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The 3rd annual π day anime and mathematics post: A symmetric group of friends of degree 5

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「ふいにコネクト」/「ものくろあくたー。」

It’s that day of the year again.

Kokoro Connect’s premise made a lot of people raise their eyebrows, because really, what good can come from body-switching shenanigans? Well, let’s think about this for a second. We have a group of five kids and every once in a while, at random, they switch into the others’ bodies at random. What does that sound like? That’s right, a permutation!

Interestingly enough, the idea of connecting body-switching with permutations isn’t new. The Futurama writers did it and apparently got a new theorem out of it. What differs in the case of Kokoro Connect and Futurama is that in Futurama, the body-switching could only happen in twos. These are called transpositions. Obviously, this isn’t the case for Kokoro Connect. This doesn’t make too much of a difference since it turns out we can write out any permutation we want as a series of transpositions, but that wouldn’t be very fun for Heartseed.

We write permutations in the following way. If we let Taichi = 1, Iori = 2, Inaban = 3, Aoki = 4, and Yui = 5, we’ll have $(1 2 3 4 5)$ representing the identity permutation, when everyone’s in their own body. If Heartseed wanted to make Aoki and Yui switch places, he’d apply the following permutation
$$ \left( \begin{array}{ccccc} 1&2&3&4&5 \\ 1&2&3&5&4 \end{array} \right) $$
While it’s helpful for seeing exactly what goes where, especially when we start dealing with multiple permutations, this notation is a bit cumbersome, so we’ll only write the second line ($(12354)$) to specify a permutation.

For the purposes of this little exercise, we’ll consider applying a permutation as taking whoever’s currently in a given body. That is, say we permute Aoki and Taichi to get $(4 2 3 1 5)$. In order to get everyone back into their own bodies, we have to apply $(4 2 3 1 5)$ again, which takes Aoki, who’s in Taichi’s body, back into Aoki’s body.

So let’s begin with something simple. How many different ways are there for the characters to body switch? Both who is switched and who they switch with is entirely random. Again, since the switches aren’t necessarily transpositions, this means that we can end up with cycles like in episode 2, when Yui, Inaban, and Aoki all get switched at the same time. This can be written as $(1 2 4 5 3)$.

But this is just the number of permutations that can happen on a set of five elements, which is just 5! = 120. Of course, that includes the identity permutation, which just takes all elements to themselves, so the actual number of different ways the characters can be swapped is actually 119.

Anyhow, we can gather up all of these different permutations into a set and give it the function composition operation and it becomes a group. A group $(G,\cdot)$ is an algebraic structure that consists of a set $G$ and an operation $\cdot$ which satisfy the group axioms:

  • Closure: for every $a$ and $b$ in $G$, $a\cdot b$ is also in $G$
  • Associativity: for every $a$, $b$, and $c$ in $G$, $(a\cdot b)\cdot c = a\cdot (b\cdot c)$
  • Identity: there exists $e$ in $G$ such that for every $a$ in $G$, $e\cdot a = a \cdot e = a$
  • Inverse: for every $a$ in $G$, there exists $b$ in $G$ such that $a\cdot b = b\cdot a = e$

In this case, we can think of the permutations themselves as elements of a group and we take permutation composition as the group operation. Let’s go through these axioms.

Closure says that if have two different configurations of body swamps, say Taichi and Iori ($(2 1 3 4 5)$) and Iori and Yui ($(1 5 3 4 2)$), then we can apply them one after the other and we’d still have a body swap configuration: $(2 5 3 4 1)$. That is, we won’t end up with something that’s not a body swap. This seems like a weird distinction to make, but it’s possible to define a set that doesn’t qualify as a group. Say I want to take the integers under division as a group ($(\mathbb Z, \div)$). Well, it breaks closure because 1 is an integer and 2 is an integer but $1 \div 2$ is not an integer.

Associativity says that it doesn’t matter what order we choose to apply our operations in. If we have three swaps, say Taichi and Inaban ($(3 2 1 4 5)$), Aoki and Yui ($(1 2 3 5 4)$), and Iori and Yui $(1 5 3 4 2)$ and we want to apply them in that order. Then as long as they still happen in that order, it doesn’t matter which one we apply first. We’d have
$$((32145)(12354))(15342) = (32154)(15342) = (34152)$$
and
$$(32145)((12354)(15342)) = (32145)(14352) = (34152)$$

The identity means that there’s a configuration that we can apply and nothing will change. That’d be $(12345)$. And inverse means that there’s always a single body swap that we can make to get everyone back in their own bodies.

