I could once again thank him for that long-ago subscription.

Math Drives Careers: Author Louis Gross

Gross jacketLouis Gross, distinguished professor in the departments of ecology, evolutionary biology, and mathematics at the University of Tennessee, is the author, along with Erin Bodine and Suzanne Lenhart, of Mathematics for the Life Sciences. For our third installment in the Math Awareness Month series, Gross writes on the role mathematics and rational consideration have played in his career, and in his relationship with his wife, a poet.

Math as a Career-builder and Relationship-broker

My wife is a poet. We approach most any issue with very different perspectives. In an art gallery, she sees a painting from an emotional level, while I focus on the methods the artist used to create the piece. As with any long-term relationship, after many years together we have learned to appreciate the other’s viewpoint and while I would never claim to be a poet, I have helped her on occasion to try out different phrasings of lines to bring out the music. In the reverse situation, the searching questions she asks me about the natural world (do deer really lose their antlers every year – isn’t this horribly wasteful?) force me to consider ways to explain complex scientific ideas in metaphor. As the way I approach science is heavily quantitative, with much of my formal education being in mathematics, this is particularly difficult without resorting to ways of thought that to me are second nature.

The challenges in explaining how quantitative approaches are critical to science, and that science advances in part through better and better ways to apply mathematics to the responses of systems we observe around us, arise throughout education, but are particularly difficult for those without a strong quantitative bent. An example may be helpful. One of the central approaches in science is building and using models – these can be physical ones such as model airplanes, they can be model systems such as an aquarium or they can be phrased in mathematics or computer code. The process of building models and the theories that ultimately arise from collections of models, is painstaking and iterative. Yet each of us build and apply models all the time. Think of the last time you entered a supermarket or a large store with multiple checkout-lines. How did you decide what line to choose? Was it based on how many customers were in each line, how many items they had to purchase, or whether they were paying with a check or credit card? Did you take account of your previous experience with that check-out clerk if you had it, or your experience with using self-checkout at that store? Was the criterion you used some aspect of ease of use, or how quickly you would get through the line? Or was it something else such as how cute the clerk was?

As the check-out line example illustrates, your decision about what is “best” for you depends on many factors, some of which might be quite personal. Yet somehow, store managers need to decide how many clerks are needed at each time and how to allocate their effort between check-out lines and their other possible responsibilities such as stocking shelves. Managers who are better able to meet the needs of customers, so they don’t get disgusted with long lines and decide not to return to that store, while restraining the costs of operation, will likely be rewarded. There is an entire field, heavily mathematical, that has been developed to better manage this situation. The jargon term is “queuing models” after the more typically British term for a waiting line. There is even a formal mathematical way of thinking about “bad luck” in this situation, e.g. choosing a line that results in a much longer time to be checked out than a different line would have.

While knowing that the math exists to help decide on optimal allocation of employee effort in a store will not help you in your decision, the approach of considering options, deciding upon your criteria and taking data (e.g. observations of the length of each line) to guide your decision is one that might serve you well independent of your career. This is one reason why many “self-help” methods involve making lists. Such lists assist you in deciding what factors (in math we call these variables) matter to you, how to weight the importance of each factor (we call these parameters in modeling) and what your objective might be (costs and benefits in an economic sense). This process of rational consideration of alternative options may assist you in many aspects of everyday life, including not just minor decisions of what check-out line to go into, but major ones such as what kind of car or home to purchase, what field to major in and even who to marry! While I can’t claim to have followed a formal mathematical approach in deciding on the latter, I have found it helpful throughout my marriage to use an informal approach to decision making. I encourage you to do so as well.

Check out Chapter 1 of Mathematics for the Life Sciences here.

Math Drives Careers: Author Oscar Fernandez

We know that mathematics can solve problems in the classroom, but what can it do for your business? Oscar Fernandez, author of Everyday Calculus, takes a look at how knowledge of numbers can help your bottom line.

