Math Goes Pop! » Et cetera 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 Some Readership Statistics http://www.mathgoespop.com/2011/07/some-readership-statistics.html http://www.mathgoespop.com/2011/07/some-readership-statistics.html#comments Sat, 09 Jul 2011 00:40:18 +0000 Matt http://www.mathgoespop.com/?p=1294 This week marks the third anniversary of Math Goes Pop!  As such, I thought it might be appropriate to engage in a bit of navel-gazing.  But since I can gaze at my own navel whenever I please, I’d like to flip the script, as it were, and turn my attention towards the collective navels of my . . . → Read More: Some Readership Statistics]]> This week marks the third anniversary of Math Goes Pop!  As such, I thought it might be appropriate to engage in a bit of navel-gazing.  But since I can gaze at my own navel whenever I please, I’d like to flip the script, as it were, and turn my attention towards the collective navels of my readership.

Our cat's third birthday is also this week. It is unclear which event he is celebrating, although the dilated pupils suggest he is celebrating a bit too hard.

I’d like to share with you some data on the geographic distribution of my US readers.  While there is a large California bias, people from all over the country seem to have stumbled upon this corner of the internet, and have hopefully enjoyed their time here.

This represents you, gentle reader. Darker green means more viewers.

Of course, a California bias shouldn’t be all that surprising.  After all, California is the most populous state in the country, accounting for roughly 12% of the country’s total population, according to this 2010 Census data.  A more interesting thing to think about, then, is not the map pictured above, but how the map contrasts with actual population data for each state.  For example, New York is a hair darker than Texas on this map, even though Texas has a larger share of our population: roughly 8% as compared to 6% for New York.

One way to compare population data with Math Goes Pop visitor data is through rankings.  This table compares the rankings, and shows the relative difference for each state (feel free to play around with the data to your heart’s content):

StateView Proportion RankPopulation Proportion RankRank Difference
California110
New York23+1
Michigan38+5
Texas42-2
Illinois550
Pennsylvania660
New Jersey711+4
Massachusetts814+6
Florida94-5
Virginia1012+2
Washington1113+2
North Carolina1210-2
Ohio137-6
Maryland1419+5
Georgia159-6
Missouri1618+2
Connecticut1729+12
Minnesota1821+3
Oregon1927+8
Arizona2016-4
Colorado2122+1
Wisconsin2220-2
Tennessee2317-6
Indiana2415-9
Washington DC2550+25
South Carolina2624-2
Louisiana2725-2
Kentucky2826-2
Iowa2930+1
Utah3034+4
Oklahoma3128-3
Alabama3223-9
Kansas33330
Nebraska3438+4
Arkansas3532-3
Nevada3635-1
Mississippi3731-6
New Hampshire3842+4
Rhode Island3943+4
New Mexico4036-4
Idaho4139-2
Maine4241-1
West Virginia4337-6
Hawaii4440-4
Vermont4549+4
Alaska4647+1
Montana4744-3
Delaware4845-3
North Dakota4948-1
South Dakota5046-4
Wyoming51510

The rankings give us some information: we see that Indiana and Alabama are not as well represented in readership as one might expect given their population rankings (both states have Math Goes Pop readership rankings 9 levels below their population rankings).  On the other hand, folks from DC, Connecticut, and Oregon are visiting this site more than would be expected based on population numbers alone; readership rankings for these states are 25, 12, and 8 levels above their population rankings, respectively.

But while the rankings tell us some things, they leave a great deal out.  For instance, while California is ranked first in both the proportion of the US population and the proportion of visitors to this site, the rankings tell us nothing about how these proportions compare to each other.  In fact, while California boasts the number 1 proportion in both categories, the proportion of California viewers to Math Goes Pop is more than twice the proportion of California’s population (26% of my viewers vs. 12% of the US population).

Comparison of the proportions in this way allows us to get a better understanding of how visitors to this site are distributed across the country, as compared to the overall population distribution.  While the population is distributed more heavily in California, the proportion of California Math Goes Pop visitors is greater than can be explained by simple population data.

If we compare the state-by-state readership proportions to overall population proportions, we get the following picture:

Click to Embiggen!

Big ups to our nation’s capital for having the largest share of viewership relative to its overall share of the country’s population.  In addition to DC, the proportion of readership from 10 states is greater than the state’s proportional population: California, Massachusetts, Michigan, New York, Connecticut, Maryland, Oregon, New Jersey, Washington, and Vermont.  For example, while Massachusetts accounts for roughly 2% of the country’s population, thus far it has accounted for nearly 3.5% of Math Goes Pop readership.