As it turns out, the group of all permutations on $n$ objects is a pretty fundamental group. These groups are called the symmetric groups and are denoted by $S_n$. So the particular group we’re working with is $S_5$.

So what’s so special about $S_5$? Well, as it turns out it’s the first symmetric group that’s not solvable, a result that’s from Galois theory and has a surprising consequence.

Évariste Galois was a cool dude, proving a bunch of neat stuff up until he was 20, when he got killed in a duel because of some drama which is speculated to be of the relationship kind, maybe not unlike Kokoro Connect (it probably wasn’t anything like Kokoro Connect at all). Among the things that he developed was the field that’s now known as Galois theory, which is named after him. What’s cool about Galois theory is that it connects two previously unrelated concepts in algebra: groups and fields.

One of the most interesting things that came out of Galois theory is related to the idea of solving polynomials. I’m sure we’re all familiar with the quadratic formula. Well, in case you aren’t, here it is:

$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$

This neat little formula gives us an easy way to find the complex roots of any second degree polynomial. It’s not too difficult to derive. And we can do that for cubic polynomials too, which takes a bit more work to derive. And if we want to really get our hands dirty, we could try deriving the general form of roots for polynomials of degree four. And wait until you try to do it for degree five polynomials.

That’s because, eventually, you’ll give up. Why? Well, it’s not just hard, but it’s impossible. There is no general formula using radicals and standard arithmetic operations for the roots for any fifth degree (or higher!) polynomial. The reason behind this is because $S_5$ is the Galois group for the general polynomial of degree 5. Unfortunately, proving that fact is a bit of a challenge to do here since it took about 11 weeks of Galois theory and group theory to get all the machinery in place, so we’ll have to leave it at that.

The 2nd annual π day anime and mathematics post

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「涼宮ハルヒの消失」/「茨乃」

Happy $\pi$ day. Once again, Nadeko will bring us in:

Snowy Mountain Syndrome is the third story in The Rampage of Haruhi Suzumiya, the fifth volume of the light novel. It’s the first story that has yet to be animated. It’s also a story that contains the dread spectre of mathematics.

So our SOS-dan is stuck in a mysterious cabin in the middle of a snowstorm on a mountain. They find a mysterious contraption that has an equation displayed:

$$x-y=(D-1)-z$$

and they are to provide $x$, $y$, and $z$. Koizumi and Kyon are confused, but Haruhi rightly identifies this equation as Euler’s polyhedron formula, which is also very often referred to as just Euler’s formula. If you’re referring to it in context, it doesn’t matter that much, but it’s useful to distinguish between all the other things that Euler discovered, which is a hell of a lot.

First, we should probably go over some basic definitions. When we talk about graphs, we’re not talking about bar graphs or pie charts or the like. We’re also not talking about graphs of polynomials on a cartesian plane or other such functions. Graphs are a mathematical structure which, when drawn, looks like a bunch of circles and lines.

Formally, a graph is a pair $G = (V,E)$ where $V$ is a set of vertices and $E$ is a set of edges. Vertices can be any old thing, but each edge is defined as a pair $(u,v)$ where $u$ and $v$ are vertices in $V$. When we draw graphs, we just draw a vertex as a circle and draw an edge as a line that connects the two vertices it’s defined as.

And that’s it! That’s the most general definition of a graph, which means we can end up with a graph that’s completely empty or a graph that’s just a bunch of vertices with no edges in between them. We can even have multiple edges going in between two vertices. Of course, often times, we’d like to add some more constraints, depending on what we want to do with our graphs. Very often, we’d like to restrict the number of edges between two vertices to one and that’s what we’ll do.

Back to the formula, usually it’s given as $\chi=v-e+f$, where $v$ is the number of vertices, $e$ is the number of edges, $f$ is the number of faces, and $\chi$ is called the Euler characteristic. That makes $x=v$, $y=e$, $f=z$ and $D-1=\chi$. Now, the only thing here that we haven’t seen defined yet is a face. Intuitively, we can see that’s just the space that’s bounded by the edges.

What I find strange is the explanation in the novel that $D$ stands for the dimension of the polyhedra. As far as I know, this only works in the three-dimensional case for platonic solids. Once we generalize the structures to other kinds of polyhedra and topological surfaces, that analogy breaks down.