Why You Should Be Learning Math Even If You Don’t Need It for Your Job

I want to tell you a short story about epic triumph in the midst of adversity. Okay, I’m exaggerating a bit, but hear me out.

A couple of years ago, I approached Boston Scientific—an S&P 500 component—with a crazy idea: let me and a team of students from Wellesley College (a liberal arts college for women) and Babson College (a business school) do consulting work for you. It was a crazy idea because what could I—a mathematician who knew nothing about their business—and some students—who hadn’t even graduated yet—possibly offer the company? Plenty, it turns out, all thanks to our common expertise: mathematics.

Mathematics, often depicted in movies as something pocket-protector-carrying people with less than stellar social skills do, is actually quite ubiquitous. I’d even say that mathematicians are the unsung heroes of the world. Alright, that’s a bit of hyperbole. But think about it. Deep in the catacombs of just about every company, there are mathematicians. They work in low light conditions, hunched over pages of calculations stained with days-old coffee, and think up ways to save the company money, optimize their revenue streams, and make their products more desired. You may never notice their efforts, but you’ll surely notice their effects. That recent change in the cost of your flight? Yep, it was one of us trying to maximize revenue. The reason that UPS truck is now waking you up at 6 a.m.? One of us figured out that the minimum cost route passes through your street.

But we’re do-good people too. We help optimize bus routes to get children to school faster and safer. We’ve spent centuries modelling the spread of disease. More recently, we’ve even reduced crime by understanding how it spreads. That’s why I was confident that my team and I could do something useful for Boston Scientific. Simply put, we knew math.

We spent several weeks pouring over data the company gave us. We tried everything we could think of to raise their revenues from certain products. Collectively, we were trained in mathematics, economics, computer science, and psychology. But nothing worked. It seemed that we—and math—had failed.

Then, with about three weeks left, I chanced upon an article from the MIT Technology Review titled “Turning Math Into Cash.” It describes how IBM’s 200 mathematicians reconfigured their 40,000 salespeople over a period of two years and generated $1 billion in additional revenue. Wow. The mathematicians analyzed the company’s price-sales data using “high-quantile modeling” to predict the maximum amount each customer was willing to spend, and then compared that to the actual revenue generated by the sales teams. IBM then let these mathematicians shuffle around salespeople to help smaller teams reach the theoretical maximum budget of each customer. Genius, really.

I had never heard of quantile regression before, and neither had my students, but one thing math does well is to train you to make sense of things. So we did some digging. We ran across a common example of quantile modelling: food expenditure vs. household income. There’s clearly a relationship, and in 1857 researchers quantified the relationship for Belgian households. They produced this graph:

fernandez 1

That red line is the linear regression line—the “best fit to the data.” It’s useful because the slope of the line predicts a 50 cent increase in food expenditure for a $1 increase in household income. But what if you want information about the food expenditure of the top 5% of households, or the bottom 20%? Linear regression can’t give you that information, but quantile regression can. Here’s what you get with quantile regression:

fernandez 2

The red line is the linear regression line, but now we also have various quantile regression lines. To understand what they mean let’s focus on the top-most dashed line, which is the 95th percentile line. Households above this line are in the 95th percentile (or 0.95 quantile) of food expenditure. Similarly, households below the bottom-most line are in the 5th percentile (or 0.05 quantile) of food expenditure. Now, if we graph the slopes of the lines as a function of the percentile (also called “quantile”), we get:

fernandez 3

(The red line is the slope of the linear regression line; it doesn’t depend on the quantile, which is why it’s a straight line.) Notice that the 0.95 quantile (95th percentile) slope is about 0.7, whereas the 0.05 quantile (5th percentile) slope is about 0.35. This means that for every $1 increase in household income, this analysis predicts that households in the 95th percentile of food expenditure will spend 70 cents more, whereas households in the 5th percentile will spend only 35 cents more.