I won’t go into an analysis of why some states might be over- or underrepresented in the blog’s readership according to this metric.  I just thought it might be appropriate to share some of this data in commemoration of Math Goes Pop!  I hope you will continue to enjoy the material posted here.  And if you live in Wyoming, Mississippi, Alabama, or any of the other 40 states I didn’t mention in the paragraph above, let’s see what we can do to get some mathematical love flowing in your neighborhood.

]]> http://www.mathgoespop.com/2011/07/some-readership-statistics.html/feed 2 Some Shameless Self-Promotion http://www.mathgoespop.com/2011/07/some-shameless-self-promotion.html http://www.mathgoespop.com/2011/07/some-shameless-self-promotion.html#comments Fri, 01 Jul 2011 21:05:14 +0000 Matt http://www.mathgoespop.com/?p=1291 Looking for a way to procrastinate before the three day weekend?  Then feel free to check out this interview I gave to the Journal of Media Literacy Education.  I gave the interview some time ago, but just happened to stumble upon it in published form this week.  If you want some behind-the-scenes perspective into how this . . . → Read More: Some Shameless Self-Promotion]]> Looking for a way to procrastinate before the three day weekend?  Then feel free to check out this interview I gave to the Journal of Media Literacy Education.  I gave the interview some time ago, but just happened to stumble upon it in published form this week.  If you want some behind-the-scenes perspective into how this blog started, and my general philosophy behind writing it, this interview is a good place to start.

Hope the long weekend treats you well!

]]> http://www.mathgoespop.com/2011/07/some-shameless-self-promotion.html/feed 2 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 Protractors for Some, Miniature American Flags for Others! http://www.mathgoespop.com/2010/08/protractors-for-some-miniature-american-flags-for-others.html http://www.mathgoespop.com/2010/08/protractors-for-some-miniature-american-flags-for-others.html#comments Wed, 11 Aug 2010 15:00:12 +0000 Matt http://www.mathgoespop.com/?p=637 Last weekend I went to the Pasadena Flea Market, self-described as “one of the most famous markets in the world.”  I had not anticipated on finding anything math related, and although I did stumble across an old adding machine, the most surprising find was what greeted me at the door. R.G. Canning produces the flea market every . . . → Read More: Protractors for Some, Miniature American Flags for Others!]]> Last weekend I went to the Pasadena Flea Market, self-described as “one of the most famous markets in the world.”  I had not anticipated on finding anything math related, and although I did stumble across an old adding machine, the most surprising find was what greeted me at the door.
R.G. Canning produces the flea market every month, but I have no idea why they were giving away protractors.  There’s furniture for sale, but I would think rulers would be the preferred measuring device when browsing through such items.  Perhaps instead they thought that August would be a good month to get rid of a surplus of protractors, with back to school around the corner?  Whatever the case, kudos to R.G. Canning attractions for their protractor giveaway bonanza.

Of course, I’m not sure how many protractors were actually taken.  Unfortunately, most people didn’t seem interested.  Their loss, I suppose.

]]> http://www.mathgoespop.com/2010/08/protractors-for-some-miniature-american-flags-for-others.html/feed 0 Mathematics Awareness Month 2010 http://www.mathgoespop.com/2010/04/mathematics-awareness-month-2010.html http://www.mathgoespop.com/2010/04/mathematics-awareness-month-2010.html#comments Sat, 01 May 2010 05:58:52 +0000 Matt http://www.mathgoespop.com/?p=306 As April comes and goes, so too does Mathematics Awareness Month.  Every year, the Joint Policy Board for Mathematics swirls mathematics with a different delightful discipline: last year it was climate, and the year before was voting.

This year’s theme is mathematics and sports, a topic which has inspired a number of articles . . . → Read More: Mathematics Awareness Month 2010]]> As April comes and goes, so too does Mathematics Awareness Month.  Every year, the Joint Policy Board for Mathematics swirls mathematics with a different delightful discipline: last year it was climate, and the year before was voting.

This year’s theme is mathematics and sports, a topic which has inspired a number of articles here on this site.  As usual, there are a number of essays that discuss this theme from various perspectives; while usual suspects such as football and baseball play a central role in many of the essays, other sports get to mingle with mathematics as well, including track, golf, and tennis (also NASCAR, if you consider that a sport).


This dude always thinks about math when he is golfing.


