Jeremy CoteScientific and mathematical thinking for the curious.
https://jeremycote.me/
The Rational Roots Theorem<p>One of the skills students learn in secondary school is to factor quadratic expressions. In particular, they learn how to solve equations like <em>x<sup>2</sup>+2x+1=0</em>. There are a slew of techniques one can use to deal with quadratics, and they mostly rely on the fact that questions asked in assignments and tests have “nice” factorizations. Most expressions have integer solutions, or at worst rational ones. This makes it straightforward to factor. Of course, this might take a while to get used to, but it’s a skill that many students pick up.</p>
<p>If they <em>really</em> have no idea how to factor an expression, they learn about the tool that deals with basically every case: the quadratic formula. This is something that most students will remember from their days in mathematics class (even if they don’t remember the formula explicitly). The nice thing about this is that, as long as you can crank the gears of arithmetic, the quadratic formula won’t fail you.</p>
<p>This is great, but the reality is that the world has much more than just quadratics and linear equations. In fact, what you learn soon after leaving secondary school is that the whole <em>class</em> of polynomials tends to be considered “nice”. Therefore, a question you might ask is, “How can I factor these expressions?” After all, there’s nothing different in principle with factoring a polynomial of degree greater than two. You still have to find the roots of the equation, though this looks more difficult when you have a polynomial with a higher degree.</p>
<p>Being able to factor larger polynomials comes up from time to time. I had to do this in my differential equations class, which doesn’t immediately suggest factoring. While one of us was bemoaning the difficulty of factoring these large polynomials, my professor stated an interesting little result that I didn’t know about. It’s called the rational roots theorem, and it made enough of an impression on me that I still remember it.</p>
<p>Here it is. The rational roots theorem tells us that if a polynomial with a constant term has a rational root in it of the form <em>a/b</em> where <em>a</em> and <em>b</em> are relatively prime, then <em>a</em> will divide the constant term and <em>b</em> will divide the coefficient of the leading term.</p>
<p>I think it’s quite instructive to see this in an example, so let’s look at one before the proof of why this works. Suppose we have the following polynomial:</p>
<p><em>4x<sup>2</sup>+9x+2</em>.</p>
<p>The question is, can we factor this nicely? Since I took a simple quadratic, you can probably figure out its factors without this method. However, if we refer to the rational roots theorem, we need to look at the constant term 2 and the coefficient of the leading term, 4. The factors of 2 are 2 and 1, while the factors of 4 are 1, 2, and 4. Furthermore, since our theorem simply says that the numbers <em>a</em> and <em>b</em> will divide the constant and leading coefficient, our values can be negative too. As such, our possible values of <em>a</em> are 1, 2, -1, 2, while the value of <em>b</em> could be 1, 2, 4, -1, -2, and -4. The possible values for a solution are given by the rational number <em>a/b</em>:</p>
<p><em>±1, ±1/2, ±1/4</em>, and <em>±2</em>.</p>
<p>This gives us eight possible solutions to the equation. If there’s a rational solution, it will be in the above list. We can then test each one and see if the output is zero (meaning it’s a root). In our case, since the coefficients of each term are positive, the only way to get an output of zero will be if the input is negative. That eliminates half the values. We can then test to find that the solutions are <em>x=-2</em> and <em>x=-1/4</em>, which are both on the list.</p>
<p>Now, you might not be too impressed with this. After all, this doesn’t guarantee we will find solutions. It just gives us possible candidates. What happens if we have a few solutions in our list, but some that aren’t there (since they aren’t rational)?</p>