Clearly quantile regression is powerful stuff. So, my team and I went back and used quantile regression on the Boston Scientific data. We came up with theoretical maximum prices that customers could pay based on the region the product was sold in. As with IBM, we identified lots of potential areas for improvement. When my students presented their findings to Boston Scientific, the company took the work seriously and was very impressed with what a few students and one professor could do. I can’t say we generated $1 billion in new revenue for Boston Scientific, but what I can say is that we were able to make serious, credible recommendations, all because we understood mathematics. (And we were just a team of 5 working over a period of 12 weeks!)

April is Mathematics Awareness Month, and this year’s theme is “math drives careers.” After my Boston Scientific experience and after reading about IBM’s success, I now have a greater appreciation of this theme. Not only can mathematics be found in just about any career, but if you happen to be the one to find it (and use it), you could quickly be on the fast track to success. So in between celebrating March Madness, Easter, Earth Day, and April 15th (I guess you’d only celebrate if you’re due a tax refund), make some time for math. It just might change your career.

Photo by Richard Howard.

Photo by Richard Howard.

Oscar Fernandez is the author of Everyday Calculus. He is assistant professor of mathematics at Wellesley College.

Math Drives Careers: Author Ignacio Palacios-Huerta

Logical thinking, analytical skills, and the ability to recognize patterns are crucial in an array of fields that overlap with mathematics, including economics. But what does math (or economics, for that matter) have to do with the world’s most popular sport? Economist Ignacio Palacios-Huerta’s recent book, Beautiful Game Theory: How Soccer Can Help Economics  made a splash during the last World Cup, showing how universal economic principles can be understood through soccer. Read on for his thoughts on why the language of modern economics, including behavioral economics, is mathematics.

The Role of Mathematics in my Life as an Economist

To describe the role of mathematics in my life as an economist, I first need to explain what, to me, Economics is all about. So let me take you to one of my favorite books, A Treatise of Human Nature, written almost 300 years ago by David Hume.

Beautiful Game TheoryIn the introduction Hume writes, “‘Tis evident that all the sciences have a relation, more or less, to human nature … Even Mathematics, Natural Philosophy, and Natural Religion, are in some measure dependent on the science of Man, [which is] the only solid foundation for the other sciences”. By the science of man Hume means the understanding of all facets of human nature, including preferences, senses, passions, imagination, morality, justice, and society. This science applies wherever men are making decisions, be it running public institutions or countries, as employees in firms, or as individuals investing in education, taking risks in financial markets, or making family decisions. This science of man is thus what one may initially be tempted to call Economics for, as George Bernard Shaw puts it in my favorite definition, “Economy is the art of making the most of life”.

But of course this definition is incomplete because other social sciences (e.g., sociology, history, psychology, political science) are also concerned with human behavior. So what makes Economics “different”? Here is the difference: the difference is not the subject matter but the approach. The approach is totally different, and a very mathematical one. As such, mathematics plays a critical role in the life of any economist.

Let me elaborate. Continuing with Hume, it turns out that he also anticipated our methodological approach in modern Economics: observation and logical arguments. Which can be translated as: data and data analysis (what we call econometrics), and mathematics, for mathematics is, after all, the language of logic. So in Economics, as in physics, we write down our ideas and theories in mathematical terms to make logical arguments, and then we use more math (statistical, econometrics, etc) to check whether the data appear to be consistent with the theoretical arguments. If they are, the evidence can be said to support the theory; if they aren’t, the theory needs to be refined or discarded. Yes, lots of math and related techniques provide what is our distinct “economics approach to human behavior.” It is not the subject matter but the approach that is different, and it heavily relies on mathematics.

To economists and other social scientists, mathematics has many methodological virtues: it can lend precision to theories, can uncover inconsistencies, can generate hypothesis, can enable concision and promote intelligibility, and can sort out complex interactions, while statistical and econometric analysis can organize and carefully interpret voluminous data.

None of this is obvious when you begin studying Economics (“Why should I take all this math, statistics and econometrics? Why all this pain?”). But I think most of us soon learn to appreciate that the language of modern economics is mathematics, and that it is rightly so. And this is not math for the sake of math (as in pure mathematics), but math with a purpose: modeling human behavior.