Anyhow, the way the formula is applied in the book is the use that I’m most familiar with, which is as a property of a planar graph. For planar graphs, $\chi=2$. In the novel, they deduce that $\chi=1$ since $D=2$ and that only works because they didn’t count the large face outside of the edges as a face, which we usually do.

But what is a planar graph? Well, if you go back to our definition of a graph, you might notice that all we’ve done is said that it’s a bunch of vertices and edges. We’ve said nothing about how to draw a graph. Usually, we represent vertices as circles and edges as lines in between those circles, but other than that, there’s really nothing telling you what order to draw your circles in or whether your lines have to be completely straight or not or how far apart everything has to be. How you choose to represent your graph is up to you, although if you draw your graph weirdly, you might make the people trying to read it angry.

Informally, a planar graph is a graph that you can draw with none of the edges crossing each other. This seems like a kind of silly thing to be worried about, because it seems like you could just keep on drawing a graph until it works out. Well, for edges with a lot of vertices and edges, it’s not obvious and even for really small graphs. For instance:

At a glance, it doesn’t look like the drawing on the right is planar, but all we have to do is drag one of the vertices into the middle to get the drawing on the left and it turns out they’re both the same graph, $K_4$, the complete graph of order 4.

That’s where Euler’s formula comes in really handy. It gives us a way of figuring out whether or not our graph is planar or not without having to fiddle around with placing edges properly and stuff. You already know how many vertices and edges you’ve got, so all you need to do is make sure you’ve got the right number of faces.

So it’s probably pretty clear at this point that you can’t draw every graph without the edges crossing. We can say something interesting about those graphs too, which just turns out to be another characterization of planar graphs, but oh well. But first, we have to introduce the concept of graph minors.

Suppose we have a graph $G=(V,E)$ and an edge $e=(u,v) \in E(G)$. If we contract the edge $e$, we essentially merge the two vertices into a new vertex, let’s call it $w$, and every edge that had an endpoint at $u$ or $v$ now has $w$ as the corresponding endpoint. Then a graph $H$ is a graph minor of $G$ if we can delete and contract a bunch of edges in $G$ to get $H$ (or a graph that’s isomorphic to $H$).

It turns out that every non-planar graph can be reduced to a minor of one of two graphs. The first is $K_5$, the complete graph of order 5:

The second is $K_{3,3}$, the complete bipartite graph 3,3:

These two graphs are the smallest non-planar graphs, otherwise we’d be able to reduce them further to get another non-planar graph. Like I mentioned before, this is a characterization for planar graphs too, since a planar graph can’t contain a $K_5$ or $K_{3,3}$ minor.

I guess I’ll end by saying that graphs are hella useful, especially in computer science. A lot of people complain about never using math like calc ever. If you’re a developer, you’ll run into graphs everywhere. It’s pretty amazing how many structures and concepts can be represented by a bunch of circles and lines.

Kenji the quantum computer, Part 2

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He will use this knowledge in less than 24 hours.

So back when I was moving in at the start of the school year, I decided to try out my BD player on my hot new Dell monitor. Since I didn’t want to take out my Kara no Kyoukai BDs from their box, I opted to crack open the cheap flimsy Summer Wars case and throw in the disc. I hadn’t actually tried playing it yet anyway. I opted to skip the OZ introduction and go straight to the opening. First of all, I was disappointed that Funimation set the credits in goddamn Arial. Seriously, what? I find it difficult to believe they didn’t have Helvetica already from one of countless projects they could’ve used it on already.

More importantly, Funimation translated the chapter title of the book that Kenji was reading on the train. It turns out he was reading about Shor’s Algorithm. Shor’s Algorithm is the quantum algorithm that can factor integers in polynomial time. I alluded to it when I talked about Summer Wars and cryptography a while back and now’s as good a time as ever to get into it, since, you know, this means Kenji really is a quantum computer and all.

In quantum computing, instead of dealing with bits that are 1s and 0s, we deal with quantum bits, or qubits, that are denoted $|1\rangle$ and $|0\rangle$. These are really just quantum states. Quantum states have interesting properties, the most famous of which is being in superposition. Essentially, what this means is that we can have qubits that are in a state where it represents 1 and 0 simultaneously.

This turns out to be a very neat property to have. If we’ve got $n$ qubits, then we’ve got like $2^n$ inputs simultaneously. That’s great, right? We can compute things exponentially fast now! Well, it turns out it isn’t that simple.