There are too many articles to discuss, so I’d encourage you to go take a look and see if anything strikes your fancy.  However, here are a few highlights:

If football is your game, Chris Jones of St. Mary’s College of California has written an article about NFL overtime rules and offers a mathematical model for determining the winner in overtime based on the winner of the coin toss at the beginning of overtime.  Since overtime ends after any team scores, one would naturally expect that winning the coin toss carries with it a significant advantage, and this is born out in the data.  Jones offers an alternative rule scheme whereby the winning team is the first one to score six points, but in this case the team which wins the coin toss still has an advantage, and it is more likely that the game will end in a tie.

Given that the NFL recently changed their overtime rules for playoff games, it’s too bad that Jones did not include this scheme into his analysis.  Perhaps, gentle reader, this would be a good exercise for you.

If your sports preferences are more varied, you may prefer the article by Rick Cleary of Bentley University, which discusses the probability of rare events in the contexts of football, baseball, and basketball.  My favorite example deals with the complaints many people have with regards to playoffs in Major League Baseball.  More specifically, the first round in MLB playoffs pits teams into a best-of-5 series, while the remaining rounds of the playoffs use a best-of-7 series.  Critics claim that the shorter first round series puts the stronger teams at a disadvantage, but in fact, a 7-series round is only slightly more advantageous for the stronger team.  In effect, Cleary argues that it’s almost incompatible to say that a best-of-5 series is unfair without also arguing that a best-of-7 series is also unfair.  The article is also well suited for a general audience.

Then again, maybe you are more interested in the intricate links between math and golf.  If that’s the case, you may want to peruse this article by Scott M. Berry, in which he analyzes the question: is Tiger Woods a winner?  In other words, does his ability to win transcend his skill level?  Does he have a mental game that helps push him to the top because of the influence he has on other players?

Berry modeled Tiger Woods’ performance with the affectionately named RoboTiger, and concluded that in fact, Woods’ record does not prove him to be a “winner” – he’s just a very skilled golfer. The jury is still out, however, on the mathematical significance of any “winning” label for Tiger woods in the bedroom.

Finally, if you’re interested in turning mathematics into cash, you may be interested in this article by Tim Chartier, Erich Kreutzer, Amy Langville, and Kathryn Pedings, which discusses different methods of predicting winners in the annual NCAA Men’s Basketball Tournament.  While I’ve discussed this topic before, this article gives more detail on a variety of methods, which, if carefully applied, will make your bracket a sure fire winner.  Just make sure no one else in your local pool is so mathematically inclined.

There are plenty of other examples illustrating the intersection of math and sports, so don’t let the magic stop here.  If you’ve ever wanted to learn how to bend it like Beckham, or if you’ve ever dreamed of somehow connecting math to NASCAR, click through to the Mathematics Awareness Month website and read on.

]]> http://www.mathgoespop.com/2010/04/mathematics-awareness-month-2010.html/feed 0 Judge v. Justices http://www.mathgoespop.com/2010/01/judge-v-justices.html http://www.mathgoespop.com/2010/01/judge-v-justices.html#comments Mon, 18 Jan 2010 18:31:00 +0000 Matt http://www.mathgoespop.com/2010/01/judge-v-justices.html . . . → Read More: Judge v. Justices]]> 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.

Yes.
]]> http://www.mathgoespop.com/2010/01/judge-v-justices.html/feed 0 A Mathematical New Years Game http://www.mathgoespop.com/2010/01/a-mathematical-new-years-game.html http://www.mathgoespop.com/2010/01/a-mathematical-new-years-game.html#comments Mon, 11 Jan 2010 06:05:00 +0000 Matt http://www.mathgoespop.com/2010/01/a-mathematical-new-years-game.html . . . → Read More: A Mathematical New Years Game]]> 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.

]]> http://www.mathgoespop.com/2010/01/a-mathematical-new-years-game.html/feed 2 More on Nerdy T-Shirts http://www.mathgoespop.com/2009/12/more-on-nerdy-t-shirts.html http://www.mathgoespop.com/2009/12/more-on-nerdy-t-shirts.html#comments Fri, 11 Dec 2009 20:54:00 +0000 Matt http://www.mathgoespop.com/2009/12/more-on-nerdy-t-shirts.html Recently I received an email imploring me to check out all of the “unique designs” available at a site called nerdytshirt.com. I’m not sure why I was the recipient of such an email – they could have found me through my university affiliations, or through this blog, but I’m not sure which. If you’ve been reading . . . → Read More: More on Nerdy T-Shirts]]> Recently I received an email imploring me to check out all of the “unique designs” available at a site called nerdytshirt.com. I’m not sure why I was the recipient of such an email – they could have found me through my university affiliations, or through this blog, but I’m not sure which.
If you’ve been reading my musings for a while, you may know of the problems I have with the intersection between mathematics and clothing. Most of what’s out there is junk. As one might expect, I was therefore quite skeptical when I received this solicitation. At the same time, I’d never heard of this site before, and so I hoped that perhaps a company that understood my frustrations had come to fruition.

