Last year, Professor Steven Strogatz of Cornell University wrote a series of op-eds for the New York Times that discussed the presence of mathematics in unlikely places. I discussed one of these columns here. Now, either those articles were well-received, or Professor Strogatz is well-connected, because this year he’s back in the Times with a much more ambitious series of articles. This time around, Strogatz is attempting to “[write] about the elements of mathematics, from preschool to grad school, for anyone out there who’d like to have a second chance at the subject.”
Preschool to grad school is a significant amount of ground to cover, but thus far Strogatz has used his articles to assault this goal with gusto. To date, he has tackled counting, patterns in addition, negative numbers, division, and basic high school algebra. This doesn’t really do justice to his content, though. Along the way he gives the reader some Sesame Street, and discusses a number of tangential topics, including the inability of Verizon employees to do math, the half-your-age-plus-seven rule, and pre-WWI European history. The latter comes about in a discussion of that old adage which is familiar to anyone who saw the first Alien vs. Predator movie: the enemy of my enemy is my friend.
Predators must be awesome at math.
While some of Professor Strogatz’s explanations are a bit hand wavy (in particular, his explanation of why (-1) x (-1) = 1 is a lacking), on the whole they are quite good. In particular, he offers a nice explanation of what it is for a mathematical argument to be “elegant.” But even more impressive than his writing is its location – to have a discussion of mathematics with as wide an audience as the New York Times readership is commendable. Even if people are not inspired to learn more mathematics after reading these pieces, hopefully they will have at least learned something. As with exercise, a little mathematics is better than no mathematics at all.
Moreover, these articles highlight aspects of math not usually seen in popular discourse. Much like Paul Lockhart’s A Mathematician’s Lament (which Strogatz references), these snack-size essays are focused on simple mathematical ideas, and the beautiful (and sometimes unexpected) results that follow. Nowhere here does Professor Strogatz multiply two really big numbers together; in fact, he’s quite sympathetic to the fact that for many people, there is nothing more tedious than calculation. By leading the conversation in this way, he’s hopefully able to give a taste of what makes math beautiful to an audience for whom such a statement might otherwise be labeled heresy.
I don’t know where this series of articles is headed, but I look forward to finding out, and hope you do to. Professor Strogatz’s articles are grouped together here.
(Hat tip to dad for sending me a few of these articles.)
Some time ago, I wrote an article on the optimal way to select a mate, assuming you know how many eligible partners exist, and that once you’ve dated someone, you can’t go back and date them again (sorry, Drew Barrymore and that dude from the Apple commercials). This is less romantically known as the secretary problem. Let me briefly recall the problem and its solution: suppose you have n candidates, from which you want to pick the best one. This applies to a variety of situations, from hiring a secretary to finding a girlfriend to apartment hunting. In either case, the outcome is the same: you should look at roughly the first n/e of them (yes, thate), and then select the first one after those n/e which is better than all that you have seen so far. While this strategy won’t guarantee you get the best choice, it will give you the best choice around 37% of the time.
The major problem with this model is that in many situations, the value of n is unknown. There are ways to circumvent this problem, which I will not discuss here. Instead, in the context of finding a mate, I offer the following method to calculate the number of partners you could reasonably expect to find in your area. This method recently gained some attention when Peter Backus, a Ph.D. candidate from the University of Warwick wrote a paper titled “Why I Don’t Have a Girlfriend.”
The basic technique involves modifying the Drake Equation, an equation used to estimate the number of potential extraterrestrial civilizations in our galaxy. For those who have never been introduced to this equation, it asserts the following:
N = R · fp · ne · fl · fi · fc · L.
According to Wikipedia, these variables represent the following quantities:
R = the average rate of star formation per year in our galaxy, fp = the fraction of those stars that have planets, ne = the average number of planets that can potentially support life per star that has planets, fℓ = the fraction of the above that actually go on to develop life at some point, fi = the fraction of the above that actually go on to develop intelligent life, fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space, L = the length of time such civilizations release detectable signals into space.
Given estimates for all of these parameters, one could then estimate the number of civilizations in our galaxy. Since we don’t know the values of any of these parameters, however, this is more of a thought experiment than anything else.