Let me conclude by saying that since the economic approach is applicable to all human behavior, any type of data about human activity can be useful to evaluate economic theories. This includes, why not, sports data, which in many ways can be just perfect for testing economic theories: the data are abundant, the goals of the participants are clear, the outcomes are easy to observe, the stakes are high, and the subjects are professionals with experience. If a theory is “correct”, sport is a good setting to check it.

So just as data involving falling stones and apples were useful to Galileo Galilei and  Isaac Newton to test for the first time theories that were important in physics, data from sports can be useful in Economics to do exactly the same. As such in some of my contributions to Economics I have used math to develop theoretical models, and further mathematical tools applied to this type of data to test them.

Mathematics Awareness Month 2015: Math Drives Careers

Internet search, pharmaceuticals, insurance, finance, national security, medicine, ecology. What is the link between these diverse career fields? Students graduating with a mathematical sciences degree can find a professional future in all of these fields, and a wide range of others as well. This year’s Mathematics Awareness Month takes a step out of the classroom to show just where mathematics can lead after graduation.

Mathematics Awareness Month is an annual celebration dedicated to increasing public understanding of and appreciation for mathematics. The event, which started in 1986 as Mathematics Awareness Week, adopts a different theme each year. This year’s theme is “Math Drives Careers,” and PUP is excited to bring you a series of guest posts from our authors. Check back all this month for posts about using math to raise revenues, to understand sports and economics, and to solve complex problems.

The organizers of Mathematics Awareness Month explain the importance of mathematics in today’s workforce:

“Innovation is an increasingly important factor in the growth of world economies. It is especially important in key economic sectors like manufacturing, materials, energy, biotechnology, healthcare, networks, and professional and business services. The advances in and applications of the mathematical sciences have become drivers of innovation as new systems and methodologies have become more complex. As mathematics drives innovation, it also drives careers.”
Check out this official Mathematics Awareness Month poster, which includes career descriptions for 10 individuals who used their love for math to find rewarding careers:



Follow along with @MathAware and take a look at Math Awareness Month on Facebook.

Don’t Beware of Math… Be Aware of It!

by Tim Chartier

[This article is cross posted from The Huffington Post]

As the last days of April unfold, we head into May and the end of the school year. Many classes focus on testing and final grades. Teachers often must focus and ready their students for end-of-the-year testing. Math classes will be asked problem after problem and question after question. In all those classrooms, a thought probably, if not often, races through someone’s mind. Yes, the thought… the one that makes pencils heavier, word problems harder and students wish they were somewhere, anywhere but where they are. There are a lot of ways that thought turns into a question. A common one: “Why study math?”

So let’s go and ask, particularly given that we are in April, which is Math Awareness Month. For some, math may be something to beware of rather than be aware of. In fact, that’s precisely the point of the month. Math has many applications, from theoretical to applied. Mathematicians continue to expand the boundaries of what we know mathematically. With the publication of each new issue of a journal, the field of math grows. NBA teams use mathematics to gain a competitive edge over their opponent. Will the better team with better mathematics win? It definitely helped the Oakland As in 2002 with the math that became known as Moneyball. Every day, credit card numbers are encrypted to allow for secure online transactions. Developing methods of encryption that simply cannot be broken with a faster computer comes from mathematics.

Studying math enables one to appreciate and possibly understand its applications. Yet one does not need to study math just so the techniques can be used in theoretical or applied settings. Mathematics teaches a way of thinking. Returning to basketball, mathematical formulas won’t pop off the court. Someone must derive them and study them to ensure their usefulness. It can take time to gain such insight.

The process toward such understanding is what probably draws many mathematicians to their field. I like to think of it as a path of wonder. For example, I’ve periodically been contacted by ESPN’s Sport Science program to aid in their analysis. They call when they are stuck. When the problem is first presented, my first thought is, “I have no idea how to do this.” And yes, every time I have found a way.

