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Published: 2006/09/20
by David Steinberg

Featured Column:Everyone Must Add and Subtract Those 18 Steps

A few months ago I was casually reading my email when I noticed the subject, "Trey New Album." I get a lot of spam these days, but I decided to take the chance that Musictoday was not involved in Nigerian scams and answered it.

It turned out that Trey had an idea for an album cover. He was releasing a bonus EP 18 Steps for people who preordered his new album Bar 17. He was looking for something mathematical that would tie the two albums together; it should be confusing but yet actually mean something. Vague memories of my math skills apparently stuck in his head, because he figured that I could get him what he wanted.

In case you’re wondering what it feels like to be put in this role, it was like being a correspondent on The Daily Show. I may not be a Senior Papal Vacancy Expert, but I was Trey’s Senior Math Expert, which was nearly as cool.

Once I had the task, I needed a plan. After an initial attempt at finding some fun facts about 17 and 18 – did you know that 17 is an unlucky number in Italy because XVII can be rearranged as VIXI which means, "I lived?" – the obvious approach came to mind. Let’s try to take 18 steps to move from 17 to 18 using as much math as possible. I broke out my old texts – hey this Lebesque Integration book is autographed by Timothy Leary, I forgot about that – and got to work.

Here are the 18 steps that I used. The graphics here are from a few different files I sent over, so you won’t be able to recreate the album cover through this column. I switched back and forth depending on which one made a better graphic for that step. I also didn’t include the addition and subtraction marks between the steps. Figuring out when I added and when I subtracted will be left as an exercise for the reader. [1]

Step 0: the starting point

Not really a step, this was more of a chance for me to engage in some mathematics humor. A 17 with a line over it would be pronounced, "Bar 17." Does that really have any meaning in that situation? No; the bar is used for repeating decimals and in set theory to signify the complement of a subset [2]. However, there are few enough opportunities in life to make math jokes; take advantage of them when you can.

Step 1: infinite summation

This step is a sum of an infinite sequence of numbers. It’s 1 +1/2 + 1/4 + 1/8 + 1/16 + ... If you look at it closely, you’ll see that this sum comes closer and closer and closer to 2 without ever reaching it; each step adds half of the distance between the current sum and 2. Since we can get arbitrarily close to 2 and we’ll never exceed 2, this series converges to 2.

Current Count: 19

Steps 2 and 3: n choose k and the distance formula

Whenever there’s a multiplication of steps, I’m going to talk about them together. The first term there is from combinatorics. Pronounced, "n choose k"- 3 choose 2 in this case – it gives you the number of ways you can choose 2 items from three non-identical items. This is the formula for the lotto. If you want to know the odds of choosing 6 numbers out of 54 (or whatever it is in your state), you’d use this formula. The way to compute this is that it’s (n!)/(k!)(n-k!). What does the ! mean? That’s step 15 below.

The second step here is a famous one. That’s the distance formula for two points on the plane; it gives the distance between (18, 77) and (15,81). Being a math geek, I had to use the fact that 32 + 42 = 52. The 3-4-5 triangle is one of the most common examples of the Pythagorean Theorem. [3]
These steps are 3 * 5 = 15.

Current Count: 4

Step 4: – a little calculus

This is the integral between 0 and 3 of 1/3 ×2. The integral is one of the basic functions of calculus. It tells us the area under the curve defined by that function. As we all know, the answer to that problem is 1/9 (3)3 – 1/9 (0)3, which is 3. If you don’t believe me, draw an infinite amount of infinitesimally small rectangles and add up the areas yourself.

Current Count: 7

Step 5: – trig time

You can just plug this one into your calculator. The sin of 180 degrees is 0. To be honest, I’m not a huge trigonometry fan as it just reeks of practical applications. However, they’re way too important to ignore.

Current Count:7

Step 6: linear algebra

Linear algebra was my introduction to real math; this was the first class I took where there wasn’t a focus on what the concepts meant in terms of the real world but rather tried to prove things about a self contained universe.

That was my second linear algebra class. My first one – where I learned about determinants – still taught us to think of matrices as representations of linear equations. All I remember about the determinant is that a matrix is invertible [4] if and only if the determinant is non-zero.

There’s an annoying formula for figuring out the determinant of a 3 by 3 matrix, or you can just plug it into a online calculator.

What can I say? I’m lazy. I used the latter to get the answer. I assume that the rest of you did it in your head to get 6.

Current Count: 13

Step 7: Stirling Numbers (of the Second Kind)

Stirling numbers of the second kind are not encounters with UFOs that will inspire you to make a model of Devil’s Tower in mashed potatoes. Rather they are the number of ways that you can group together a set of n distinct elements into k different groups; in this case, it’s how many ways you can get 3 groups out of a 4 element set.

Those are ((1, 2), (3), (4)), ((1, 3), (2), (4)), ((1, 4), (2) (3)), ((1), (2, 3), (4)), ((1), (2, 4), (3)), and ((1), (2), (3, 4)). So S(4, 3) is 6.