Nevertheless, the idea can be easily modified to try and find the number of eligible mates in a given area. Peter Backus’ approach is fairly specific to him, but he links to a more general approach discussed here, in which the following equation is presented:
n = P · ft · fo · fc · A · R
In this case, the parameters are given by
P = the population. This could be the population of your university, your city, or your country, depending on how ambitious you are. ft = the fraction of that population which you would want to mate with, in broad terms. If you’re a straight male, this would be the fraction of females. If you’re a gay male, it would be the fraction of males, and so on. fo = the fraction of the population you’d want to mate with which wants to mate with you. For example, if you’re a straight male who wants to mate with females, this will compensate for the fact that some females will be lesbian and therefore unwilling to mate with you. fc = author Raymond Francis labels this the out fraction, and describes it as the answer to the question “Of the people in your target gender and orientation, how many of them are open enough about their sexuality to engage in a relationship of the sort you’re hoping for?” If you are straight, this value is likely 1, or very close to it. If not, things can be a little bit fuzzier. A = the fraction of those remaining who fall within your desired age range. This is, of course, personal to you – if you’d like a socially acceptable age range, you could follow the “half your age plus seven” rule. R = factors for any remaining filters you wish. Do you want your partner to have a certain level of education, or a certain income? Do you need a non-smoker, or demand a euphonium player? Here’s where you can fold that into the mix.
Yes, Wikipedia has a graph illustrating the half your age plus seven rule. Amazing.
With all these parameters accounted for, N will give you the number of potential mates in your area.
Let’s take this equation for a spin, shall we? Suppose you are a straight male living in Los Angeles, and looking for a girl to date in Los Angeles. According to Wikipedia, the estimated population of LA as of 2008 was 3,833,995. Of those, let’s say that 51% are female, and of the females, let’s posit that 90% are straight or bisexual. fc should be high in this case – to be conservative, let’s put it at 95%.
To estimate the age filter, one can obtain some data from this site. Suppose you are a 25 year old man – then, absent any personal preference, the socially acceptable age range of women for you to date is between 19.5 and 36. According to census data, in 2000 there were 3,694,820 people in Los Angeles, and of them, 974,004 were between the ages of 20 and 34. Additionally, there were 251,632 people between the ages of 15 and 19, and 584,036 people between the ages of 35 and 44. If we make the assumption that ages are roughly uniformly distributed within these brackets, this gives us an additionlal 141,970 people either between the ages of 19.5 and 20, or between the ages of 35 and 36. Combining this gives a total of 1,115,974 people between the ages of 19.5 and 36 in Los Angeles in 2000, or roughly 30% of the population. Let’s use this for our value A.
Assuming you have no other restrictions (i.e. taking R = 1), this gives us n = 3,833,995 · .51 · .9 · .95 · .3 = 501,544. That’s a lot of ladies out there for the taking. Of course, taking R = 1 is probably unrealistic. It’s unlikely you want to date women who are married, for example, and everyone has their own personal taste that will decrease the pool even further. Once you’ve calculated your personal value for R, however, you then know how many eligible mates will be in your area. Given that, you’ll know how large n/e is, and then you’ll know how many people you should date before you think about settling down.
Although Peter Backus has received a fair amount of buzz for the short paper he has written on this idea, he readily admits that he is not the first to think of applying Drake’s equation in this situation. I’ve discussed mostly Raymond Francis’ approach here, but Backus has links to many other people that have discussed the idea on his website. In particular, here’s an exchange from CBS’s The Big Bang Theory (don’t worry, there’s a laugh track so you know when things are supposed to be funny).
In summary, not only can the Drake Equation be used to consider the existence of extraterrestrial life, it can also be used to consider potential mates right here on Earth. The next step, of course, is obvious: we must combine these two equations to calculate the number of potential extraterrestrial mates. Undoubtedly the number will be small, but one should never underestimate the power of love.