Part of this stems from my awareness of that path of mathematical wonder. You don’t have to simply know the answer to a math problem to solve it. In fact, math is usually more interesting when you don’t know how to solve a problem. Would a jigsaw puzzle be fun if it had only two or three pieces? You never know exactly how to fit a 1,000-piece puzzle together when you start, and you won’t always try to fit connecting pieces. It’s a puzzle, so you explore and experiment.

Math can be the same way. As such, there is a certain sense of mystery to math. You step into a question and simply stand in the unknown. Then you begin to explore, looking for pieces that fit together. This type of thinking is helpful for life, as it offers its unknowns. In life, you may be forced to stand in the unknown. What questions do you want to explore, and what pieces do you want to try to fit together?

Some math ideas are developed through a similar process of exploration. For example, about 10 years ago, I learned how Robert Bosch, Adrianne Herman and Craig Kaplan were creating pictures like the one that I made (after learning their ideas) below.


The image above is a portrait of Martin Gardner, who we’ll return to momentarily. Later, it occurred to me that I could make mazes with these images if I used a math formula developed by Leonhard Euler, who lived in the 1700s. Seeing that I could fit these two ideas together — one about a decade old and another hundreds of years old — enabled me to create mazes for my book Math Bytes. Returning again to the NBA, here is such a maze:

Click here if you’d like a larger version of the maze.

This creative edge of math engages me. It makes teaching math every day at Davidson College a great job. And it makes answering that question “Why learn math?” a question I look forward to being asked.

But does this sound like the mathematics you know? If not, then you might want to spend some of these last days of April exploring the Mathematics Awareness Month website. The theme for April 2014 is Mathematics, Magic and Mystery. Each day of the month an engaging idea of mathematics has been unfolded. See the ones already shared and await those yet to come. Learn secrets of mental math, mathematics of juggling, optical illusions, and many more interesting ideas and the math behind them! Want to dig deeper? Note that the theme was chosen as 2014 marks what would have been the 100th birthday of Martin Gardner. Simply put, he engaged millions in his mathematical writing and made mathematicians and children alike aware of the wonders and mysteries of math.

So be aware of math! It has many applications, from magic to sports to the theoretical to the historical. I often tell my students in class that if you don’t like math, it may simply be that you haven’t discovered the area of math that fits the way you think! Be careful of sampling from only one part of the math buffet and walking away. A great place to sample many engaging ideas of math is every April with Math Awareness Month. This April, you can learn math and soon engage friends and family with ideas in the mystery and magic of mathematics!

So why study math? It develops your mathematical sense, which enables you to see life through that lens. In the process, you hone your ability to think in ways that can make you more aware of life itself. So enjoy these last days of April and be aware of math!

Follow Tim Chartier on Twitter:

Exclusive content from Princeton University Press



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Who are these mathematicians?

Leave your guesses in the comments.

THIS IS MATH: Beautiful Geometry

Since this is still April, I will direct you back to the Math Awareness Month Calendar to the window marked The Beautiful Geometry of Crop Circles. You can use a compass and ruler to make beautiful geometric patterns and you can use other media as well. Many of you probably have already done this using a Spirograph.

To find out more about the connection between art and geometry, I will point you to Beautiful Geometry. Eli Maor, who is a mathematician, and Eugen Jost, who is an artist, teamed up to illustrate 51 geometric proofs and assorted mathematical curiosities.

Let’s start with one that most people know about—the Pythagorean theorem or a2 + b2 = c2. No one knows exactly how many proofs there are but Elisha Loomis wrote a book that includes 367 of them. The following illustration is a graphical statement of the theorem that if you draw a square on each of the three sides of a triangle, you will find that the sum of the areas of the two small squares equals the area of the big one.

pythagorean 1

If you look at the colorful figure below by Eugen Jost, you will see something similar, but much more interesting to look at. The figure above is a 30, 60, 90 degree triangle whereas the one below is a 45, 45, 90 degree triangle.

