Math Goes Pop! » pi http://www.mathgoespop.com Ruminations on the Intersection Between Mathematics and Popular Culture Sat, 04 Feb 2012 02:21:45 +0000 en hourly 1 http://wordpress.org/?v=3.2.1 Wedding Mathematics, Part 3 http://www.mathgoespop.com/2011/09/wedding-mathematics-part-3.html http://www.mathgoespop.com/2011/09/wedding-mathematics-part-3.html#comments Tue, 20 Sep 2011 02:50:09 +0000 Matt http://www.mathgoespop.com/?p=1391 Today I would like to wrap up my series on mathematics and weddings (a series begun here and continued here) with a little advice for soon-to-be brides and grooms who are looking to integrate some math into their celebrations.  If this describes you, then congratulations – not only on your upcoming nuptials, but also on the . . . → Read More: Wedding Mathematics, Part 3]]> Today I would like to wrap up my series on mathematics and weddings (a series begun here and continued here) with a little advice for soon-to-be brides and grooms who are looking to integrate some math into their celebrations.  If this describes you, then congratulations – not only on your upcoming nuptials, but also on the classy way you are looking to celebrate them.

For our own wedding, my bride and I decided it would be natural to incorporate some mathematics into the table numbers.  There is some freedom in how one decides to do this.  For example, we initially toyed with the idea of using numbers for the tables that were somehow significant to us and our relationship, but found it too difficult to come up with examples meeting this criterion.  If one wants intrinsically interesting numbers, there are many examples among the whole numbers (I was particularly fond of using the smallest whole number expressible as the sum of cubes in two different ways).  In the end, though, we decided to expand the realm of p0ssibilities beyond the range of whole numbers.  This turned out to be a good decision, both aesthetically and educationally.

Table number e. Hat tip to Caroline for the shot.

If you are looking for a way to incorporate some math into your celebration, the table numbers are certainly one option.  At each of our tables we had a small placard, with the number on one side and a brief description of the number (and some table exercises!) on the reverse.  I tried to have sympathy for our audience, and give descriptions that a general audience would be able to understand, though I gave myself more flexibility with a table occupied by other math students.  For sake of completeness, here are all the numbers we used, along with their descriptions (see if you can tell which table had the math students!).  In no particular order:

1. \pi (see here for more).

The ratio of a circle’s circumference to its diameter, \pi is perhaps the most famous irrational number. Historically, \pi has also been known as Archimedes’ constant, and Archimedes himself proved that 3\frac{10}{71}<\pi<3\frac{1}{7}.

More than one trillion digits of the decimal expansion of \pi have been computed, and folks with nothing better to do than recite those digits come together each \pi day (March 14th, naturally) to see who has memorized the longest string of numbers in the decimal expansion. If you’re looking for more interesting properties of \pi, though, here are a few to mull over:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots,

\frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{2+\sqrt{2}}}{2}\cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \ldots,

 

\frac{\pi}{2} = \frac{2}{1}\cdot \frac{2}{3}\cdot \frac{4}{3}\cdot \frac{4}{5}\cdot \frac{6}{5}\cdot \frac{6}{7}\ldots .

Table exercises!

1. Use geometry to show that 2\sqrt{2}<\pi<4. These bounds are not as good as those of Archimedes, but they are easier to derive.

2. (Harder!) Explain why \pi is irrational, i.e. why it cannot be written as a fraction p/q where p and q are integers.

2. e (see here for more).

e, a.k.a. Euler’s number, a.k.a. Napier’s Constant, is an irrational number of fundamental importance. While it lacks the general public awareness of a number like \pi, I assure you it is no less charming. Typically defined as the limit

e:=\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n},

e enjoys many other identities, including

e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\ldots,

and

e=\lim_{n\rightarrow\infty}\frac{n}{\sqrt[n]{n!}}.

e also determines the base of the exponential function e^{x}, unique among all exponential functions in the study of calculus because it is equal to its own derivative.

Table exercises!

1. Use one of the identities above to verify that e < 3.

2. Use one of the identities above to verify that e is irrational, i.e. that it cannot be written as a ratio p/q where p and q are integers.

3. Suppose each of you has brought a hat to this wedding. Everyone leaves his or her hat inside, and when a person leaves, he can’t be bothered to search for the hat he brought, and simply takes one from the hat pile at random. Show that the probability nobody ends up with the hat they came in with tends to 1/e as the number of people increases.

3. \zeta(3) (see here for more).

Take all the perfect cubes (1^{3}=1, 2^{3}=8, 3^{3}=27, and so on), take the reciprocals of all those perfect cubes, and add them all together. You will end up with a number that is sometimes called Apéry’s constant, and is written

\zeta(3) = 1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\frac{1}{4^{3}}+\ldots \approx 1.202\ldots .

The constant is named in honor of Roger Apéry, who proved in 1978 that this number is irrational. Intuitively, one can interpret \frac{1}{\zeta(3)} as the probability that three randomly chosen whole numbers will have no prime factors in common.

One can consider more general numbers as well. For example, for any whole number k bigger than 1, the sum

\zeta(k)=1+\frac{1}{2^{k}}+\frac{1}{3^{k}}+\frac{1}{4^{k}}+\ldots

will yield some finite value. When k is even, one has nice formulas for the values, for instance \zeta(2)=\frac{\pi^{2}}{6}, \zeta(4)=\frac{\pi^{4}}{90}.

In fact, it is possible to let k take on quite a large range of values. The function one gets is called the Riemann zeta function, and lies at the center of one of the most famous unsolved problems in mathematics.

Table exercises!

1. Show that \zeta(1)=\infty.

2. Given that \zeta(2)=\frac{\pi^{2}}{6}, show that 1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\ldots=\frac{\pi^{2}}{8}.

4. \gamma (see here for more).