Late last year, a study was published in Proceedings of the National Academy of Sciences which tried to pin down origins for the gender gap in mathematics education. As I’ve discussed before, the gender gap in math education is shrinking, and has been shown to be less about biology and more about culture – in cultures where gender equality is weaker, the gender gap is stronger. Nevertheless, even in American culture, the gender gap still persists, and this study by Sian Beilock and others has tried to figure out how, if the gender gap is culturally based, it comes about in young students. The original study can be found here, while a discussion of the study that was featured in the news can be found here.
Professor Beilock and her colleagues tried to correlate young students’ math anxiety with the math anxiety of their teachers. In particular, they looked at 1st and 2nd grade students, of whom a vast majority (over 90%) have teachers who are female. The study assessed the math anxiety of the teachers and measured the math achievement of the students at the beginning and end of the year. Here are the results, taken from the introduction to the paper:
There was no relation between a teacher’s math anxiety and her students’ math achievement at the beginning of the school year. By the school year’s end, however, the more anxious teachers were about math, the more likely girls (but not boys) were to endorse the commonly held stereotype that “boys are good at math, and girls are good at reading” and the lower these girls’ math achievement. Indeed, by the end of the school year, girls who endorsed this stereotype had significantly worse math achievement than girls who did not and than boys overall.
These findings make intuitive sense, and lend further support for the need to better our mathematics education at all levels, or at the very least require primary educators to study mathematics more seriously. Teaching mathematics with confidence is not something that comes automatically, even for those who may have been good at math in their early years.
It’s interesting that boys weren’t more likely to endorse the view that boys are good at math and girls are good at reading if their teacher had math anxiety. I’d be curious to see what the case is in a classroom led by a male teacher, both with and without math anxiety. Given the dearth of male primary educators, however, this type of data may be harder to acquire. In any event, the lesson here is clear: if you want your daughter to not fear math, it wouldn’t hurt to demand that her teachers not fear it either. Or at the very least, demand that any math fear be exhibited only by male teachers. That may be a cheaper solution.
I’d also be interested in knowing whether this trend can be reversed by a suitably competent teacher. If a group of 2nd grade girls is taught math by a woman who is unqualified, but in 6th grade is taught by a woman who is exceptional, can this help undo the damage that the 2nd grade teacher has done? I would hope so.
In the aftermath of the Super Bowl, some of you fans may be dreading the next six months. To kick off this football drought, I’d like to highlight this article, which was featured on Yahoo yesterday. The article says that Saints quarterback Drew Brees should hope to lose the coin toss at the start of the game, because in the past 43 Super Bowls, the team that won the coin toss had only won 20 times.
An unlucky coin? Unlikely.
Um…what? Who cares? While 20/43 is slightly less than the expected 50%, this difference is not even close to being statistically significant. Actually, the fact that this ratio is only 1 1/2 games shy of the mean is pretty good. Matt Springer has posted an article that discusses why we shouldn’t really care about this difference.
Of course, the sample size is naturally restricted by the small number of Super Bowls, but if the author (Mark Pesavento) had really been interested in the question of whether or not the coin toss is correlated with the winner in a football game, he could’ve easily collected data over a couple of seasons and obtained an answer to the question. At the very least, he could’ve owned up to the fact that his analysis is worthless, but instead, to the critics he offers only the following rebuttal: “because of the small sample size, some statisticians argue that the win-loss record of coin-toss winners is statistically insignificant.”
This is completely disingenuous, because it suggests that there would be a debate among statisticians about the significance in the data Pesavento uses, when no such debate exists. Anyone with even a rudimentary background in statistics would understand that the sample size here would be too small to draw the conclusion he draws.
Moreover, Pesavento falls for one of the most common traps in statistics: mistaking correlation for causation. Even if the data was much stronger in indicating that the coin toss winner is at a disadvantage, this would not imply that Brees should hope to lose the toss. A correlation between these two effects does not imply a causal relationship between the two. I feel like I’ve discussed this before, but just in case, here’s a thorough discussion of this misconception.
Here this point is moot, since we don’t even have a correlation. I thought no one would need to point out that “No correlation does not imply causation,” but apparently we do.
Thankfully, most of the comments on Pesavento’s post are scathing in regards to his methods. But that’s cold comfort in light of the fact that the article was deemed fit for posting on the front page of Yahoo.