<p>This is a valid concern, but the reason I’m discussing this theorem is because it has practical value in class. When factoring polynomials, chances are the solutions will be rational. For quadratics, I wouldn’t employ this method since I’m used to eyeballing the solution (through lots of practice). However, when the degree of the polynomial is larger, this theorem comes in handy.</p>
<p>The idea isn’t to necessarily find <em>all</em> the factors in one go. In fact, the context that my professor mentioned this was when he explained that factoring a polynomial becomes easier as soon as you find one factor. That’s because you can then exploit long division to make the polynomial smaller, which is easier to work with.</p>
<p>So what is the proof of this result? Thankfully, there isn’t anything too difficult with it. The crux of the proof lies in rearranging the equation for the roots.</p>
<p>To begin with, let’s consider a polynomial <em>p(x)</em> of the form:</p>
<p><em>p(x) = a<sub>n</sub>x<sup>n</sup> + … + a<sub>1</sub>x + a<sub>0</sub></em>.</p>
<p>For convenience, we’ll say that the leading coefficient <em>a<sub>n</sub></em> isn’t zero (or else we’ll just consider the next leading coefficient). One other requirement is that the constant term <em>a<sub>0</sub></em> isn’t zero either. This is crucial for the proof, as we will see below.</p>
<p>Now, consider a rational root to <em>p(x)</em>. Furthermore, since we can always simplify a rational number until the numerator and denominator are relatively prime, let’s call the rational root <em>u/v</em>. Substituting this into the polynomial gives:</p>
<p><em>p(u/v) = a<sub>n</sub>(u/v)<sup>n</sup> + … + a<sub>1</sub>(u/v) + a<sub>0</sub> = 0</em>.</p>
<p>We then want to see if this imposes any conditions on <em>u</em> or <em>v</em>. To start with, we can move the constant term <em>a<sub>0</sub></em> to the other side of the equation, and then multiply both sides by <em>v<sup>n</sup></em>, since this will remove any fractions. Doing so gives:</p>
<p><em>a<sub>n</sub>u<sup>n</sup> + … + a<sub>1</sub>uv<sup>n-1</sup> = -a<sub>0</sub>v<sup>n</sup>.</em></p>
<p>At this point, look at the left side of the equation. Each term contains at least one <em>u</em>, which means the whole of the left side is divisible by <em>u</em>. However, since the two sides are equal, this means <em>u</em> also divides the right side. In terms of the above equation, this looks like:</p>
<p><em>u(a<sub>n</sub>u<sup>n-1</sup> + … + a<sub>1</sub>v<sup>n-1</sup>) = -a<sub>0</sub>v<sup>n</sup>.</em></p>
<p>We know a bit more than this though. Since the root for <em>p(x)</em> was <em>u/v</em>, we know that <em>u</em> can’t divide <em>v</em>. Why? Because <em>u</em> and <em>v</em> are relatively prime, which means they don’t share any common factors. Therefore, one definitely can’t be a multiple of the other. Since <em>u</em> doesn’t divide <em>v</em>, this <em>also</em> means <em>u</em> can’t divide <em>v<sup>n</sup></em>. This gives us only one final possibility, which is that <em>u</em> must divide <em>a<sub>0</sub></em>!</p>
<p>We can play a similar game with our equation of <em>p(u/v) = 0</em> by rearranging the terms for <em>v</em>. Doing so gives us:</p>
<p><em>a<sub>n</sub>u<sup>n</sup> = -a<sub>0</sub>v<sup>n</sup> - a<sub>1</sub>uv<sup>n-1</sup> - … - a<sub>n-1</sub>u<sup>n-1</sup>v = v(-a<sub>0</sub>v<sup>n</sup> - a<sub>1</sub>uv<sup>n-1</sup> - … - a<sub>n-1</sub>u<sup>n-1</sup>).</em></p>
<p>Again, we that there is an overall factor left in the polynomial terms, so we can pull out a <em>v</em> from each term. This means the whole right side is divisible by <em>v</em>. As such, we get that the left side is divisible by <em>v</em>. Then, we can note that the same argument applies as before, which lets us conclude that <em>v</em> must divide <em>a<sub>n</sub></em>. This gives us the desired result we wanted.</p>
<p>Note here that is was very important to have a constant term in our polynomial. Without it, we couldn’t bring a term onto the other side of the equation and then multiply through by <em>v<sup>n</sup></em>.</p>
<p>There’s one more thing we can note here. If the leading term in the polynomial is <em>a<sub>n</sub> = 1</em>, this implies <em>v = 1</em>, since it’s the only integer that divides 1. (Strictly speaking, we could also consider <em>v = -1</em>, but we can just absorb the sign in the numerator.) What this means is that, if the leading term of a polynomial has a coefficient of one, the roots will either be integers or irrational numbers.</p>