\gamma, a.k.a. the Euler-Mascheroni constant (not to be confused with Euler’s number e), is perhaps best introduced geometrically. Consider the following figure:

The black portion of the area pictured above is found by drawing rectangles between two integers n and n + 1 with height 1/n (the rectangle between 1 and 2 has height 1, the rectangle between 2 and 3 has height 1/2, and so on), and subtracting the area under the graph of the function y = 1/x.  The total black area, if this picture were to be extented out to infinity, would represent the number \gamma.

\gamma can be approximated by its decimal expansion, \gamma\approx0.5772\ldots, and while this number comes up quite naturally in number theory and mathematical statistics, surprisingly little is known about it. For example, it is unknown whether or not \gamma is a rational number (unlike constants such as \pi or e, which are known to be irrational).

Table exercises!

1. Using geometry and the figure above, show that \gamma>\frac{1}{4}+\frac{1}{12}+\frac{1}{24}+\frac{1}{40}+\ldots.

2. Show that the sum on the right hand side of the inequality in the first exercise equals \frac{1}{2}, so that \gamma>\frac{1}{2}.

5. \infty (see here for more).

\infty is a concept of central importance in mathematics, and ergo, a concept of central importance in all things. While the figure-eight symbol for infinity is known and loved by all, it was not introduced until the year 1655, though many ancient cultures grappled with the idea of the infinite.

Though \infty may seem like a single idea, great minds have shown that not all infinities are created equal. For example, the mathematician Georg Cantor showed that even though there are infinitely many whole numbers, and there are infinitely many real numbers, there are (in a sense that can be made rigorous) infinitely many more real numbers than counting numbers.

On a related note, the love Matt and Meg feel for you all for standing with them on this day is undoubtedly infinite. How this compares to their love for one another, however, is a problem that has yet to be investigated.

Table exercises!

1. Show that there are infinitely many prime numbers.

2. How does the number of even integers compare to the number of integers? Are there more of one type of number?

3. Suppose a set is finite with N elements. Show that the set of subsets of the original set is finite with 2^{N} elements.

6. \varphi (see here for more).

Suppose two line segments have length a and b, with a larger than b. If the ratio of a to b is the same as the ratio of a + b to b, this ratio is called the golden ratio, and is written \varphi. In other words,

\varphi=\frac{a}{b} = \frac{a+b}{a} = 1 + \frac{1}{\varphi}.

This, in turn, implies that \varphi^{2}-\varphi-1=0, or (by the quadratic formula)

\varphi=\frac{1+\sqrt{5}}{2}\approx1.618\ldots.

The golden ratio has a rich history, both mathematically and artistically. It is also closely related to the Fibonacci sequence, the sequence of numbers whose first two terms are 0 and 1, and where all subsequent terms are found by adding the previous two terms. In other words, the sequence begins 0,1,1,2,3,5,8,13,\ldots. If we let F_{n} denote the n^{th} Fibonacci number (so F_{0}=0, F_{7}=13, and so on), then

\varphi=\lim_{n\rightarrow\infty}\frac{F_{n+1}}{F_{n}}.

Table exercises!

1. Show why the above limit formula for \varphi is true.

2. Show that F_{n}=\frac{\varphi^{n}-(1-\varphi)^{n}}{\sqrt{5}}.

3. Show that for any n, F_{0}+F_{1}+F_{2}+\ldots+F_{n}=F_{n+2}-1.

7. \Lambda (see here for more).

The de Bruijn-Newman constant, the value of which is currently unknown, is intimately connected to the Riemann Hypothesis. There exists a class of functions H_{t}(x), one for each real number t. H_{0}(x) is essentially the Riemann \xi function, and in particular, the Riemann Hypothesis is true if and only if H_{0}(x) has only real zeros.

Here are some properties of the family of functions H_{t}(x):

1. H_{t}(x) has only real zeros for any t\geq1/2.

2. If H_{t}(x) has only real zeros, then for any t^{\prime}\geq t, H_{t^{\prime}}(x) has only real zeros too.

3. There exists a real value t_{*} such that H_{t_{*}}(x) has at least one non-real zero.

These properties combine to show the existence of a constant \Lambda, lying somewhere in the range -\infty<\Lambda\leq1/2, such that H_{t}(x) has only real zeroes if and only if t\geq\Lambda. This is how the de Bruijn-Newman constant is defined. Moreover, the Riemann Hypothesis is equivalent to the statement that \Lambda\leq0.

The current best estimates for \Lambda state that

-2.7\times10^{-9}<\Lambda\leq1/2,

so if the Riemann Hypothesis is true, it is, in some sense, “just barely” true. In particular, it’s possible that \Lambda=0, in which case you are really just sitting at the 0 table. But while your table may be marked as such, you should know that none of you are zeros in our hearts.

Table exercises!

1. Prove or disprove the Riemann Hypothesis.

8. i (see here for more).

i, more formally known as the square root of -1, is defined to be one of two solutions to the equation x^{2}=-1 (the other solution being -i).

While this might seem like an arbitrary construction, in the larger context of history, it makes perfect sense. Just as the whole numbers are perfectly good for solving basic counting problems, but may be insufficient for problems involving debts or losses (where negative numbers play a prominent role), or problems involving rates or ratios (where fractions take the spotlight), the extension of numbers to include i leads to a wide variety of applications. This include (but are not limited to) applications in electrical engineering, signal processing, and fluid dynamics.

i is also one of the key ingredients in Euler’s identity, one of the most popular formulas in mathematics. This formula states that e^{i\pi}+1=0, and is noted for its unification of five constants of fundamental importance in mathematics: e, \pi, i, 1 and 0.

Table exercises!

1. Show that i^{n}=1 whenever n is divisible by 4.

2. Find all x satisfying the equation x^{4}-1=0.

3. The set of complex numbers is defined as the set of all a + bi, where a and b are real numbers. 1 + i is a complex number, as is \sqrt{2}-7i. Can you define an addition law on the set of complex numbers? A multiplication law?

9. \rho (see here for more).

The plastic constant \rho can be viewed as a cousin to the golden ratio \varphi (see the \varphi table for more information). Formally, \rho is equal to the real root of the equation x^{3}=x+1. The value of \rho is

\rho=\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}\approx1.3247\ldots.