This is the third in a series of posts about pools used for betting on the outcome of football games (part one can be found here, and part two here). Let me briefly recall the setting, which is probably familiar to anyone who has been to a Super Bowl party. Typically, one bets on the outcome of a football game using a 10 x 10 grid. People can buy any number of the 100 squares on the grid, and when all the squares have been purchased, each row and each column is assigned a random digit from 0 to 9.
Suppose, for example, that you buy four squares, and after the rows and columns have been labeled, you find that you own square 3-7, square 2-5, square 9-0, and square 6-6. You will win money if, at the end of any one of the four quarters, the last digit in each team’s score matches your pair. For example, if the score after the 3rd quarter is 13-27, you will win some money, since the last two digits are 3 and 7, and you own square 3-7. There are variants of this: some pools only pay out every half, not every quarter, and usually the payouts vary by quarter, so that having the right square at the end of the game wins you more money than having the right square at the end of the first quarter.
Here's an example of a football pool which has been tagged in the four squares mentioned above.
In the first part of this discussion, we introduced a new way to conduct the pool: rather than looking at the last digit of a team’s score, we looked instead at the digital root of the team’s score. Recall that the digital root of a team’s score is obtained by adding the digits in their score. If that sum is between 1 and 9, we stop – if it is larger than 9, we compute the digital root again, until we get a digit between 1 and 9. For example, the digital root of 14 is 1 + 4 = 5, while the digital root of 38 is 2, since 3 + 8 = 11, and 1 + 1 = 2. We then analyzed the distribution of scores, and found that the digital root of a team’s score is more evenly distributed between 1 and 9 than the last digit of a team’s score is evenly distributed between 0 and 9 (this is subject to the convention that we assign 0 a digital root of 9, since 0 is the only number with digital root equal to 0).
In the second part of the discussion, we tackled questions of independence. Namely, we asked whether the last digit in one team’s score is independent of the last digit of the other team’s score, and similarly we asked whether the digital root in one team’s score is independent of the digital root of the other team’s score. In both cases, we found the answer to be negative.
The subject of this article is based on the following observation: when you have wagered in a traditional football pool, it’s not uncommon for a small number of squares to be hit with high frequency during the course of a game. For example, suppose you watch a game in which one team scores 7 points, then 3, then 7, then 3, while the opposing team never scores. This means that the game’s score will go from 0-0, to 7-0, to 10-0, and then to 17-0. So, while there are four unique scores in the game, with the usual football pool, only two squares will be hit: the 0-0 square, and the 7-0 square. However, with the digital root pool, four squares will be hit: again using the convention that we assign 0 a digital root of 9, the squares will be 9-9, 7-9, 1-9, and 8-9.
The reason the digital root pool hits more squares in this case is because whenever one team increases its score by 10, the last digit of their score will return to a previous value. However, with the digital root method, if a team increases its score by 10, the digital root increases by 1. Because a score increase of 10 is a relatively common occurrence in football (all one needs is a touchdown, extra point, and field goal), one may therefore guess that using the digital root pool, more squares should be hit during the course of the game.
Whether one would want more squares to be hit or not is up for debate, but I see certain benefits. For example, if more squares are hit during the game, then more people will have something invested in the game as it airs. If you are sitting on the square that represents the current score, you want the score to remain the same through the end of the quarter so that you can reap the rewards – but if the winning squares keep bouncing around between a small number of people, there may be fewer people actively invested in the score as the game progresses. This is especially true in Super Bowl parties, when many of the attendees are less interested in the game than they should be.
In other words, I’m of the belief that if more squares are hit, it’s a good thing. It therefore becomes natural to ask whether or not the digital root pool actually does hit more unique squares than the traditional pool. Thankfully, we have a wealth of data which we can use to answer this question.