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<p>What I find so interesting about this theorem is how the leading coefficient and the constant term impose “constraints” on the roots of a polynomial. It’s not <em>really</em> surprising when you understand that when you expand the factored form of a polynomial, the “information” from the roots is encoded in the constant term. I’m not sure how rigorous this interpretation is, but I like the heuristic.</p>
<p>Like most interesting results, there’s no amazing practical use of it. Yes, it can make your life easier if you find yourself needing to factor large polynomials, but other than that I just find it makes for an interesting theorem.</p>
<p>As a final note, there is a clever application of this theorem that I want to mention. In the essay, I didn’t mention the straightforward implication of the result: if you find that there’s a solution to the equation which <em>doesn’t</em> fall into the rational category, the solution is irrational. This might sound sort of obvious, but it means we can “test” to see if certain numbers are irrational by constructing polynomials with them as solutions. I learned about this from a Mathologer <a href="https://www.youtube.com/watch?v=D6AFxJdJYW4">video</a>, so I highly recommend checking it out if you want more details.</p>
Fri, 24 May 2019 00:00:00 +0000
https://jeremycote.me/rational-roots-theorem
https://jeremycote.me/rational-roots-theoremMathematics Isn't Just Numbers<p>We often equate mathematics with numbers, as if mathematics doesn’t extend further than doing arithmetic. Each time this happens, I have to restrain myself from going on a rant. I want to grab the person by the collar and exclaim, “There’s <em>so</em> much more to mathematics than just numbers! It’s like saying that running is just a bunch of one-legged hops. While that might be technically true, it’s not the way most people would describe their experience. In the same way, mathematics is way larger than just numbers.”</p>
<p>Even within my own family, some of them still see mathematics as essentially just a bunch of numbers with the associated arithmetic. I’m <em>in</em> a mathematics program, and this still isn’t clear to them. I think that’s clearly a failure on my part to share the diverse aspects of mathematics.</p>
<p>I encounter (and recognize) mathematics everywhere in my life. I know that it’s responsible for a lot of what I see on the web in terms of illustrations and graphics, algorithms rule our lives both online and offline, and mathematics is present in all the engineering I see. Most of these examples aren’t <em>purely</em> about numbers. Sure, it’s difficult to get away from numbers in mathematics, but they aren’t always the primary players. And yes, you can find plenty of examples of mathematics in the “real world” which aren’t cringe-inducing (like you might see in textbooks).</p>
<p>As a mathematics student, I’ve learned <em>so</em> much that doesn’t directly involve numbers. I’ve learned a bunch of geometry, analysis, probability and statistics, abstract algebra, discrete mathematics, and graph theory. While numbers are present in each of these topics, they only serve to make the concepts easier to handle.</p>
<p>If I wanted to draw “mathematics” and “numbers” as two sets, the former would encompass the latter. In other words, numbers are a <em>part</em> of mathematics, but they aren’t everything. This is something that I want to make more clear to a general audience. Mathematics isn’t all about numbers. In particular, if you just look at the area of geometry, there is so much you can learn without even worrying about numbers. This is an especially fertile ground for those with a passing interest in mathematics.</p>
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<p>The lesson is simple: there’s a lot more to mathematics than numbers. Of course, numbers are present almost everywhere in mathematics, but they aren’t the point in and of themselves (unless you’re studying number theory, perhaps). Mathematics is a lot richer than that, so there’s no reason to put it off if you’ve “never been good with numbers”.</p>
Mon, 20 May 2019 00:00:00 +0000
https://jeremycote.me/mathematics-isnt-just-numbers