When I first started this project, I was really excited to use these. Why? Because James Stirling amuses me. He had the kind of ego that I can only aspire to. He discovered two important concepts in combinatorics. He named one "Stirling numbers of the first kind," and the second, "Stirling numbers of the second kind." The notation of the first kind is s(n, k) – s is for Stirling of course – and the second uses S(n, k). This is what mathematicians did before we could get our work onto album covers.

Current Count: 7

Steps 8-9: limits and set theory

This is another multiplication of two steps. The former is a simple application of L’Hopital’s Rule. As I’m sure you never cared to know, this rule states that if you want to find the limit of a function that is either 0/0 or infinity/infinity at that number, you can take the derivative of both the numerator and the denominator, plug in the number into that function, and if that gives you a result, that limit is the same as the limit of the original function. [5]

This is a really useful technique for calculus students. It’s so useful, that people always want to use it for functions where it doesn’t apply. You can only get away with that once or twice before the teacher catches you.

Anyway, if you apply that rule, the limit as x goes to 2 is 4.

Step 9 is one of my favorites. All of those squiggly lines. What could those possibly mean? It’s residue of the attempt to put all of mathematics on a formal footing.

Rather than have numbers be a squishy concept based on reality, they were redefined in a much better manner. Starting with only a few basic axioms about set creation, it is also decreed that there exists a (unique) empty set {}. Since that contains 0 elements, that can be thought of as zero. Using that as a building block, we can have a set { {} } that contains the empty set. Call that 1. Continuing, 2 is the set that contains the empty set and the set containing the empty set: { {}, { {} } }. That’s clearly more obvious that looking at two apples and getting the idea of two from that, right?

Current Count: 15

So if you’re reading this far, you’re either into math, intellectually curious, or a groupie. Personally, I’m rooting for the groupie theory; I see no reason why I can’t charge admission for people to see me elegantly prove the irrationality of the square root of 2. Of course, that probably explains why I don’t have groupies…

Step 10: it’s big, it’s heavy, it’s wood.

Recall that log1594019679 asks what power of 159 would give you 4019679. Well, of course, 4019679 is 159 cubed, so the answer is 3. You knew that, right?

Current Count: 12

Steps 11-12: the many faces of algebra

Anyone who has taken an algebra class has seen the quadratic formula. It’s a way of figuring out the answer to an equation of the form ax2 + bx + c = 0, for some integers, a, b, and c. Step 11 finds the solution for x2 + 2x + 1, which has one root, -1.

That’s the quadratic formula. Much uglier formulas were found for cubic equations and quartic equations. The question then turned towards the quintic equation. Could there be a general formula that would solve all equations of the form ax5 + bx4 + cx3 + dx2 + ex + f =0?

The famous mathematician Galois found an answer to that question. He devised an incredibly beautiful structure involving ideas in abstract algebra. I took a Galois theory class in my second year of grad school and it was the crack cocaine of mathematics. I spent a semester of my life going, "What? I’m so lost. This makes no sense. Oh WAIT! If you look how this idea relates to this other one, there’s a connection between them! OH WOW! THAT’S SO COOL! I CAN’T BE… Damn, I lost it again…"

Galois theory builds and builds to a goal – the fundamental theorem of Galois theory which states… errrr… something or other about the interesting structure of these objects. It made sense to me for about 20 minutes in 1994. While I don’t pretend to understand the theory, I do remember the corollary to it – there is no general solution for any quintic or higher equation.

That – in addition to inspiring the old Irish mathematicians drinking song [5], "I’m a rambler and a gambler/And that’s not an evasion. / I know there’s no solution to a quintic equation. / I eat when I’m hungry and drink when I’m dry. / And if the moonshine don’t kill me, I’ll calculate pi." – is one of the most important discoveries in math. It also is what Galois theory was designed to do, and once done, all of this beautiful structure doesn’t really serve any other purpose. People study it because it’s amazing, but no one actually does anything with it anymore.

So, what exactly is the meaning of step 12? That Q with a line through it is shorthand for the rational numbers [6]. The idea here is to take a field [7] containing rational numbers and then throw in an irrational number – in this case, the cube root of 752701181. That creates a larger field. The order of this field over the rationals is the same as the order of the smallest polynomial with rational coefficients that will have that number as a solution. Since 752701181 is prime, that would be x3 – 752701181 = 0. Therefore, the order of this extension is 3.

Three. That’s a nice, friendly number. Why don’t we look at that for a while and forget all of this splitting field nonsense…

Current Count: 9

Steps 13-14: trying to use a constant that’s completely free

The Golden Ratio occurs over and over in random places. You would think that there wouldn’t be anything too interesting about a constant that equals one plus the square root of five divided by 2, but you’d be wrong. The Greeks believed that rectangles that had sides 1 and (1 + sqr(5))/2 were the most aesthetically pleasing to the eye. The Fibonacci Sequence – popularized in The Da Vinci Code – has a relationship with this number. If you divide any number in the sequence by the number before it, you get closer and closer and closer to this ratio, the further you progress in the sequence.

All of that is interesting, but that’s not why I included this ratio. The Greek letter used to symbolize this constant is phi. This letter is pronounced "fee."[8] I managed to sneak a Junta reference into the cover of a Trey album. Who says math isn’t fun?