I looked at all the games from this current season, and counted the number of boxes that would have been hit in each game using the traditional pool and the digital root pool. Averaged over 331 games (this includes preseason and postseason), the number of squares hit using the traditional pool is approximately 6.84. By comparison, the number of squares hit using the digital root pool is 8.43 – an increase of 1.59 boxes, or an increase of about 23%. This effect is amplified when one considers the fact that the digital root pool uses only 81 squares, as opposed to the traditional pool’s 100. This means that as a proportion of the total number of squares, the traditional pool hits about 6.84% of its squares, while the digital root pool hits 10.4% – here we have an increase of over 50%!
This is strong evidence that the digital root pool hits more squares than the tradition pool. In fact, the data shows that an average game will have a change in the score approximately 8.73 times, which is only a bit higher than the average number of boxes hit by the digital root pool. This makes sense when we slice the data another way: of the 331 games analyzed, in 252 of them the number of squares hit with the digital root pool was equal to the number of changes of score, meaning that no square got hit more than once. The same cannot be said of the traditional pool – in this case, the number of games in which no square got hit more than once was only 62.
The data has convinced me that the digital root pool may be better suited for festive gathering, where wagering on football will be but one of many activities designed to induce merriment. At the very least, it’s hard to argue that the traditional pool will hit as many squares as the digital root pool. Some may balk at a break from football pool tradition, but that’s ok. I won’t watch football games with them anyway.
This is just a quick note to welcome you to the new Math Goes Pop! We are still tweaking the look of the site, but hope you enjoy the changes.
If you haven’t already done so, I’d encourage you to subscribe to the RSS feed. If you’re already subscribed, please check your feed URL, as it has now changed from the blogspot address. The new URL is http://www.mathgoespop.com/feed. Of course, you may find it easier to just subscribe via the link up top. I encourage you to do so!
If you come here regularly, you know of my complaints regarding so-called “math holidays” that get plenty of press, but rarely have anything to do with actual mathematics. The most well known is pi day, celebrated here in the states on March 14th, also known here as 3/14.
Aside from the mathematical arguments one can make for or against this holiday, there is a larger problem. It’s all well and good to celebrate pi day on the date representing the first three digits of pi, but this is only possible if we write dates in the MM/DD format. Most of the world, however, uses the (more logical) DD/MM format, therefore depriving them of such a delicious play on numbers. Many loyal international fans of this holiday no doubt decry the fact that April has only 30 days, for otherwise they could simply celebrate pi day on 31/4. As it is, they are left with two options: Celebrate on 3/14 like those of us in the states, or enjoy a neutered version of this play on numbers by celebrating on 3/1.
Today I would like to propose an alternative to those for whom the DD/MM notation is standard. Rather than trying to work with imperfect solutions to the pi day problem, take a different number and celebrate it in your own way: the number e.
While e may not be as popular as its irrational sibling pi, it is no less important. No doubt many would argue that it is more important. It is certainly not as well-known in popular discourse, and so highlighting it, I would argue, is more important than highlighting the attention-whore known as pi. Moreover, since the decimal expansion of e begins with 2.71828183…, countries that use the DD/MM format could celebrate e day today, January 27th. Sadly, since February does not have 71 days, and since there are not 27 months in a year, people in America would be unable to celebrate in quite the same way – but given all the press that pi day has received over the past few years, I think that’s fair.
Of course, in order to celebrate the holiday properly, one needs activities. Topics could include the ways in which this fantastic number arises naturally, or a discussion of exponential growth (and orders of magnitude in general). One could also prove that e is irrational, a fact which follows quite easily from the Taylor series expansion of the exponential function ex at x = 1. Perhaps I’m being overly optimistic though – such a holiday would probably include less exciting activities, such as a recitation of the decimal expansion of e to a certain number of digits (a mind numbing activity which is practiced without fail every pi day).
Special consideration needs to be given to a replacement for the act of eating pie, which seems like a suitable activity to do on pi day, but not on e day (especially since the surfaces of pies are circular). I’m not sure what natural analogue exists – there is one thing that comes to mind when one wants to celebrate a day called “e day,” but I don’t want to promote drug use. Perhaps instead one could eat foods that start with the letter e, like eclairs, eggplants, and elephants. But these foods don’t work on a higher level, in that they don’t really relate to the number e in the way that the circular shape of a pie can be related to the number pi itself.