https://jeremycote.me/mathematics-isnt-just-numbersDegeneracy of the Quantum Harmonic Oscillator<p><em>Note: This post uses MathJax for rendering, so I would recommend going to the site for the best experience.</em></p>
<p>I just love being able to find neat ways to solve problems. In particular, there’s something about a combinatorial problem that is so satisfying when solved. The problem may initially look difficult, but a slight shift in perspective can bring the solution right into focus. This is the case with this problem, which is why I’m sharing it with you today. Don’t worry if you don’t know any of the quantum mechanics that goes in here. The ingredients themselves aren’t important to the solution of the problem.</p>
<h2 id="energy-of-the-quantum-harmonic-oscillator">Energy of the quantum harmonic oscillator</h2>
<p>If you have taken a quantum mechanics class, there’s a good chance you studied this system. The quantum harmonic oscillator is one that can be solved exactly, and allows one to learn some interesting properties about quantum mechanical systems. Briefly, the idea is that the system has a potential that is proportional to the position squared (like a regular oscillator). In the quantum mechanical case, the aspect we often seek to find are the energy levels of the system. This is what is of interest in our problem here.</p>
<p>In one dimension, the energy is given by the relation $E_n = \left(n+1/2 \right) \hbar \omega$, where <em>n</em> is an integer greater or equal to zero, and the terms outside of the parentheses are constant (Planck’s constant and the angular frequency, respectively). However, what’s nice is that this extends into any number of dimensions in a straightforward way. If we want to look at the harmonic oscillator in three dimensions, the energy is then given by:</p>
<script type="math/tex; mode=display">E_{n_x,n_y,n_z} = \left(n_x + n_y + n_z +3/2 \right) \hbar \omega.</script>
<p>In other words, there’s a <em>n</em> value for each dimension. We can even consider the harmonic oscillator in <em>N</em> dimensions, and the energy would change in the same way. We would just add a new <em>n</em> index, and throw in an extra factor of 1/2. Furthermore, it’s important to know that each combination of <em>n</em>’s gives a <em>different</em> physical system.</p>
<p>What you might notice from the three dimensional case is that there are different combinations of <em>n<sub>x</sub></em>, <em>n<sub>y</sub></em>, and <em>n<sub>z</sub></em> that give rise to the same total energy. For example, we can note that the combinations (1,0,0), (0,1,0), and (0,0,1) all give the same total energy. This is called <em>degeneracy</em>, and it means that a system can be in multiple, <em>distinct</em> states (which are denoted by those integers) but yield the same energy. In this essay, we are interested in finding the number of degenerate states of the system.</p>
<h2 id="the-counting-problem">The counting problem</h2>
<p>Here’s the question. Given a certain value of <em>n</em> (which in the three-dimensional case is <em>n = n<sub>x</sub> + n<sub>y</sub> + n<sub>z</sub></em>), how many different combinations of those three numbers can you make to get the same energy? If we want to be more general, for a given <em>n</em> and <em>N</em> (the number of dimensions), what is the degeneracy?</p>
<p>If we do a few examples, we will see that the degeneracy in three dimensions is one (no degeneracy) for <em>n = 0</em>, three for <em>n = 1</em>, six for <em>n = 2</em>, and ten for <em>n = 3</em>.</p>
<p>I don’t know if you’re seeing a pattern here, but it’s not super clear to me. I definitely don’t see how to generalize this to any <em>n</em>, let alone for more dimensions. As such, we’re going to look at this in a whole different way.</p>
<p>The method I’m going to discuss is one I found <a href="https://physics.stackexchange.com/questions/317323/degeneracy-of-the-isotropic-harmonic-oscillator">here</a>, and is called the <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)">“stars and bars” method</a>. It’s a beautiful technique that captures exactly what this problem is asking.</p>
<p>We start by thinking about how we can represent this problem. For a given <em>n</em>, we want to find a way to split this number into separate parts. Say we want to split the number into four parts. Then, we would need to introduce <em>three</em> splits in the number <em>n</em> so that there are four “pieces”.</p>