Plate 5 NEW

25 + 25 = 49, Eugen Jost, Beautiful Geometry

Using the Pythagorean formula, we know that 52 + 52  should equal 72. Now this means that

52  + 52 = 72

25 + 25 = 49

I think we all know that is just not true, yet we know that the formula is correct. What is going on here? It seems that the artist is having a bit of fun with us. Mathematics must be precise but art is not bound by the laws of mathematics. See if you can figure out what happened here.


Where’s the Math?

We know that there are at least 367 different proofs for the Pythagorean theorem but the most famous of them is Euclid’s proof. Eli Maor will walk you through it below, and, he will not try to trick you.



Important Note: We are going to assume you agree that all triangles with the same base and top vertices that lie on a line parallel to the base have the same area. Euclid proved this in book I of the Elements (Proposition 38).

Before he gets to the heart of the proof, Euclid proves a lemma (a preliminary result): the square built on one side of a right triangle has the same area as the rectangle formed by the hypotenuse and the projection of that side on the hypotenuse. The figure above shows a right triangle ACB with its right angle at C. Consider the square ACHG built on side AC. Project this side on the hypotenuse AB, giving you segment AD. Now construct AF perpendicular to AB and equal to it in length. Euclid’s lemma says that area ACHG = area AFED.

To show this, divide AFED into two halves by the diagonal FD. By I 38, area FAD = area FAC, the two triangles having a common base AF and vertices D and C that lie on a line parallel to AF. Likewise, divide ACHG into two halves by diagonal GC. Again by I 38, area AGB = area AGC, AG serving as a common base and vertices B and C lying on a line parallel to it. But area FAD = 1⁄2 area AFED, and area AGC = 1⁄2 area ACHG. Thus, if we could only show that area FAC = area BAG, we would be done.

It is here that Euclid produces his trump card: triangles FAC and BAG are congruent because they have two pairs of equal sides (AF = AB and AG = AC) and equal angles ∠FAC and ∠BAG (each consisting of a right angle and the common angle ∠BAC). And as congruent triangles, they have the same area.

Now, what is true for one side of the right triangle is also true of the other side: area BMNC = area BDEK. Thus, area ACHG + area BMNC = area AFED + area BDEK = area AFKB: the Pythagorean theorem.



THIS IS MATH: Magic Squares, Circles, and Stars

If you have been following the opening of the windows in the Mathematical Awareness Month Poster, you might want to go back to window #1 and review Magic Squares. If you haven’t been there yet, please take a look at it. You will learn how to amaze your friends with your magical math abilities.

Magic squares come in many types, shapes, and sizes. Below you will see a magic square, a magic circle, and a magic star. If you would like to see hundreds more, you might want to check out The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions by Clifford Pickover.

Normal Magic Squares

This is a third-order normal magic square where all of the rows, columns, and diagonals add to 15.



Is this the only solution to this magic square? Can you find others?

You could also have a 4 x 4 square or a 5 x 5 square and so on. How big of a square can you solve?


Magic Circles

Below you will see a magic circle composed of eight circles of four numbers each and the numbers on each circle all add to 18.  The thing that makes this magic circle special is that each number is at the intersection of four circles but no other point is common to the same four circles.


Magic Stars

The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.


So, where’s the math?

Well, you should have noticed already that there are numbers on this page. However, there is more to math than numbers. Let’s add at least one equation.

If we go back to the normal magic square you should know that all these magic squares have the same number of rows and columns, they are n2. The constant that is the same for every column, row, and diagonal is called the magic sum and we will call it M.  Now we can figure out what that constant should be. If we use our 3 x 3 square above, we know that n = 3. If we plug our n into the given formula below we will find what our constant has to be.


 Since our n = 3, the formula says M = [3 (32 + 1)]/2, which simplifies to 15. For normal magic squares of order n =  4, 5, and 6 the magic constants are, respectively: 34, 65, and 111. What would M be for n = 8? See if you can solve this square. (The figure for the normal square is from Wikipedia.)






THIS IS MATH!: Amaze your friends with The Baby Hummer card trick

Welcome to THIS IS MATH! a new series from math editor Vickie Kearn.