Just as the golden ratio is intimately related to the Fibonacci sequence, the plastic constant is related to a sequence known as the Padovan sequence. The first three numbers in the Padovan sequence are given by

P_{0}=P_{1}=P_{2}=1,

and the nth term is given by adding two earlier terms in the sequence:

P_{n}=P_{n-2}+P_{n-3}.

For example, the first few terms in the sequence are given by 1,1,1,2,2,3,4,5,7,9,\ldots.

One can similarly construct a sequence known as the Perrin sequence. This sequence is similar to the Padovan sequence, but in this case, the equations needed to get started are A_{0}=3,A_{1}=0,A_{2}=2,A_{n}=A_{n-2}+A_{n-3}. In either case,

\lim_{n\rightarrow\infty}\frac{A_{n+1}}{A_{n}}=\rho=\lim_{n\rightarrow\infty}\frac{P_{n+1}}{P_{n}}.

Table exercises!

1. Show why the limit formulas given above are true.

2. Show that the first few terms of the Perrin sequence are 3,0,2,3,2,5,5,7,10,\ldots.

3. Show that if p is a prime number, A_{p} is divisible by p.

10. \sqrt{2} (see here for more).

Along with \pi, \sqrt{2} is probably the most well known number on display here. While it may seem mundane, \sqrt{2} has an interesting mathematical history, notably because it was one of the first examples of an irrational number (i.e. a number that cannot be expressed as a fraction p/q where p and q are both integers). An early proof of this fact is attributed to the Greek thinker Hippasus, a follower of Pythagoras; legend has it that when he discovered \sqrt{2} was irrational, the result was so controversial that he was thrown out to sea by his colleagues and drowned.

These days, mathematics is (for the most part) less fraught with peril. The following elegant identities involving \sqrt{2} have been met with much less controversy:

\sqrt{2} = 1 + \frac{1}{2+\frac{1}{2+\frac{1}{2+\ldots}}},

\sqrt{2} = \left ( 1+\frac{1}{1} \right )\left ( 1-\frac{1}{3} \right )\left ( 1+\frac{1}{5} \right )\ldots,

\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}} = 2.

Table exercises!

1. Prove that \sqrt{2} is irrational (make sure you are removed from any large bodies of water).

2. Try to prove the identities written above.

3. For which whole numbers m is \sqrt{m} a rational number?

Enjoy the table exercises!

]]> http://www.mathgoespop.com/2011/09/wedding-mathematics-part-3.html/feed 0 Second Annual Tau Day: Interview and Ideas! http://www.mathgoespop.com/2011/06/second-annual-tau-day-interview-and-ideas.html http://www.mathgoespop.com/2011/06/second-annual-tau-day-interview-and-ideas.html#comments Tue, 28 Jun 2011 20:19:06 +0000 Matt http://www.mathgoespop.com/?p=1277 Last year marked the dawn of a new era in mathematical holidays.  Spearheaded by Dr. Michael Hartl, Tau Day (celebrated today, June 28th) is an attempt to draw awareness to what he sees as a fundamental error in the definition of the beloved circle constant .  In particular, he (and others) argue that the more natural . . . → Read More: Second Annual Tau Day: Interview and Ideas!]]> Last year marked the dawn of a new era in mathematical holidays.  Spearheaded by Dr. Michael Hartl, Tau Day (celebrated today, June 28th) is an attempt to draw awareness to what he sees as a fundamental error in the definition of the beloved circle constant \pi.  In particular, he (and others) argue that the more natural choice of the circle constant should be 2\pi, which he affectionately dubs \tau.  I outlined the reasons for this in a post last year, though if you have the time, I highly encourage you to read Hartl’s Tau Manifesto.

This year, I thought it would be nice to talk with Dr. Hartl in more detail about his inspirations for Tau Day, and where he envisions it in the future.  He was gracious enough to agree to a brief interview, which I humbly submit to you here.

*

Q: When did you first discover that \pi was “wrong”?  Did you have an intuition that something was amiss before reading Bob Palais’s 2001 article in The Mathematical Intelligencer?

A: I don’t remember how deep my suspicions about \pi ran before I encountered that article, but “\pi Is Wrong!” definitely opened my eyes, and every section of The Tau Manifesto owes it a debt of gratitude.

Q: What inspired you to write your own manifesto on the subject?
A: I saw that “π Is Wrong!” was getting noticed on social news sites like reddit and Hacker News, but it hadn’t crystallized into a movement. I perceived the opportunity to write an article with a dramatic narrative arc–combined, of course, with an official holiday, Tau Day–that could spark such a movement. In short, I saw the potential for a social hack, and it was too good an opportunity to pass up.
Q: What has the response been like to your manifesto?  In general, would you say people have been supportive, or are pi devotees too large in number?

A: Support has been overwhelmingly positive. I monitor Twitter mentions of Tau Day, and nearly every commenter has something nice to say.

Q: What is your ultimate goal with this project?  Would you like to see tau replace pi in textbooks?  Would it be enough for students to be exposed to tau concurrently with pi when they learn trigonometry?  Given that pi is such an ingrained part of mathematics education, do you have any thoughts on how best to steer this massive ship towards a new definition of the circle constant, especially for students who are first being exposed to trigonometry?

A: As a social hack of geek culture, the project has already exceeded my expectations. At technical conferences, people often recognize me as “that tau guy”. That said, the problem with pi is real, and I do believe that adding tau to the elementary curriculum would make mathematics more intuitive and more fun.  Since the installed base of pi users is so big, the only hope from my perspective is a grassroots effort, which based on reader feedback does seem to be happening. Someday, perhaps the American Mathematical Society and its foreign equivalents will get their act together and we can have a top-down effort as well, but for now it’s bottom-up all the way.

Q: Besides this debate over the circle constant, are there any other anachronisms in math and science education that you feel ought to be addressed (for example, something along the lines of Ben Franklin’s choice for the sign of electric charge, which you mention in your manifesto)? Aside from the mathematics itself, what can students learn from these discussions over which choices are more natural than others?