Suppose that we do end up running an algorithm or something on our $2^n$ states. What happens when we want an answer? Well, we measure the state. The problem with that is that on measuring a quantum state, you won’t get all $2^n$ results. You’ll only get one of those results with a certain probability. What’s more is that once you measure a quantum state, you’re stuck with whatever it gave you and you can’t go back and check for another result.

That is obviously no good, because we want not just any answer, but, you know, the right answer. So it looks like approaching our problem from the naive interpretation doesn’t seem to work. And in fact, that immediately tells us that quantum computers aren’t a magic bullet that can help us solve SAT in a reasonable amount of time.

What we can do is manipulate our resultant quantum state so that the actual correct result we want is the one with very, very high probability and reduce the probability of getting any of the other results to near-zero. Of course, the problem is that we don’t know which result is the correct one to begin with. It’s actually this exact hitch that prevents us from efficiently solving general NP-complete problems.

Luckily, if we want to factor some numbers, we can take advantage of this. We can reduce the problem of factoring any old number into prime factors to the problem of order finding, which I’d mentioned before. For an integer $N$, there’s another integer $m$ such that $k^m=1\bmod N$ and we call $m$ the order of $k$ in the integers modulo $N$.

So suppose we have $k\bmod N$ with order $m$ and we decide to keep on taking powers of it, like $k\bmod N,k^2\bmod N, k^3\bmod N$ and so on. Eventually, we’ll get up to $k^m\bmod N$, at which point, that’s just $1$ and we start our cycle over again. The thing to note here is that this sequence is finite and periodic, which narrows down what we have to search through.

So what do we do? We create a quantum state for our period that’s in a superposition over all the powers of $k$ in $\bmod N$. We don’t know what our period is (that’s what we’re trying to find), so we just choose a large enough one that our order will be in the period. Now, in order to find this order, we do some magic with a Quantum Fourier Transform.

What?

So I mentioned at the beginning that when we look at a quantum state in superposition, we can think of it as returning one of the possible states with a certain amount of probability. How the probabilities for these quantum states work is actually more complicated and it’s from this that we can massage an answer out of our magic quantum algorithm.

Quantum states have something called amplitudes, which are kind of like probabilities, except that amplitudes can be basically any complex number as long as all of the amplitudes $a_i$ for a quantum state satisfy $\sum|a_i|^2=1$. If we’re just measuring a quantum state, this really doesn’t change anything, we just get state $|i\rangle$ back with probability $|a_i|^2$. But when we’re manipulating quantum states, there’s more to the amplitudes to consider.

If we think about amplitudes in terms of waves, we can see that we can cancel waves out. For instance, the states $\alpha|1\rangle$ and $-\alpha|1\rangle$ have the same probability if we were to measure them (since $|\alpha|^2=|-\alpha|^2$), but if we put them together and add them, they’d cancel each other out. What’s more is that since amplitudes are complex numbers, so we can think of them as having a direction (like a vector on the complex plane) and we end up with weird stuff like having things partially cancel out.

So just like a regular Fourier transform extracts some information about frequencies from a sequence, we can use the Quantum Fourier Transform to extract some information about all of this amplitude stuff that we can’t see just by measuring the state. And this is exactly how we can manipulate our state so that it increases the probability of the answer we want when we go to measure it and depress the probabilities of all the other answers.

Now after rambling about amplitudes and Fourier transforms and orders, we should probably step back and see what we’ve done in perspective. What we have is a way to find the order $m$ of an element $k \bmod N$. Well, actually, we don’t even have that quite yet. What we get out of all of that quantum stuff is just some information about $m$ that we have to manipulate further, except we just do that with plain old number theory instead of quantum computation. But, what we’ve kind of stumbled through is kind of a vague description of Shor’s algorithm.

Like I mentioned in the first Summer Wars post, this is the only way we know of so far that lets us factor large numbers (especially the ones that are used in cryptography) within a feasible amount of time. Again, Kenji (or at least his brain) is basically a quantum computer. Coincidentally, I played around a bit with what the implications could be if our brains were quantum computers in an earlier post on Kaiba.

But even if Kenji isn’t a quantum computer, he’s goddamn smart for a high school student. The theory behind the computing part of quantum computing is really just a ton of linear algebra, so an understanding of quantum mechanics isn’t really necessary (which is great, because I don’t know any of that crap). The quantum computing course that I took in my undergrad only had second year linear algebra as a formal prerequisite, but that was mainly to allow students from different departments to take it and also implies that you’ve got first and second year math down at the very least.