This is the first of a series of essays on interesting ways you can use math. You just may not have thought about it before but math is all around us. I hope that you will take away something from each of the forthcoming essays and that you will pass it on to someone you know.

3-28 Diaconis_MagicalApril is Math Awareness Month and the theme this year is Mathematics, Magic, and Mystery. There is a wonderful website where you will find all kinds of videos, puzzles, games, and interesting facts about math. The homepage has a poster with 30 different images. Each day of the month, a new window will open and reveal all of the wonders for that day.

Today I am going to elaborate on something behind window 3 which is about math and card magic. You will find more magic behind another window later this month. This particular trick is from Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks by Persi Diaconis and Ron Graham. It is a great trick and it is easy to learn. You only need any four playing cards. Take a look at the bottom card of your pack of four cards. Now remember this card and follow the directions carefully:

  1. Put the top card on the bottom of the packet.
  2. Turn the current top card face up and place it back on the top of the pack.
  3. Now cut the cards by putting any amount you like on the bottom of the pack.
  4. Take off the top two cards (keeping them together) and turn them over and place them back on top.
  5. Cut the cards again and then turn the top two over and place them back on top.
  6. Give the cards another cut and turn the top two over together and put them back on top.
  7. Give the cards a final cut.
  8. Now turn the top card over and put it on the bottom of the pack.
  9. Put the current top card on the bottom of the pack without turning it over.
  10. Finally, turn the top card over and place it back on top of the pack.
  11. Spread out the cards in your pack. Three will be facing one way and one in the opposite way.
  12. Surprise! Your card will be the one facing the opposite way.

This trick is called the Baby Hummer and was invented by magician Charles Hudson. It is a variation on a trick invented by Bob Hummer.

So where’s the math?
The math behind this trick covers 16 pages in the book mentioned above.

THIS IS MATH! will be back next week with an article on Math-Pickover Magic Squares!


Celebrate Math Awareness Month with Us

April is Math Awareness month and this year the theme is Mathematics, Magic, and Mystery. To kick off the celebration, visit the Math Awareness web site where you can “open” the days on an advent calendar revealing wonderful math and magic tricks. Today, for example, you can learn a bit about Geometrical Vanishes which make everyday objects appear to … disappear. The videos show how to make everything from dollar bills to chocolate disappear. Tomorrow we’ll start a new series of posts called This Is Math! in which our acquisitions editor for math titles will explain the various ways we encounter math in our everyday lives…and perhaps even add a few tricks of her own!

In the meantime, here’s another Geometrical Vanish courtesy of Tim Chartier, author of Math Bytes:

Play along by printing and cutting out your own set of vanishing PUP Logos. Cut along the solid lines and reverse the top two sections to see a logo magically disappear and reappear. if you have a suggestion for something else you would like to make appear and disappear, leave a comment below and I’ll see if we can get more of these print outs made (keep it clean please!).




Click the smaller images above to open full size images.

New Mathematics Catalog!

Be among the first to browse and download our new mathematics catalog!

Of particular interest is Undiluted Hocus-Pocus: The Autobiography of Martin Gardner. Gardner takes readers from his childhood in Oklahoma to his college days at the University of Chicago, his service in the navy, and his varied and wide-ranging professional pursuits. Before becoming a columnist for Scientific American, he was a caseworker in Chicago during the Great Depression, a reporter for the Tulsa Tribune, an editor for Humpty Dumpty, and a short-story writer for Esquire, among other jobs. Gardner shares colorful anecdotes about the many fascinating people he met and mentored, and voices strong opinions on the subjects that matter to him most, from his love of mathematics to his uncompromising stance against pseudoscience. For Gardner, our mathematically structured universe is undiluted hocus-pocus—a marvelous enigma, in other words. Undiluted Hocus-Pocus offers a rare, intimate look at Gardner’s life and work, and the experiences that shaped both.