Have my prayers been answered? Sort of. Let’s consider a few examples.

Despite claiming to have “unique designs,” the shirts at nerdytshirt.com are all variations on one theme: put a formula on the shirt, and below that make a pun related to the formula. Sometimes the results of this pairing are good:

wokka wokka wokka!

A good gift for Sarah Palin, perhaps?

Other times, however, the jokes just fall flat.
Is this even a joke?

On the plus side, the website doesn’t restrict itself to just math jokes; they also offer shirts for chemistry, physics, and statistics. The underlying principle never changes, though: take an equation, expression, or sequence, then add a pun underneath.

I certainly don’t think this is a bad strategy – on the contrary, I find some of them quite clever (as clever as one can find a pun to be, I suppose). But by restricting themselves to this one type of design, the whole enterprise seems a little one-note. Not only are they robbing us of the full power of their creative juices by using this one format, but they also exhaust this one idea fairly quickly. There are some good shirts, it’s true, but there are duds as well, and so one is left feeling that in their rush to claim that they are “now offering more than 100 unique Nerdy T-Shirt designs,” they have placed quantity over quality.

Their site is still in its infancy, and so one hopes that the masterminds behind the site will not be afraid to branch out in their shirt designs. There is potential here, and I wouldn’t mind wearing some of the shirts they have for sale, but it may be dangerous to give one of these shirts as a gift unless you are able to separate the wheat from the chaff. Also, the link to their website in small fond on the front of every t-shirt may be a dealbreaker for some.

At the very least, one could use the following t-shirt for Halloween, should one choose to be the monster group (as I’ve discussed before).

That large number is the size of the monster group.
]]> http://www.mathgoespop.com/2009/12/more-on-nerdy-t-shirts.html/feed 3 Mathematics Awareness Month 2009 http://www.mathgoespop.com/2009/04/mathematics-awareness-month-2009.html http://www.mathgoespop.com/2009/04/mathematics-awareness-month-2009.html#comments Fri, 01 May 2009 04:30:00 +0000 Matt http://www.mathgoespop.com/2009/04/mathematics-awareness-month-2009.html . . . → Read More: Mathematics Awareness Month 2009]]> With April on its way out, it behooves me to take a moment and mention the focus of this year’s Mathematics Awareness Month. April has been bestowed with this glorious title every year since 1986 – last year the topic was Mathematics and Voting, which I discussed at some length in three earlier posts (see here, here, and here).

This year’s focus is on Mathematics and Climate. On the homepage you can find links to a variety of articles, most of which focus on the difficulty in coming up with mathematical models that can accurately reflect the complexity of the interconnected world in which we live. This is perhaps best summarized by Professor Pat Kenschaft, who writes the following in her essay, “Climate Change: A Research Opportunity for Mathematics?”:

How do we analyze the dynamics of the atmosphere, the oceans, the solid earth (especially volcanic emissions) and the biosphere (the system of plants, animals, and other living things)? Scientists have studied pieces of these systems, cutting them both conceptually and geographically, but even the pieces are not tractable by current mathematics, and the challenges as we try to understand the interplay of all phenomena involved are far beyond current conceptual and computational capabilities.

This is a theme that comes up in quite a few of the articles related to this year’s focus on the intersection of math and climate. As we begin to demand more from our models, those models will necessarily need to become more sophisticated. This requires mathematicians to create models that not only reflect reality, but are also optimized so that we can obtain results within a reasonable time frame.

There are a host of other articles discussing the interplay between climate and mathematics. Some of the articles cover related topics as well – for example, Professor Margot Garritsen’s article “Mathematics in Energy Production” provides a good example of the essential role mathematics plays in our current methods for procuring gas and oil, and briefly discusses the relationship between math and alternative energies.

With city-sized blocks of ice crumbling off of the Antarctic, there can be little doubt that climate change is happening, even if we don’t understand everything that underlies it. Will mathematics come to our rescue? Don’t worry – if it doesn’t, I’m hopeful that Captain Planet will.

Captain Planet: Math Spokesman for the 21st century?
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