Suugaku Girl supplementary handout for chapter 2: On definitions

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An introduction to mathematical writing

In chapter 2 of Suugaku Girl, we’re introduced to the third component of the little love triangle that’s forming. Tetra is the underclassman that the main character is tutoring and she’s one of the many people who think they might like math but school eventually beats that silly thought out of them.

Anyone who’s taken a real math class will know that by the end of it, your notes are essentially a giant list of definitions, theorems, and proofs along with an example thrown in once in a while. By a real math class, I mean a math class that’s actually concerned about reasoning about how we get our theorems instead of focusing on how to use them for practical applications. I did my undergrad at a school where math has its own faculty and where there’s a lot of rivalry between the various faculties. Everyone in math often jokes about how an example suffices as a proof for engineers and their ilk.

I checked your textbook for a proof and it said that we’ve done enough examples for it to be plausible. Must have been written by engineers.
— Vanderburg, PMATH 340 (F09)

Here, Tetra falls into the same trap when the main character asks her to define what prime numbers are. So MC-kun has to explain how definitions work. First of all, definitions have to be precise, just like theorems. It’s pretty easy to lose marks for misstating theorems and definitions by leaving off an edge case for 0 or 1 or something silly like that. Of course, that leads Tetra to question why these things need to be so precise and arbitrary.

One of the things I realized about math is that it’s all about trying to do things with the definitions you start out with. You can kind of see that in how the number system is built up. We can start with plain old numbers that we use to count things. And then we can add things to it. Like, what happens if we have less of a thing, how would we represent that? Tada, we’ve got negative numbers. Okay, now what if we have a part of a thing? Now we’ve got fractions. And so on and so forth. We realize that not every number can be written as a fraction and suddenly we’ve defined real numbers.

I’m sure we’ve all wondered why complex numbers work out the way they do. It’s because we’ve defined everything to work out like that. Some guy tried to take the square root of a negative number and found that it didn’t work out very well, so we defined the square root of -1 to be $i$. Now we have this $i$ thing, what can we do with it? Well, we can just start writing crazy things like $3 + i + 4i^2 + 5i^3 + 9i^4 + 2i^5 + 6i^6$, but then we realize that it just becomes $2-2i$. Well, okay that’s kind of neat.

But now we know that all complex numbers can be written as $a+bi$ with $a,b\in\mathbb{R}$. So someone along the line must’ve been like, what would happen if we tried to graph these things? So we treat $a$ as the $x$ component and $b$ as the $y$ component and it turns out we can think of complex numbers as other structures like a vector or just a 2-tuple or something. And suddenly, this gives us a way to compare complex numbers, by taking the length of the vector that they define. And now that we have vectors, we can do some weird geometry stuff with them. We can think of these things as the length of the vector and the angle that they form. And then you can go crazy and talk about roots of unity or what multiplication of complex numbers might mean.

The point is that all of mathematics is built up like this. You start off with some definitions or premises and you go nuts. Of course, you don’t have to use the same definitions as everyone else. The problem with doing that is you might not end up with anything interesting. Like, what if we did something nuts and defined 1 to be a prime number like so many people do when they leave off that condition? Well, not much, we’d probably just rephrase all of our theorems to exclude 1.

On the other hand, sometimes when we play around with definitions, we do end up with something interesting. For instance, we can define something called the extended complex numbers, which is just the set $\mathbb{C}\cup\{\infty\}$. Yep, we just say okay, infinity is a number now, deal with it. So what can we do now?

Well, we can divide by 0 now.

I imagine there might be a few people who might flip out at this notion, but yes, since $\infty$ is included in our number system, we can define $\frac{1}{0}=\infty$. Of course, we can’t do everything — $0\times\infty$ and $\infty-\infty$ still don’t mean anything. But if you’ve been paying attention, you might be going, okay well, we can divide by 0, but what else can we do? Dividing by 0 is kinda meaningless if there’s nothing new we can do.

As it turns out, the extended complex numbers defines a very different geometric object. If we remember from the example of the complex numbers, we can basically define every complex number as a vector over a two-dimensional plane. Here, with the extended complex numbers, we can define every number as a line passing through something called the Riemann sphere. And like the complex plane, this sphere lets us create weird relationships between numbers and angles and stuff. This turns out to have interesting properties in complex analysis and quantum mechanics.