Also be sure to note Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction by Charles L. Adler. From teleportation and space elevators to alien contact and interstellar travel, science fiction and fantasy writers have come up with some brilliant and innovative ideas. Yet how plausible are these ideas—for instance, could Mr. Weasley’s flying car in the Harry Potter books really exist? Which concepts might actually happen, and which ones wouldn’t work at all? Wizards, Aliens, and Starships delves into the most extraordinary details in science fiction and fantasy–such as time warps, shape changing, rocket launches, and illumination by floating candle—and shows readers the physics and math behind the phenomena.

And don’t miss out on Beautiful Geometry by Eli Maor and Eugen Jost. If you’ve ever thought that mathematics and art don’t mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important and beautiful branches of mathematics.

Even more foremost titles in mathematics can be found in the catalog. You may also sign up with ease to be notified of forthcoming titles at Your e-mail address will remain confidential!

If you’re heading to the Joint Mathematics Meeting in Baltimore, MD, January 15th-18th, come visit us at booth 407. We’ll be hosting the following book signings:

The 5 Elements of Effective Thinking, Edward B. Burger and Michael Starbird
Wednesday, January 15th 4:30-5:30 p.m.

Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction, Charles L. Adler
Thursday, January 16th 11:00 a.m.-12:00 p.m.

Also stop by 629, the Martin Gardner Centennial Booth. Staffed by a team of enthusiasts who have long been inspired by Gardner, there will be interactive activities and different handouts and puzzles to enjoy each day. Don’t miss “Martin Gardner’s Outreach in His Centennial Year: Mathematics Awareness Month 2014,” a short talk by Colm Mulcahy, Bruce Torrence, and Eve Torrence, Saturday, January 18th at 1:00 p.m. in Convention Center room 346.

Follow @MGardner100th on Twitter for more updates throughout the year, and #JMM14 and @PrincetonUnivPress for updates and information on our new and forthcoming titles throughout the meeting. See you there!

How Mathematical Models Make Sense of Big Data

Tim Chartier, co-author with Anne Greenbaum of Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, explains how to make sense of big data with numerical analysis.


You submit a query to Google or watch football bowl games as we enter a new year. In either case, you benefit from mathematical methods that can garner meaningful information from large amounts of data. Such techniques fall in the field of data mining.

Massive datasets are available with every passing minute in our world. For example, during the Oscars in February, the Cirque du Soleil performance resulted in 18,718 tweets in one minute according to TweetReachBlog. While tweets cannot exceed 140 characters in length, their average length is 81.9 characters according to MediaFuturist. So, in one minute, approximately 1.5 million characters zoomed through Twitter. From Wikipedia, we’ll take the average length of a word (in English) to be 5.1 characters. Assuming these Oscar tweets are written in English and conform to the standard length of words, 300,000 words were tweeted in one minute. This is approximately the number of words contained in the entire Hunger Games Trilogy!

Mathematical models and numerical analysis play important roles in data mining. For example, a foundational part of Google’s search engine algorithm is a method called PageRank. In Anne Greenbaum and my book, Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, published by Princeton University Press, we discuss the PageRank method– both its underlying mathematical model and how it is computed on a computer.

In an exercise in the text, you can develop a system of linear equations in a manner similar to that used by the Bowl Championship Series to rank college football teams (editor – or college basketball teams for March Madness). An important part of this problem is developing the linear system. Our text also discusses the computation challenges of such problems and what numerical methods result in the most accurate results.

Many techniques utilized to solve the large linear systems of data mining are also utilized in engineering and science. The book discusses how large linear systems (containing millions of rows) can derive from problems involving partial differential equations. Again, the book analyzes the efficiency and accuracy of the methods utilized to solve such systems. Such techniques led to the computed animated figures we enjoy in modern movies and aid in simulating the aerodynamics of a car created with computer-aided design.

As stated at the opening of Chapter 1 of the text, “Numerical methods play an important role in modern science. Scientific exploration is often conducted on computers rather than laboratory equipment. While it is rarely meant to completely replace work in the scientific laboratory, computer simulation often complements this work.” As such modern science demands the use and understanding of numerical methods.