A: Fixing the sign of electric charge (in short, electrons, not protons, should be positive) is virtually impossible, since all the old textbooks would have to be rewritten. In contrast, switching from pi to tau can happen incrementally. There are some other anachronisms, but I’m not sure they’re worth fixing. (The temperature scale, for instance, is subtly broken, but what we have is probably good enough.)

Students can learn from this subject that notation matters, and that even geniuses (e.g., Euler) sometimes make mistakes. They can also learn that just because (nearly) everyone believes something, that doesn’t make it true.

Q: A big contributing factor to Pi Day’s success has undoubtedly been the food.  Besides eating twice as much pie, do you have any ideas on how to build Tau Day into a distinct mathematical holiday?

A: Tau Day happens during the summer, so perhaps we could add a distinctive outdoor component. Tau Day at the beach? I’m certainly open to suggestions!

*

Though I’m usually a curmudgeon when it comes to mathematical holidays, Tau Day does present a somewhat unique educational opportunity, and since it is still new to the scene, there is ample opportunity for people to contribute to future traditions.  It is in this spirit that I offer the following suggestion for today (and future Tau days!):

1. Embrace the season.

I agree with Dr. Hartl here.  Kids are out of school, and this might seem to put Tau Day at a distinct disadvantage.  On the other hand, a summertime holiday naturally lends itself to outdoor activities (at least in this hemisphere). Since tau is all about relating the circumference of a circle to its radius, there are many ways to explore this relationship in an outdoor setting.  If you’re celebrating at the beach, you could have a circle drawing contest, where each contestant is given a line in the sand and tries to draw a perfect circle with the given line as its radius.  The circle for which the ratio of circumference to radius is closest to tau would be declared the winner.  Or, if you are celebrating by a lake, you could attempt to measure the circumference of the lake, and use it to determine the size of a circle with equal circumference.  Planned carefully enough, one could hint at the isoperimetric inequality (though perhaps not too explicitly, depending on how excited your kids are to do math during the summer).  Any activity involving some kind of perimeter measurement could work here.

2. Cut the memorization.

As my readers know, I am no big fan of the recitation contests that have somehow become a Pi Day tradition, in which people compete to see how many digits of pi they can recite.  Reasons for my objection can be found here.  Given that pi and tau are so closely related, it might be tempting to introduce a similar contest for Tau Day.  But these contests offer little in the way of actual mathematical learning, and are terrible PR for mathematics in general.  In order to help Tau Day mature into its own independent entity, I would advocate for removal of any recitation contests.  If the focus is on a mathematical constant, let’s focus on some real mathematical insights – this would be more educational, and could be more fun too.

What would these activities look like?  There’s plenty of freedom here.  If you have kids interested in computers, one of my readers wrote up some Tau Day activities related to formal proof writing and machine automated proof verification.  There is some cool stuff here, though sitting for too long in front of the computer may run counter to the first suggestion.  Whatever you decide, the purpose should be to emphasize mathematics as a creative pursuit full of ideas, not one that relies solely on blind memorization.

3. Take the food to the next level.

Non-math students who enjoy Pi Day probably enjoy it for the food.  If we are to hook people on Tau Day, food will probably remain an important component.  But if you advocate that tau should take the throne from pi, then it seems only natural that the food on Tau Day needs to be cranked up to 11.

As tau is nothing more than two times pi, pie still remains a natural food choice – simply make twice as much.  I think we can do better, though.  One idea: the Tau Day Pumpple.


(Images courtesy of Foodaphilia via Gawker.)

The Pumpple consists of two pies – one pumpkin, and one apple.  This takes care of the pun.  To really take it to the next level, though, the two pies are then baked inside of a cake.  I can think of no better way to celebrate.

Now, given that it is the summertime, perhaps a pumpkin pie isn’t entirely appropriate.  With so much fruit in season, one has tremendous choice in selecting a dessert appropriate for today’s festivities.  One could bake a Chapple, perhaps (cherry and apple), or maybe even a Bleach (blackberry and peach).  As long as two pies are baked inside of a cake, the spirit of the holiday will be honored.

Any other suggestions for Tau Day festivities?  This has the potential to be the only math holiday I’d willingly support, so I hope some truly exceptional traditions take root.

(Thanks to Michael Hartl for taking the time to answer some questions, and to Jim for the Wikiproofs link!)

]]> http://www.mathgoespop.com/2011/06/second-annual-tau-day-interview-and-ideas.html/feed 3 Math Clock Showdown http://www.mathgoespop.com/2011/02/math-clock-showdown.html http://www.mathgoespop.com/2011/02/math-clock-showdown.html#comments Sat, 05 Feb 2011 05:41:19 +0000 Matt http://www.mathgoespop.com/?p=1068 When shopping for gifts for someone, there are a few wells from which one frequently draws inspiration.  A person’s favorite TV show, for example, or favorite band; such preferences can often provide good fodder for gift ideas.  One’s career can also be included in this list – in my case, the result is that I am . . . → Read More: Math Clock Showdown]]> When shopping for gifts for someone, there are a few wells from which one frequently draws inspiration.  A person’s favorite TV show, for example, or favorite band; such preferences can often provide good fodder for gift ideas.  One’s career can also be included in this list – in my case, the result is that I am frequently the recipient of math-themed paraphernalia.

I’ve written before about my mixed feelings regarding math t-shirts.  Today, though, I’d like to tackle a different type of gift: the math clock.  This is inspired, in part, by a gift I received from my grandmother (bless her heart) over the holiday.  The gift, pictured below, was an analog clock in which the numbers have been replaced by (what one would hope to be) mathematically equivalent expressions.

Figure 1: Clock with a black background.

Don’t tell her, but we haven’t yet put this clock up in our apartment.  In my own defense, this is mostly because we already have a math clock displayed prominently in the kitchen.  My future wife says that she gave me our first clock, and this is a claim I have no reason to doubt.

Figure 2: Clock with a white background.

