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. (see here for more).

The ratio of a circle’s circumference to its diameter, is perhaps the most famous irrational number. Historically, has also been known as Archimedes’ constant, and Archimedes himself proved that .

More than one trillion digits of the decimal expansion of have been computed, and folks with nothing better to do than recite those digits come together each 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 , though, here are a few to mull over:

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 , I assure you it is no less charming. Typically defined as the limit

e enjoys many other identities, including

and

e also determines the base of the exponential function , 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. (see here for more).

Take all the perfect cubes (, 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

The constant is named in honor of Roger Apéry, who proved in 1978 that this number is irrational. Intuitively, one can interpret 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

will yield some finite value. When k is even, one has nice formulas for the values, for instance , .

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 .

2. Given that , show that .

4. (see here for more).

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 .

can be approximated by its decimal expansion, , 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 is a rational number (unlike constants such as or e, which are known to be irrational).

Table exercises!

1. Using geometry and the figure above, show that .

2. Show that the sum on the right hand side of the inequality in the first exercise equals , so that .

5. (see here for more).

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 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 elements.

6. (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 . In other words,

.

This, in turn, implies that , or (by the quadratic formula)

.

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 . If we let denote the Fibonacci number (so , and so on), then

Table exercises!

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

2. Show that .

3. Show that for any n, .

7. (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 , one for each real number t. is essentially the Riemann function, and in particular, the Riemann Hypothesis is true if and only if has only real zeros.

Here are some properties of the family of functions :

1. has only real zeros for any .

2. If has only real zeros, then for any , has only real zeros too.

3. There exists a real value such that has at least one non-real zero.

These properties combine to show the existence of a constant , lying somewhere in the range , such that has only real zeroes if and only if . This is how the de Bruijn-Newman constant is defined. Moreover, the Riemann Hypothesis is equivalent to the statement that .

The current best estimates for state that

so if the Riemann Hypothesis is true, it is, in some sense, “just barely” true. In particular, it’s possible that , 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 (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 , and is noted for its unification of five constants of fundamental importance in mathematics: e, , i, 1 and 0.

Table exercises!

1. Show that whenever n is divisible by 4.

2. Find all x satisfying the equation .

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 . Can you define an addition law on the set of complex numbers? A multiplication law?

9. (see here for more).

The plastic constant can be viewed as a cousin to the golden ratio (see the table for more information). Formally, is equal to the real root of the equation . The value of is

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

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

For example, the first few terms in the sequence are given by .

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 . In either case,

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. Show that if p is a prime number, is divisible by p.

10. (see here for more).

Along with , is probably the most well known number on display here. While it may seem mundane, 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 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 have been met with much less controversy:

Table exercises!

1. Prove that 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 a rational number?

Enjoy the table exercises!

It was the first time in 18 years that such a quirky thing happened with a full schedule. On Aug. 10, 1993, all seven NL games featured one team scoring precisely two runs, STATS LLC said.

The last time it occurred with five or more runs was July 20, 1955, when all four AL games had at least one team score exactly six, STATS LLC said.

When I read this article, some questions immediately came to mind: exactly how rare is it for one team in a collection of 7 baseball games to have a common score of 5?  Also, if 7 teams in 7 games have the same score, which score are they most likely to share?  Are the 7 games with a common score 0f 2 more or less likely to occur than the 7 games with a common score of 5?

We can answer these questions with some (relatively) simple probability models, given some caveats.  I’d like to estimate these probabilities using only one parameter: the average number of runs a team scores during a game.  Of course, that average will vary from team to team, and also from year to year (in particular, runs per game have declined from the heyday of steroid-mania that gripped baseball at the turn of the millennium).  Due to different rules, there may also be variation between the American and National Leagues.  Let me ignore this, though, and consider only an average number of runs per game overall – what we lose in precision we will more than make up for in clarity.

Ahh, the late 90's, when it was easier to sock a few dingers.

The question remains: how many runs are scored on average in a baseball game?  I found some data online which is somewhat outdated, but I’ll stick to it for convenience (and, more importantly, out of laziness) – any alteration in this number is easy to propagate throughout the following discussion.  In this article from 2005, the author tabulated the average number of runs per game in MLB over a 5 year span from 2000-2004 (that’s over 12,000 games!).  He has a nice looking graph of the distribution of scores as well:

A savvy probability student might see the long tail of this probability distribution and liken it to the Poisson distribution, a distribution encountered in many probability courses, and which is frequently motivated by a desire to model “rare events.”  I put the term in quotations since what constitutes “rare” is frequently left undefined, and in any event, is not really pertinent to this discussion.

Let us suppose, then, that the number of runs scored per game by each team follows a Poisson distribution.  French aside, this means that the probability a team will score n runs is equal to

,

where A is the average number of runs scored per game – in this case, 4.82, and e is the unsung hero sometimes known as Euler’s number.  Don’t worry too much about this formula; if you prefer, the graph of the function looks like this (courtesy of Wolfram Alpha):

Note that the fit isn’t perfect – this graph starts much lower at 0 than the graph of the actual data pictured above, for example – but there is precedence for using the Poisson distrubtion to model runs in a baseball game (this article provides one such example, but a subscription is required to view it in its entirety).  More careful analysis is possible, and can be found in resources like this one, but again, I want to keep things relatively simple.

So, let us suppose that the probability that a team scores n runs is .  What then, is the probability than in a baseball game, one of the teams will score n runs?  Either team A can score n runs or team B can score n runs, but they can’t both score n runs since baseball games can’t end in a tie.  This means that the probability of A or B scoring n runs is simply the probability that A scores n runs plus the probability that B scores n runs, or

For the odds that this happens 7 times, we then multiply this number by itself 7 times (lurking under this is the assumption that runs scored in different games are independent, which seems like an entirely reasonable assumption to make).  To summarize, we estimate the probability that one team in each of 7 games scores n runs is

If n = 5 (as it did earlier this month), the probability is roughly .064%.  In other words, if 7 AL games were played every day, you would expect this outcome once every 1,560 days or so.  Having said that, with more careful analysis it’s possible to show that in fact, if 7 games will have teams scoring the same number of runs, 5 is the most likely number.  For comparison, when n = 2 the probability is only a paltry 0.00812%, making what happened on May 8th over 75 times more likely than what happened on August 10, 1993.  Of course, it’s not fair to compare these records to the 6 run record in 1955, since in that case only 4 games were played, rather than 7.  Nevertheless, it’s not difficult to adjust this model from 7 games to 4 games (or an arbitrary number of games).

So, rather than some murky intuition telling us this event should be unlikely, with a little more effort we can attempt to quantify exactly how unlikely this event should be.  More sophisticated models for runs could be used, but perhaps that is a topic I will save for another day.

The joke ran at the end of a muppet-themed segment of the show.  In an homage to Sesame Street, after the segment finished (but before the somewhat racy joke involving a very physical muppet Moe) an announcer stopped to give thanks to the sponsors of the show.  Unlike Sesame Street, however, which is sponsored every day by two letters and a number, this episode of The Simpsons was sponsored by one symbol and one number that looks like a letter:

In case you’re late to the party (since I don’t think that clip will be online forever), let me quote: “Tonight’s Simpsons episode was brought to you by the symbol umlaut, and the number e. Not the letter e, but the number, whose exponential function is the derivative of itself.”

Kudos to the writers for incorporating some choice math humor into the tail end of this episode (I’m willing to overlook some qualms with their wording).  Perhaps Simpsons aficianados would can begin preparations for next year’s e day.