Eggs for e day?
There are obstacles to overcome, that much is certain. But if we’re going to celebrate holidays related to math, we may as well do a halfway decent job of it. So happy e day to you – don’t do anything I wouldn’t do.
Just as you can’t judge a book by its cover, it is not always easy to determine a person’s mathematical background based on his or her occupation. Sure, a burger flipper at McDonald’s may not look like the next Einstein, but how can you be sure she’s not just working a summer job to afford university? Conversely, just because someone is highly educated doesn’t mean he knows the difference between a prime and a composite number (although I’d argue that it should).
Case in point: Supreme Court justices may or may not know the meaning of the word orthogonal. Here’s a snippet from the oral arguments in the case of Briscoe v. Virginia (courtesy of blog The Volokh Conspiracy):
MR. FRIEDMAN: I think that issue is entirely orthogonal to the issue here because the Commonwealth is acknowledging -
CHIEF JUSTICE ROBERTS: I’m sorry. Entirely what?
MR. FRIEDMAN: Orthogonal. Right angle. Unrelated. Irrelevant.
CHIEF JUSTICE ROBERTS: Oh.
JUSTICE SCALIA: What was that adjective? I liked that.
MR. FRIEDMAN: Orthogonal.
CHIEF JUSTICE ROBERTS: Orthogonal.
MR. FRIEDMAN: Right, right.
JUSTICE SCALIA: Orthogonal, ooh.
(Laughter.)
JUSTICE KENNEDY: I knew this case presented us a problem.
(Laughter.)
MR. FRIEDMAN: I should have — I probably should have said -
JUSTICE SCALIA: I think we should use that in the opinion.
(Laughter.)
MR. FRIEDMAN: I thought — I thought I had seen it before.
JUSTICE SCALIA: Or the dissent.
(Laughter.)
MR. FRIEDMAN: That is a bit of professorship creeping in, I suppose.
While Friedman uses “orthogonal” in a bit of a metaphorical sense, this use is far from unprecedented – indeed, this use is even documented in the venerable internet database ubrandictionary.com, which defines orthogonal as a term that is “used to describe two things that are independent of one another. One does not imply the other.” Claiming that this usage is just a “bit of professorship” sounds a bit like a cop out. I wish Friedman had embraced it more completely.
In any event, the mathematical definition of orthogonal should be given in any halfway decent high school geometry course, if only as a synonym for perpendicular. The fact that Scalia and Roberts seem so unfamiliar with the concept is, at the very least, a little disappointing.
But all is not lost. On the other hand, last weekend Fox aired a special commemorating 20 years of The Simpsons, appropriately titled The Simpsons Anniversary Special: In 3-D! On Ice!. Several people contributed interviews to the special, including Mike Judge, creater of Beavis and Butthead and King of the Hill, among other comedic gems. Watch the clip below for a bombshell revelation:
That’s right – without The Simpsons, Judge believes he would be a math teacher. In fact, after doing some research online, I discovered that Judge didn’t begin playing with animation until the age of 26, while he was doing graduate studies in mathematics in the hopes of becoming a teacher.
Does this mean that Beavis and Butthead are smarter than Roberts and Scalia? Of course, some may cry out that this is an unfair comparison, but I think I can provide a fair answer.
First, let me begin by wishing a happy 2010 to you all. If you celebrate the holidays the way I do, then the past few weeks have seen you spending time with friends and family. And if you really celebrate the holidays the way I do, then some of that time with friends and family will have been spent with mathematical puzzles.
Very recently I was with a group of friends, discussing all that would come to pass in this new year. One friend, whose anonymity I will preserve by referring to him only as “Smith,” was in the enviable position of being the only one among us whose age divided the current year (I won’t embarrass him by revealing his age, but given that it’s a divisor of 2010, this certainly restricts the possibilities). Once we realized this, it became natural to ask how common an occurrence this should be. In other words, how often can you expect your age to divide the current year? Of course, implicit in this is a choice of calendar – for our purposes, we will stick to commonly used Gregorian calendar, although the results would be equally valid under a different choice (e.g. the Hebrew calendar or Islamic calendar). For example, if you were 1, 7, 41, or 49 last year, your age divided the year (of 2009). Next year, only the one year olds will win out, since 2011 is a prime number.