Sadly, our apartment is simply not big enough for two mathematically themed clocks.  The question becomes, then, which one should be kept in-house, and which one should be relegated to the dungeon of an office I hold at the math department?  It seems that the most natural way to answer this question is to compare the mathematics of each clock side by side.

Let’s start at the top and work our way around.

12: 6 \cdot 2 vs. \sqrt[3]{1728}

The black clock has a simpler expression, but perhaps it’s a little too simple.  At least the white clock asks you to work a little for it.  Point: white clock.

1: 102,413 – 102,412 vs. B^{\prime}_{L}

The white clock’s expression for 1 is a little to esoteric for my taste.  The notation is meant to symbolize Legendre’s constant – this number is related to the asymptotic behavior of prime numbers, and historically it was believed to be greater than 1 for some time, based on numerical evidence.  But if you didn’t know all that, there’s no way you could make the connection between the notation and the number 1.  At least the expression in the black clock makes the connection to 1 clear.  With a little more information so that the casual time-teller could have learned something about primes, the white clock may have had the upper hand.  As it stands, though, I must side with the black clock here.

2: \sqrt{4} vs. \sum_{i=0}^{\infty}1/2^i

Here I must tip my hat to the infinite sum.  The square root is nice, but the sum is nicer, and if you’re trying to impress a date who doesn’t know about geometric series, this will provide you with an excellent opening.  Point: white clock.

3: 198 \div 66 vs. some XML garbage.

Come on, white clock.  That isn’t even math!  Point: black clock.

4: 50/2 = 100/x vs. 2^{-1}(mod7)

I prefer the poetry of including “clock arithmetic” on the face of a clock.  Plus, modular arithmetic (as it is more professionally known) is a topic that the general population is not always exposed to, even though it’s not hard to explain.  I’ll take any opportunity I can get for a clock to educate the masses.  Point: white clock.

5: 630 \div 126 vs. (2\varphi - 1)^2

Given my stance on 1, this may seem a little hypocritical, but I’m going to give the edge to the white clock here.  Part of the reason is that the black clock loses steam pretty quickly – out of the 12 numbers, 3 are expressed in terms of long division.  Come on, guys.

Besides, since we know that the expression on the white clock equals 5, this allows us to solve for \varphi and obtain the golden ratio.  This is something someone could discover on his or her own, perhaps with the aid of something like Wolfram Alpha.  So the comparison to the 1 o’clock entry isn’t quite apples to apples.  Or at least, that’s what I’ll keep telling myself.

6: \frac{1}{8}\cdot\frac{96}{2} vs. 3!

The factorial is a little less conventional, but every student should encounter it at some point.  Here I’m giving the edge to the white clock again.

7: 52 - x^2 + x = 10 vs. 6.\overline{9}

Presumably, the black clock wants us to solve for x, using the quadratic formula or something.  I don’t get it, though – if they’re going to express 7 as an unknown in a quadratic function of x, why would they write an equation that has two solutions, one of which isn’t 7? Since 52 - x^2 + x = 10 is the same as 42 - x^2 + x = 0, here’s a graph of 42 - x^2 + x, so you can see the two roots:

I realize I’m being a little pedantic (after all, there isn’t any negative 6 o’clock), but it would’ve been just as easy to write a quadratic that had only 7 as its root.  Here’s one: x^2 - 14x + 50 = 1.

Besides, the white clock’s entry for 7 is good in its own right.  No contest here, white gets the point.

8: \sqrt{64} vs. some dots.

I’ll give it to the black clock here.  The white clock is expressing 8 in base 2, but I don’t know why they don’t do it using digits.  Probably because they also play around with the base in the next number, where they write 9 in base 4 as 21_4.

9: 3(\pi - .14) vs. 21_4.

If I could give negative points, I would give them to the black clock here.  Their expression doesn’t evaluate to 9; instead, the clock only perpetuates common misunderstandings about the number \pi.  Admittedly, the black clock does give a fairly good approximation, but I’ve never heard of 9.004778… o’clock.

10: -8 = 2 - x vs. \begin{pmatrix}5\\2\end{pmatrix}

These are both worthy contenders.  For the sake of fairness, since I gave the white clock the point for 3!, I’ll choose the black clock here in favor of the white clock’s binomial coefficient.

11: 1221 \div 111 vs. some hexadecimal representation of 11.

The long division is redundant, but in a sense, so is the white clock’s entry – we’ve already seen two other cases of representing a number in a different base.  In this case, I’ll defer to the one that’s clearer.  Point: black clock.

By my count, the final score is 7 points for the white clock, 5 for the black.  It was a close match, but it looks like a decision has been made.  Regardless of the outcome, though, both clocks have their share of problems.

I should point out that, somewhat surprisingly, these are not the only math clocks on the market.  Here are even more examples.  The one which speaks to me the most, though, is probably the last one.

]]> http://www.mathgoespop.com/2011/02/math-clock-showdown.html/feed 2 Watch Out for The Secret Number http://www.mathgoespop.com/2011/01/secretnumber.html http://www.mathgoespop.com/2011/01/secretnumber.html#comments Thu, 06 Jan 2011 07:24:08 +0000 Matt http://www.mathgoespop.com/?p=999 I would like to offer my somewhat reserved congratulations to the helmers of the upcoming film project titled The Secret Number, whose Kickstarter project ended today having exceeded its fundraising goal of $10,000 (I’ll also point out that this isn’t the first time Kickstarter has made an appearance on this blog).  The film, a senior thesis . . . → Read More: Watch Out for The Secret Number]]> I would like to offer my somewhat reserved congratulations to the helmers of the upcoming film project titled The Secret Number, whose Kickstarter project ended today having exceeded its fundraising goal of $10,000 (I’ll also point out that this isn’t the first time Kickstarter has made an appearance on this blog).  The film, a senior thesis for director Colin Levy, is based on a short story of the same name, and is the reason behind my inclusion of the word “reserved” in the sentence above.  By way of introduction, please take a look at the filmmakers’ fundraising video:

As you can see, the story centers around a mathematician who claims to have discovered an integer between 3 and 4.  Forgetting the mathematical particulars for a moment, the source material worries me, mostly because the mathematician featured in the story has been hospitalized following a nervous breakdown brought on by his work (if you have a chance, I’d encourage you to read the story for yourself – the whole thing comes in at under 2,000 words).  Do we really need another film exploring the psyche of the brilliant but frail mathematical mind?