<p>How many parts do we want for our particular problem? Well, it depends on the dimension we’re working in! For example, if the dimension is three, we want to split <em>n</em> into three parts. This means we need to “cut” the number twice.</p>
<p>Words don’t describe this as well as an explicit visual example. Let’s pretend we have <em>n = 5</em>, and we are working in three dimensions. We will represent the number five by circles, and the splitting will be done using vertical bars. Then, here’s a way we can “cut” n.</p>
<p><img src="https://res.cloudinary.com/dh3hm8pb7/image/upload/c_scale,q_auto:best,w_615/v1556381168/DegeneracyCut.png" alt="A way to break the number 5 into three pieces." class="centre-image" /></p>
<p>As you can see, this is just another way to word our original question. What’s neat though is that the construction doesn’t actually “know” about the way the number <em>n</em> is being split. In other words, all we’ve done is introduce a second kind of object into the mix (the vertical bars). <em>We</em> recognize those vertical bars as dividing the number into three pieces, but the mathematics doesn’t care.</p>
<p>If you’ve taken some discrete mathematics, you may know where we’re going with this. We’ve reduced the question to finding the number of ways we can arrange objects. This is a common combinatorial problem, and one that is well-studied. The answer is ripe for the taking.</p>
<p>For our scenario, how many objects do we have in all? Well, there are <em>n</em> objects, and we have to also include the number of bars. But the number of bars is just <em>N-1</em>. Therefore, the total number of objects is <em>N-1+n</em>. Then, from those objects, we want to fix the position of the bars. Therefore, we get the usual combinations formula. If we want to label the degeneracy as <em>g<sub>n</sub></em>, we get:</p>
<script type="math/tex; mode=display">g_n = {N-1+n \choose N-1}.</script>
<p>In particular, we can now solve this question for three dimensions. Substituting <em>N=3</em>, we get:</p>
<script type="math/tex; mode=display">g_n = {2+n \choose 2} = \frac{(2+n)!}{2! n!} =\frac{(n+1)(n+2)}{2} .</script>
<p>All I can say is that this is slick. I remember first trying to solve this on my own, and getting stuck. Even when I was given the answer, it didn’t feel satisfying to me. I knew that there <em>had</em> to be some better explanation for the degeneracy. I felt like there should be some combinatorial argument for the degeneracy, and it turned out that I was right! I hope that this argument helps clear things up for other students who were wondering about the formula and how to get it. In my mind, this is one of the clearest ways to get it.</p>
Fri, 17 May 2019 00:00:00 +0000
https://jeremycote.me/quantum-harmonic-oscillator-degeneracy
https://jeremycote.me/quantum-harmonic-oscillator-degeneracyBeing Happy With Being Repetitive<p>One of the most difficult things to do in life is to focus. Maybe you’re different than me, but I have a lot of trouble sitting down and focusing on one task or idea. Instead, my mind buzzes with activity while my hands do another. I’m always switching between ideas, and it takes a lot of energy to focus on just one.</p>
<p>On a more macro level, my trouble with focus manifests in the types of activities I want to do. There are so many wonderful things I <em>could</em> spend my time on. I could become a better runner, I could type words all day, I could learn to program, or I could work on teaching others what I know in mathematics or physics. This is just a small sampling of the activities which interest me, and the difficulty is being able to focus on just one (or at most, a few). After all, I don’t want to just <em>do</em> these activities. I want to to be great at them!</p>
<p>This is where reality comes knocking at the door. We simply don’t have enough time to do all the activities we wish we could do. Furthermore, it’s likely that you don’t even <em>like</em> a lot of those activities. Instead, you think you would enjoy them, but you haven’t actually tested that theory out. The result is that you have a bunch of ideas flying around in your head that are either not grounded in reality or aren’t practical to do all at once.</p>