Depending on the year you were born, you may find that this happens quite frequently, or not very frequently at all. For example, if you were born in the year 0, you’re in luck, because your age will divide the current year for at least part of the year for every subsequent year. The phrase “part of the year” is important, because in a given year you will be two different ages – the age before your birthday, and the age on and after your birthday. Of course, this isn’t an issue if you were born on January 1st or December 31st, but we will ignore this (simpler) case.
Let’s take a more detailed example. Suppose you were born in 1982. In 1983, after your first birthday, your age will divide the year (since 1 divides everything). Similarly, in 1984, your age will divide the year after your 2nd birthday, since 1984 is even. And in 1986 your age will divide the year until your 4th birthday, since 1986 ÷ 3 = 662. Unfortunately, you will be too young to appreciate this arithmetic coincidence at any of these opportunities, and unless you live to be 661, you’ll never again be able to say that your age divides the year.
However, if you were born just a few years earlier, in 1979, you’ll find that your age divides the year quite frequently. In fact, by the year 2000, the only years in which your age wouldn’t have divided the year at all would have been 1987, 1988, 1993, 1994, 1996, 1997, and 1999.
Why is it that some years allow for one’s age to be divisible by the year quite frequently, while other years do not? The answer is quite simple. Suppose we let b denote the birth year, and we let a denote a person’s age. That person will be a years of age from their birthday in year b + a until their birthday in year b + a + 1. Therefore, your age will divide the year from your birthday until the end of the year if a divides b + a, or from the first of the year until your birthday if divides b + a + 1. So, the question becomes: when does a divide b + a, and when does it divide b + a + 1?
In the first case, since a always divides a, we know that a divides b + a if and only if a divides b. By the second same argument, we see that a divides b + a + 1 if and only if a divides b + 1. In other words, we conclude the following:
Your age will divide the current year if, and only if, either (i) it is between January 1st and your birthday, and your age divides the year after you were born, or (ii) it is between your birthday and December 31st, and your age divides the year you were born. To put it more simply, your age will divide the year for at least part of the time you are at that age if and only if that age divides the year of your birth or the year after your birth.
With this knowledge, it’s easy to see why people born in 1979 will have their age divide the current year more frequently than people born in 1982. In the former case, determining the set of ages which will divide the current year is equivalent to finding the divisors of 1979 and 1980. 1979 is a prime number, so it will never be the case that your age will divide the year between your birthday and December 31st (except after your 1st birthday); on the other hand, 1980 has a prime factorization of 2 x 2 x 3 x 3 x 5 x 11, which gives it a large number of small factors, and consequently a large number of solutions to the problem.
By contrast, if you were born in 1982, you won’t get many factors either way: 1982 factors as 2 x 991, and 1983 factors as 3 x 661. This is why, if you are born in 1982, your age won’t divide the current year after you’re 3.
While it’s not often that mathematics comes up when I’m with my friends at home, I certainly relish every opportunity. I hope that this may serve as an example to all of you who would like to make mathematics more of a part of your everyday life, especially in social circles into which math rarely intrudes. Single guys looking for first date conversation material are especially urged to keep this sentiment in mind.
This past September, a very strange thing happened. The worlds of mathematics and comics combined to give birth to the graphic novel Logicomix, written by Apostolos Doxiadis and Christos Papadimitriou, and illustrated by Alecos Papadatos and Annie Di Donna. The book gives a slightly fictionalized account of Bertrand Russell’s life, and uses this storyline as a gateway to explore the ideas in mathematical logic which were developed around the turn of the last century.
Combining mathematics and comics may sound like a recipe for disaster, but Logicomix has achieved a remarkable level of success. Not only has the critical response been exceedingly positive, but the book has also made the New York Times bestseller list. I’m assuming it was quite a popular gift item as well, because up through Christmas eve it was on back order at Amazon.com. It’s certainly rare for anything so fundamentally imbued with mathematics to break into the mainstream.