Regarding the mathematical content itself, that frustrates me a little bit in the same way that a film like Pi frustrates me.  Often times, the lack of mathematical understanding is used to create a false illusion of mystery.  There isn’t really anything interesting about the question “What if there was an integer between 3 and 4?” in the same way that there isn’t really anything interesting about the question “What if bananas tasted like apples?”  It’s not as if there is a lack of open mathematical questions that even the layperson could understand (think of the twin prime conjecture, for example, or Goldbach’s conjecture) and which might supply the necessary mystique without sacrificing mathematical authenticity.  Of course I realize this is not a problem unique to mathematics – any specialist will likely find fault with a film focusing on that specialty.  Some faults are more apparent than others, however, and the potential for a misstep when working with this source material seems quite large.

Hopefully this film will eschew the use of the slinky.

Nevertheless, kudos to the filmmakers for exceeding their fundraising goal.  To have a student film budget in excess of $10,000 is quite a feat, and the publicity from their campaign can’t hurt either.  I look forward to seeing their final product, and I hope they can avoid the cliches that so frequently arise when telling these types of stories.  Only time will tell.  I will admit, though, that some of the production art is pretty slick.

(Hat tip to Meg for sending a link to this project my way.)

]]> http://www.mathgoespop.com/2011/01/secretnumber.html/feed 4 Pi, I Shake My Fist at You http://www.mathgoespop.com/2010/11/pi-i-shake-my-fist-at-you.html http://www.mathgoespop.com/2010/11/pi-i-shake-my-fist-at-you.html#comments Wed, 24 Nov 2010 22:02:33 +0000 Matt http://www.mathgoespop.com/?p=953 A couple of days ago I watched a video that really depressed me.  Here‘s a link to a local news story from Ankeny, Iowa – I’d encourage you to take a look at the news clip there (unfortunately, I can’t embed it here).  The story concerns a 6th grade student who has memorized the decimal expansion . . . → Read More: Pi, I Shake My Fist at You]]> A couple of days ago I watched a video that really depressed me.  Here‘s a link to a local news story from Ankeny, Iowa – I’d encourage you to take a look at the news clip there (unfortunately, I can’t embed it here).  The story concerns a 6th grade student who has memorized the decimal expansion of pi to 340 or so digits.

In and of itself, this might not seem like a particularly newsworthy achievement – as any Pi Day aficionado can tell you, there are people who have memorized more digits.  Perhaps what makes it newsworthy is the fact that the student is only twelve years old, or, more perversely, the fact that his accomplishment came in response to the challenge of his math teacher, who asked his students to memorize as many digits of pi as possible.  By far the most depressing part of the video is a brief clip that shows all the students in the classroom mindlessly rattling off successive digits of pi.  The lack of enthusiasm is almost palpable.

Now, I don’t want to come off as too much of a curmudgeon here.  I have no doubt that this student is stoked that he made it on to TV for an academic achievement, regardless of the actual merits of that achievement (at least the student is aware enough to remark that the information he’s memorized will probably never be put to use).  That’s fine – he has every right to be proud of himself for making it onto the local news.  What really gets my goat is the fact that this teacher thought it would be a good idea to make students memorize digits of pi.  I can think of few better ways to dampen a natural enthusiasm for mathematical learning than by asking students to memorize a series of digits that will have no practical value for any of them, ever.  It would be like having an English teacher ask students to memorize a random string of words which, taken collectively, didn’t teach the student anything about vocabulary or grammar.

Is there any benefit to this exercise?  According to the teacher, “The ability to memorize that much stuff has to help tremendously.”  Well, ok.  But aren’t there more important things to learn about in math class?  Is math class really the best venue to discover a talent like this?  I am fairly certain that students in Singapore aren’t spending class time and homework time memorizing digits of pi.  I’m sure this teacher has good intentions, but I fail to see much value in this apparently newsworthy event.  The mystique of the number pi, I suppose, never fails to attract attention.

If this exercise is what gets this sixth grader interested in math, then by all means he should memorize as many digits of pi as he can.  For the vast majority of students, however, such an exercise is probably beyond tedious.  I can only hope that this news story doesn’t inspire other teachers to compel other students to do the same thing.

]]> http://www.mathgoespop.com/2010/11/pi-i-shake-my-fist-at-you.html/feed 1 Happy Tau Day? http://www.mathgoespop.com/2010/06/happy-tau-day.html http://www.mathgoespop.com/2010/06/happy-tau-day.html#comments Tue, 29 Jun 2010 00:52:28 +0000 Matt http://www.mathgoespop.com/?p=483 In the past, I’ve used this blog as a platform to make clear my mixed feelings about Pi Day, a math themed holiday celebrated every year on March 14th (3/14, har har) in honor of the beloved mathematical constant .  My thoughts on the subject can be found here.

It would seem that I am not alone . . . → Read More: Happy Tau Day?]]> In the past, I’ve used this blog as a platform to make clear my mixed feelings about Pi Day, a math themed holiday celebrated every year on March 14th (3/14, har har) in honor of the beloved mathematical constant \pi.  My thoughts on the subject can be found here.

It would seem that I am not alone in my frustration.  Michael Hartl, an educator and entrepreneur (as well as a Ph.D. graduate from Caltech), has just today launched a website in favor of Tau Day as a replacement for Pi Day.  However, his argument (based on a 2001 paper by Bob Palais) goes a step farther – he argues that \pi day shouldn’t be celebrated because \pi isn’t the fundamental constant we should be considering!  Rather, he argues that the true fundamental constant is 2\pi, which is approximately 6.283185… .  Hartl argues that this should be the fundamental constant of interest, and renames it \tau (for reasons given on the website).