So yes, definitions are often arbitrary. Why? Because it’s just useful and interesting that way. You could argue that it’s because nature forces us to define things a certain way. Kind of like, of course you can’t take a square root of a negative number, you just can’t do it! What happens, though, is that we always seem to end up finding useful things that line up with our mathematics rather than inventing our mathematics to do useful things with. After all, mathematicians have been playing around with imaginary numbers for at least a few decades before electromagnetism was even discovered.

Suugaku Girl supplementary handout for chapter 1: Sequences

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That really isn't enough terms to identify as Fibonacci

Suugaku Girl is a manga about people who really like math. The great thing about it is that it contains a substantial amount of math. This is also great because it will give me plenty to blog about.

The first chapter of Suugaku Girl is about the main character’s initial meeting with Milka, our socially retarded genius meganekko. She just starts spouting out numbers and, for whatever reason, he feels compelled to guess what comes next. And that is apparently the beginning of this love story.

In the chapter, we’re introduced to a few sequences, some of which are famous and some of which you might not have considered a sequence. It’ll probably help to understand what a sequence is beyond just being a bunch of numbers in a prescribed order.

Formally, we define a sequence as a function $a:S\to R$. The set $S$ is basically our index and is $\{1,2,…,n\}$ if $a$ is a finite sequence or the set of natural numbers $\mathbb{N}$ if we’re dealing with an infinite sequence. Normally, functions are written as $a(n)$, but, as was alluded to before, we refer to terms by index as in $a_n$.

However, $R$ can be any set. In the case of Suugaku Girl, it seems to be sticking with integers, but we can have sequences of bitstrings, vectors, complex numbers, functions, or whatever. What this also means is that a sequence doesn’t necessarily need to have a “pattern”, but can really be any ordered list of numbers (or functions or vectors or…).

Milka also brings up the idea of infinite sequences. A lot of the time, people will try to “solve” a sequence by completing it when they’re given only the first few terms. But, like Milka suggests, what they’re doing in that case is assuming that the rest of the sequence goes on as initially implied. Really, we can define any sequence we like with any behaviour we like. Again, remember that a sequence can be anything we want it to be. In fact, the sequence that Milka defines using the digits of π is kind of like that in that it’s completely arbitrary and doesn’t really have the kind of sequence definitions we’re used to seeing.

For finite sequences, we can just list all of the terms of our sequence like $(1,1,1,1,1,1000000,1)$. Obviously, we can’t do that for an infinite sequence. Luckily, a lot of the time we define sequences that have some sort of useful pattern that we can represent in a succinct way. Sometimes, like the digits of π sequence, this is harder.

We can try to formally define all of the sequences that were given in the manga. For instance, the Fibonacci sequence $(a_n)_{n=1}^\infty$ is commonly defined as $a_n = a_{n-2} + a_{n-1}$, but we have to give the first two terms $a_1 = 1, a_2 = 1$. The second sequence (which we’ll call $(b_n)_{n=1}^\infty$) takes a bit more work to define. We’ll need to define $p_n$ to be the $n$th prime number and then we can define $b_n = p_n \times p_{n+1}$. The third sequence $(c_n)_{n=1}^\infty$ is easy, it’s just $c_n = n^n$. And we can formalize the last one, which I’ll call $(d_n)_{n=1}^\infty$, just like the first two with a few more words. We let $\pi = q_1q_2q_3\cdots$ be the decimal expansion of π. Then $d_n = 2\cdot q_n$.

So that’s all fine, but what exactly are sequences used for? I’m pretty sure everyone’s learned about arithmetic and geometric sequences in grade school. Obviously, we can study sequences and their behaviour on their own. We can talk about whether they increase or decrease or how fast they grow or whether they converge. Apart from that, I don’t remember seeing sequences used for something besides sequences until analysis.

Analysis is basically the field of pure math that formalizes the concepts that we’re introduced to in calculus and generalizes them to spaces. Limits are a fundamental idea in calculus and analysis and these are defined by how a sequence converges. And this is where those weird sequences of vectors or functions comes into play, since we can talk about the convergence of a sequence of vectors or a sequence of functions in these other spaces that we want to do analysis in.

That’s probably the easiest example of an “application” of sequences. For myself, over the last few months I’ve read about automatic sequences, which are sequences that can be generated by a deterministic finite automaton. This gives us a way to relate automata theory to number theory and algebra. For instance, once we have k-automatic sequences, we can talk about k-regular sequences and come up with power series with respect to certain rings and fields and bla bla bla.

If you ever want to find out what crazy sequence you might have a part of, check out the Online Encyclopedia of Integer Sequences.