Here’s a video trailer for the book.
Having just finished the book, I understand why it has received so much praise. Unlike most works created for mass consumption that try to tango with mathematics, Logicomix offers a rare example of a work that is able to give some insight into what mathematics is about, and what mathematicians do, without making things too opaque for a general audience to follow along.
Russell, logicomicized.
Certainly the pretty pictures help. But more than that, by placing Bertrand Russell in the pilot’s seat, the reader is given a front row seat in the drama of early 20th century mathematical progress. Mathematics is given humanity: we see great thinkers struggle with fundamental problems, we see how their devotion to mathematics affects their relationships, and we see their work in a historical context as the world decays into war. Great mathematical ideas are not dissociated from their logical origins, but instead the reader sees in an organic way how a sizable chunk of mathematics from this time period was developed.
Russell in real life.
For example, a significant amount of time (some would consider too much) is spent explaining Russell’s paradox. There are many approaches to explaining this paradox for a non-mathematician: Logicomix tries to do it by means of the Barber paradox, but one can think of other ways to interpret it as well. For example, I stole the following explanation from Wikipedia:
Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library’s books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.
Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs – one of all the catalogs that list themselves, and one of all those that don’t.
The question is now, should these catalogs list themselves? The ‘Catalog of all catalogs that list themselves’ is no problem. If the librarian doesn’t include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.
However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the ‘Catalog of all catalogs that don’t list themselves’, the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves.
Notice that problems arise because of the presence of self-reference: asking whether the catalog of all catalogs that don’t list themselves should itself be listed in the catalog of all catalogs that don’t list themselves is a question of whether or not that catalog should reference itself. Russell’s proposed solution to the paradox involved creating a hierarchy so that such self-reference cannot be exploited. Later mathematicians offered their own solutions to this problem. Cleverly, the structure of Logicomix plays with self-reference as well, for there are many instances when the action cuts from early 20th century Europe to modern Greece, where the authors and illustrators are discussing their progress and their choices in bringing the relevant mathematics to life.
Logicomix also gives the reader some insight into Gödel’s incompleteness theorem, and shows the profound impact it had on the search for a logical foundation for mathematics. Most importantly, everything is explained in a way that is easy to understand.
If you watched the video above, you may have noted that the authors admit to being interested in this material because so many of the mathematicians they discuss ended up going insane. Thankfully, the book points out that this is the exception rather than the rule, and in fact the common thread that combines the mathematicians under their consideration is that all of them studied logic, in an attempt to put the foundations of mathematics onto solid footing. While their efforts to explain this connection are full of conjecture, it is nice to see them say at the outset that lunacy is not the necessary end point for all those who study mathematics.
My biggest complaint about Logicomix (if it could be called a complaint) is that it necessarily tackles only one slice of a very large pie. Certainly the developments in mathematical logic during this time offer a rich piece of mathematical history, and laid the foundation for the development of computers in the later part of the 20th century. But there are just as many other areas of mathematical history that would also be ripe for graphic novelization. Of course, I wouldn’t expect the authors to try and tackle everything in a single volume – instead, my hope would be for new volumes that use the same framework as Logicomix to tell stories about the development of other areas of mathematics. Should the authors be so inclined, I would encourage them to consider other figures to focus on for future work, so that other areas of mathematics can be illuminated. In doing so, they could bring into focus another feature of mathematics that is often misconstrued by the general public: the wide expanse of it all. This book certainly shows that mathematicians do more than multiply large numbers, but it would take several more books to show exactly how much more we do.
All things considered, this book is quite a success. That such a book could be created and executed so well gives me hope for the future of mathematics in popular culture. Logicomix shows that mathematical ideas don’t always need to be dumbed down to be made interesting or comprehensible – with the right explanation (and maybe some pretty pictures), the audience should have no trouble rising to the occasion, and with any luck they may learn some mathematics in the process.
My apologies for not writing this up sooner, as this book would have made a great gift for the mathematician in your family. But it would also make a good gift for anyone with an interest in mathematics, at any age and with any level of experience. You may be skeptical, but I encourage you to give this book a chance. There is a lot to enjoy here.