Why should this be viewed as a more fundamental constant?  Recall how \pi is defined – it is the ratio of a circle’s circumference to its diameter.  But a circle itself is more naturally defined in terms of the radius, i.e. as the set of points whose distance from the center is equal to the radius.  Because of this, doesn’t it seem more natural to consider the ratio of a circle’s circumference to its radius, rather than the ratio of circumference to diameter?  Put another way, isn’t a more natural constant given by the circumference of a circle with radius 1 rather than the circumference of a circle with radius 1/2?  He offers plenty of other aesthetic examples for why 2\pi should be viewed as more fundamental, including references to the Bernoulli numbers and simple quadratic forms.

On the one hand, this may seem like a trivial issue – after all, the difference between \pi and \tau is only a factor of 2, and different normalizations of quantities are quite common in mathematics.  On the other hand, Hartl does make a convincing argument from a pedagogical point of view.  His strongest argument comes from trigonometry.  When students learn to convert between radians and degrees, they learn that 2\pi corresponds to full revolution.  From this, one sees that half of a revolution corresponds to an angle of \pi, 1/4 of a revolution corresponds to an angle of \pi/2, and so on.  But if we define the fundamental quantity to be \tau, then in radians, half a revolution is \tau/2, a quarter of a revolution is \tau/4, and the measure of c revolutions is given by c\tau for any number c.

Hartl concludes the following: “The unnecessary factors of 2 arising from the use of \pi are annoying enough by themselves, but far more serious is their tendency to cancel when divided by any even number. The absurd results, such as \pi/2 for a quarter circle, obscure the underlying relationship between angle measure and the circle constant. To those who maintain that it “doesn’t matter” whether we use \pi or \tau in teaching trigonometry … from the perspective of a beginner, using \pi instead of \tau is a pedagogical disaster.”

It’s an interesting argument, and one I think students would benefit from seeing.  \pi is fairly entrenched, so I’m not sure how much of a following Hartl will gain, but even if \pi remains the standard, offering students this viewpoint can only help them as they learn trigonometry.  For that reason, I for one will be endorsing Tau Day (6/28, get it?).  It certainly doesn’t sound as delicious as Pi Day, and the fact that students are out of school is a bit of a problem, but today is apparently the inaugural Tau Day, and these are wrinkles that I’m sure can be ironed out.

So happy Tau Day to you, no matter your preference!

(Big ups to James Hawkins for sending me the Tau Day link.)

]]> http://www.mathgoespop.com/2010/06/happy-tau-day.html/feed 4 Knowing http://www.mathgoespop.com/2010/04/knowing.html http://www.mathgoespop.com/2010/04/knowing.html#comments Fri, 23 Apr 2010 21:25:56 +0000 Matt http://www.mathgoespop.com/?p=297 Nicolas Cage commands a powerful fan base.  On the one hand, this should be expected of any man with the foresight to see how awesome a film The Rock would turn out to be, but on the other hand, some of his more recent outings (I’m thinking of Bangkok Dangerous, Next, Ghost Rider, and Knowing) have . . . → Read More: Knowing]]> Nicolas Cage commands a powerful fan base.  On the one hand, this should be expected of any man with the foresight to see how awesome a film The Rock would turn out to be, but on the other hand, some of his more recent outings (I’m thinking of Bangkok Dangerous, Next, Ghost Rider, and Knowing) have met with less than critical praise.  Nevertheless, support for Nicolas Cage has, from my perspective, only seemed to grow over the past few years.  Perhaps it’s because of the National Treasure series, or because, according to Wikipedia, he named his youngest son Kal-El after Superman.  Or perhaps people feel sorry for him because of his tax problems after spending too much money buying castles and islands.  Whatever the case, this love for Nicolas Cage manifests itself in a variety of ways, from the usual fan sites such as cagefactor.com, to the less standard celebrity homage known as Nic Cage as Everyone, in which the faces of celebrities are replaced by Cage’s charming mug.


Big Poppa, or Big Daddy?

For purposes of this discussion, however, I will try not to get pulled into the ocean of Nicolas Cage adulation, and will instead try to focus on the mathematical content in his 2009 film, Knowing.  This turns out to be a fairly simple task, as you will soon see.

In Knowing, Cage stars as John Koestler, a professor of astrophysics at MIT (already, the potential for mathematics is promising).  He is a widower with a young son, named Caleb, who goes to a nearby elementary school.  Near the beginning of the film, Caleb’s school holds a celebration in honor of its fiftieth birthday, and as part of the festivities they unearth a time capsule that was buried 50 years prior.  The time capsule is filled with drawings from former students, but Caleb gets a little short-changed: instead of receiving paper with a drawing, his paper just has a long sequence of numbers.



As might be expected by the presence of a long list of numbers, Knowing is less concerned with math than it is with numerology.  Don’t let John Koestler’s profession fool you – aside from one scene in the classroom where you can spot Maxwell’s equations if you know where to look, there is very little in the way of mathematics in this film.  But the more important question to ask is whether this is a bad thing.

While the lack of mathematics may make the trailer seem a bit disingenuous, Knowing is aptly named in the sense that it knows the film’s conceit has nothing to do with mathematics.  Unlike a film such as Pi, which attempts to pass off numerological voodoo as actual mathematics, Knowing is very much aware that Koestler’s analysis of the numbers from the time capsule does not constitute research in mathematics.  At one point his colleague even points this out to him, saying “Whoa. Just step back. Have another look at it! Systems that find meaning in numbers are a dime a dozen. Why? Because people see what they want to see.”

Another difference between this film and Pi is that here, the pattern of the numbers is discovered fairly early on, and it becomes quite predictable.  There is less mystery in the numbers themselves than there is in how a schoolgirl 50 years ago came to write the numbers down.  Thankfully, the answer to the latter question is resolved as well (spoiler alert: it all has to do with angel aliens).

Does this film pertain to mathematics?  Not really.  But nor does it aspire to be.  In that sense, then, I would certainly consider this a better film than Pi.  Should they remake Pi with Nicolas Cage, however, all bets are off.

