Math Goes Pop! » riemann hypothesis http://www.mathgoespop.com Ruminations on the Intersection Between Mathematics and Popular Culture Tue, 07 Feb 2012 04:49:20 +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 Happy Birthday, Riemann Hypothesis! http://www.mathgoespop.com/2009/11/happy-birthday-riemann-hypothesis.html http://www.mathgoespop.com/2009/11/happy-birthday-riemann-hypothesis.html#comments Thu, 19 Nov 2009 04:03:00 +0000 Matt http://www.mathgoespop.com/2009/11/happy-birthday-riemann-hypothesis.html . . . → Read More: Happy Birthday, Riemann Hypothesis!]]> Big ups to Liz Landau for bringing attention to one of the most important unsolved math problems of our time, the Riemann Hypothesis. Over at the CNN SciTechBlog, she has written a nice article on the problem aimed at a general audience.
This year marks the 150th anniversary of the publication of Riemann’s manuscript, where he proposed the now famous conjecture on the zeros of the Riemann-zeta function, and November was the month in which it was published. However, as Landau points out, the exact date of publication isn’t known, which makes having a birthday celebration a little tricky. The American Institute of Mathematics picked today to celebrate, and in honor of Riemann talks were held all around the world.

The Riemann Hypothesis has held the attention of the mathematical community for a century and a half, but it’s also made occasional forays into the realm of popular culture. For starters, there are quite a few books on the conjecture that are aimed for a general audience (one of the more recent ones is by Dan Rockmore). It has also made cameos in television shows such as Law and Order and Numb3rs. The Numb3rs episode is particularly notable because in it, the Riemann Hypothesis is solved by none other than Neil Patrick Harris. While he may not be a major contender to solve the problem in real life, it may be a bit too early to make a final verdict.

This is what NPH looks like when he thinks about mathematics.

While the Riemann Hypothesis may not be as simple to state to a general audience as, say, the statement of Fermat’s Last Theorem, there is ample room for its inclusion in popular discourse. This is important not just because the conjecture excites mathematicians, but also because it shows that the work mathematicians do involves more than multiplying large numbers and being awkward in social situations.

In an ideal world, we would celebrate the Riemann Hypothesis every November 18th, just as we celebrate Pi Day on March 14th. The difference would be that 11/18 would commemorate something substantive within mathematics. For that reason, the idea probably won’t take hold. But at least a man can dream. Thanks again, Liz, for doing your part!

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