Unfortunately, having an irrational number floating around would cause all sorts of problems if I wanted to get to 18. I had to get rid of it somehow. Modular arithmetic to the rescue.

Modular arithmetic takes a number and associates it with its remainder when you divide it by another number. 17 (mod 2) is equivalent to 1, because if you divide 17 by 2, you get a remainder of 1. The best way of thinking about this is clocks. If it’s 11 and you’re supposed to meet someone in two hours, unless you’re in the military, you wouldn’t wait for 13:00. Rather, you’d say that 13 is equivalent to 1 mod 12 [9], so I’ll meet them at one.

Since 8250210 is a multiple of 18 – it’s 458345 times 18 if you were wondering – 8250210 (mod 18) is 0. Multiplying anything by zero gives you zero, so the Phish joke doesn’t mess up the math fortunately.

Current Count: 9

Step 15: factorials

Way back in step 2, I promised to explain the exclamation point and I’m a man of my word. 3! (pronounced "3 factorial") is 1 * 2 * 3. In general n! is 1* 2 * 3 *.. * (n-1) * n. This function got its punctuation notation due to the fact that it grows so quickly. 3! might only be 6 but 10! is already 3,628,800.

Current Count: 15

Steps 16-17: matrix multiplication and the return of trig

Back in step 6 I used the determinant of a matrix. Step 16 returns to the linear algebra well to multiply two matrices. There’s a trick to doing this. You multiply the individual cells and then add them together. In this case it is (25 * 12) + ( 5 * -30) + (-37 *4) or 300 – 150 – 148 which is 2.

Ok, technically, it’s a 1 by 1 matrix with the single value 2 in it, but give me some poetic license here.

As for step 17, it’s time to pull out the old calculator again, first making sure to switch over to radians instead of degrees. The cosine of 2*pi is simply 1.

Current Count: 17

Step 18: returning to my roots

So far, we’ve used 17 steps and most of the interesting numeric concepts in modern mathematics, yet we’re back to where we started. How can we get to our goal? What would be worthy enough to be the 18th step? How about using the equation that inspired me to write an entire column?

Way back in 1998, the second issue ever of Jambands contained a column by me in which I rhapsodized about how cool it was that e(pi)(i) + 1 = 0. There are times in my life where that equation almost convinced me that there was a god of some sorts; how else could the three weirdest constants somehow cancel each other out in such a perfect way.

e(pi)(i) had to make an appearance, and so it did. If you solve for it, you find that e(pi)(i) = -1, so taking the absolute value, gives you 1. Add it to our running total and we find…

Current Count: 18

Eighteen steps to get to 18. It wasn’t the easiest of all paths, but I hope it was worth it. I think it’s safe to say that no other album released this year will contain so much mathy goodness on the cover.

If nothing else, I learned what a mathematics education is good for. The glut of PhD’s in the field might make it hard to find a teaching job, but apparently it can lead to you making album covers for one of your favorite musicians. If that doesn’t make you want to pick up Hungerford’s Algebra and turn to the chapters on Galois Theory, I don’t know what will.

You can order 18 Steps (oh, and that other album with the cover by some redheaded guitarist) from http://stores.musictoday.com/store/dept.asp?band_id=1105&dept_id=7800&sfid=2

[1] Solution will be provided on the cover of 18 Steps of course.

[2] The complement of a subset is every element that’s in the larger set that’s not in the subset. So if A = {1, 2, 3, 4, 5} and a subset B is {1, 3, 4}, its complement is {2, 5}.

[3] If you’re wondering how the distance formula is connected to a right triangle, think of the plane. If you want to find the distance between (18,77) and (15,81), you could draw a straight line between those points. If you also drew a line between (18,77) and (15,77) and then one between (15,77) and (15,81), you’d have a right triangle. Get some graph paper and draw it.

The hypotenuse of the triangle – the c in a2 + b2 = c2 – would be our distance. If you solve that equation for c, you get the distance formula.

[4] A matrix A is invertible if there exists another matrix B such that A*B is the identity matrix. The identity matrix has nothing to do with what you learn after eating the red pill.

[5] That’s really hard to describe without some graphics. If you’re curious about this, the Wikipedia page is pretty informative. Among other things, you can learn that L’Hopital’s Rule was actually discovered by Bernoulli. Sure, that’s not really fair, but it’s a lot of fun to type, "L’Hopital’s Rule."

[6] Written by me on a really boring day at Bard…

[7] Why Q? It would be R but R is used for the real numbers. The "Q" stands for quotient; a rational number, after all, is any number that can be expressed as a fraction.

[8] A field is a sort of abstraction of the concepts of addition and multiplication.

[9] Or "fi" depending who you talk to, but it’s not funny then.

[10] Well ok, you wouldn’t but I might if I were wanting to annoy people.

David Steinberg got his Masters Degree in mathematics from New Mexico State University in 1994. He first discovered the power of live music at the Capitol Centre in 1988 and never has been the same. His Phish stats website is at www.ihoz.com/PhishStats.html

He is the stats section editor for The Phish Companion and is on the board of directors for the Netspace Foundation. You can read more of his thoughts at www.livejournal.com/users/thezzyzx.

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