]]> http://www.mathgoespop.com/2010/04/knowing.html/feed 0 Pi Day http://www.mathgoespop.com/2009/03/pi-day.html http://www.mathgoespop.com/2009/03/pi-day.html#comments Thu, 12 Mar 2009 05:04:00 +0000 Matt http://www.mathgoespop.com/2009/03/pi-day.html Hot on the heels of Square Root Day comes Pi Day, a day held in honor of arguably the most famous mathematical constant, π. And like Square Root Day, I am forced to approach this holiday with a certain degree of hesitation.

There is no doubt that Pi Day is the most prestigious mathematical holiday, but . . . → Read More: Pi Day]]>

Hot on the heels of Square Root Day comes Pi Day, a day held in honor of arguably the most famous mathematical constant, π. And like Square Root Day, I am forced to approach this holiday with a certain degree of hesitation.

There is no doubt that Pi Day is the most prestigious mathematical holiday, but this recognition usually only serves to illustrate the sad state of mathematical literacy in this country. For example, one year I remember reading a news article about Pi Day where the author described π as a number whose decimal expansion “was believed to go on forever.” Of course, belief has nothing to do with it – this is a simple consequence of the irrationality of π, a fact which is apparently lost amidst the pie eating hubbub of this holiday.
Unfortunately, this is not an isolated incident – for as much as Pi Day aims to educate people about π, it seems to do just as good a job of showing how little people actually know. Searching Google News for articles on the upcoming holiday, it’s possible to find a number of stories that say a whole lot of garbage. For example, there’s this quote from a Pi Day article on SF Gate:

Pi, as [Pi Day co-organizer Ron] Hipschman noted, is strange because it’s both an irrational number (its decimal expansion never ends or repeats) and yet the number is also transcendental (no finite sequence of algebraic functions could ever produce it).

To a physicist like Shaw, that kind of contradiction and beauty was all the inspiration he needed to contemplate a Pi Day.

This sort of writing is like nails on a chalkboard to anyone who knows better. Forgetting the convoluted definition of a transcendental number given above, the more important point is that there is no “contradiction” in the statement. There is nothing special about the fact that π is both transcendental and irrational – as is immediate from the definition of a transcendental number, any transcendental number is automatically irrational.

No doubt there will be other examples of this mathematical butchery as Pi Day draws near. Here are a couple more. From the Times Online:

[S]ince 1988 mathematicians across the land have been celebrating pi day each year by tucking into a feast of [sticky pudding]. The number has obsessed generations of mathematicians for millennia, and not because it’s an excuse to eat pudding.

As I’ve said before, there’s not a working mathematician today (nor can I think of one over the past several hundred years) who has made a career studying the number pi. No mathematician is “obsessed” with this number – although numerologists and Max Cohen certainly may argue otherwise.

Some sources don’t even seem to know what π is. From Jacksonville, FL:

The next big day to celebrate in the math community this year [after Square Root Day] is pi day, March 14th. It represents 3.14 – a common mathematical expression.

And from Montgomery, AL:

On March 14 math lovers can celebrate Pi, the mathmatical [sic] formula used to find the circumference of a circle’s diameter, which is 3.14.

Not only is π defined incorrectly in both of these quotes, but it’s clear that the authors don’t know that π is neither a formula, nor an expression, any more than the number 12 is a formula or an expression.

One could argue that perhaps I am just nitpicking. For the general reader, such details are of no consequence, you may say. Unfortunately, history has shown that misinterpretations of the number π can lead to quite embarrassing consequences. One need look only to the good people of Indiana for proof.

As discussed in the article linked above, around the turn of the last century, a man named Edwin J. Goodwin claimed to have done what mathematicians already knew was impossible: he claimed to be able to square the circle (i.e., he claimed he had found a way to construct a square with the same area as a given circle, using only a compass, straightedge, and a finite number of steps).

Not content to keep the discovery to himself, Dr. Goodwin decided to share his discovery with his fellow countrymen in Indiana:

The stalwart Hoosier determined that the great state of Indiana should be the first to benefit from what he fervently believed to be a “new mathematical truth.” He would allow the state to use his discovery and to put it in the school textbooks free of charge. There would be no need for Indiana to ever pay him any royalties.

On January 18, 1897, after emerging from the House Swamplands and Education Committees (legislatures sometimes work in mysterious ways), Indiana House Bill 246 was introduced to codify Dr. Goodwin’s discovery. Legislators freely admitted they did not understand the jargon-filled bill, although they were certain it had something to do with circles. Of course they passed it unanimously.

While his heart was certainly in the right place, his mathematical rigor was not. His construction relied upon the unfortunate claim that π = 3.2. Thankfully, the error was pointed out before the bill was able to do any damage. The story does go to show, however, that we have a long history of not understanding π.

Eating pie is certainly an activity I can support, but other than that, I’m not really sure of this holiday’s purpose. On the official Pi Day website, for instance, the three questions on the front page up for discussion are: “Why do you like Pi?”, “What are you doing at your school to celebrate Pi Day?”, and “How many digits of Pi have you memorized?” Note that two of these questions actually have nothing to do with the number π, and the one that does deal with π doesn’t ask about any actual mathematics.

If you’re going to delve into this number, at least ask some interesting questions. How about “How can you show that π is irrational?” (here is a simple proof that only requires some basic calculus) or the related question, “How can you show that π is transcendental?” For younger students who may not understand or appreciate such proofs, how about “Where are some unusual places that π appears?” (You could show them the infinite series π/4 = 1 – 1/3 + 1/5 – 1/7…, which is usually quite surprising to a first time viewer). Or, for a more philosophical question, “Why does π appear in so many places in mathematics?”

I’m also a reluctant supporter of this holiday because I don’t really see a reason for π to steal all the limelight from other constants that may not have the PR that π does. There are other constants equally deserving of our attention. This is a slippery slope, of course, and once we say this, it’s natural to say that there are certainly more important concepts in mathematics, each one deserving of its own day to celebrate.

Perhaps in time, we will see more effective use of these “math holidays.” For now, though, I think that this is about the best we can expect to get:

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