<p>With only finite amount of time in a day, we can only make meaningful progress in a few of those areas.</p>
<p>This sounds limiting. When I encountered this obstacle, my instinct was to rebel and try to find a loophole. Surely <em>I</em> was different? Maybe most people can’t focus on a bunch of hobbies and projects, but I’m sure that I can.</p>
<p>Perhaps you are different than me, but I found that I can’t do it. If I try to take on multiple projects or hobbies, I can’t sustain it. I might be able to do them, but I won’t improve and get to a level of excellence that I want. It just doesn’t happen.</p>
<p>Instead, I’ve learned to embrace repetition.</p>
<p>Instead of looking to do a bunch of activities, choose a few of your favourites. Perhaps one or two to start. Then, focus <em>all</em> of your attention on these pursuits. If you need to, block out the sources which broadcast what you’re “missing”. They aren’t helping you if you feel the need to go do those activities every time you see them.</p>
<p>This isn’t easy. It’s <em>very</em> difficult, but it is also freeing. When you remove the other sources of activities you could be doing, you don’t have to fight a mental battle each day to choose what to do. Instead, you can focus on the few pursuits which you want to do for this season of your life.</p>
<p>If you want to really get better though, you need to <em>embrace</em> the repetition. Not only will you take small steps every day to improve in your pursuit, you will look forward to the very act of doing the thing you want. This might cause some confusion. Aren’t I doing this thing because I love it already? That may be true on a mental level, but you have to feel it.</p>
<p>For myself, this is the difference between feeling like I “have” to go on a run every day because that’s how I’ll get faster and remain a lifelong athlete versus feeling <em>lucky</em> to go run. When I go to bed at night, I look forward to the run the next morning, even if the weather is supposed to be terrible. I’m looking forward to the run of <em>today</em>. Not the results in three months, but the activity in the moment. This is what I mean by embracing repetition. It’s more than just doing the same thing over and over. It’s finding joy in the daily act, not in the long term results.</p>
<p>I must confess: this didn’t come automatically to me. For a long time, I was pursuing activities as a means to an end. I had one eye on the moment and one towards the result that I was seeking. However, I now think this is the wrong way to go about it. If you want to improve and get really good at your thing, start by loving the act of doing it every day. If you enjoy writing, learn to love the experience of facing your keyboard and putting down words on the page.</p>
<p>Why? Because that’s all you will ever get.</p>
<p>Yes, you might someday be published in your dream publication. Yes, you might bring a book into the world. However, that won’t change the fact that the act of writing <em>isn’t</em> seeing the published work, but the actual work of crafting words. There are a lot of details and specifics depending on exactly what you do, but the essence of writing is putting words on the page. That’s it. If you can really enjoy doing that each day, you will improve at your craft.</p>
<p>Furthermore, when you start enjoying the daily act of your work, it’s easier to ignore those other calls for attention. Sure, it might be fun to try out this new pursuit, but if I already love what I’m doing right here, why bother changing it up?</p>
<p>I want to be clear. There’s no problem in changing things up with your pursuits. In fact, that’s a great thing, because it lets you expand your skills and grow as a person. However, this can be taken too far, and that results in switching from pursuit to pursuit, never quite satisfied.</p>
<p>The problem is that a surface-level familiarity with a pursuit isn’t enough. To <em>really</em> enjoy it, you have to dig deeper. This takes time and focus. Only then will you start to see that this pursuit is meaningful, perhaps just as much as that “other” pursuit which society loves.</p>
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<p>I’ve experienced what happens when you switch from pursuit to pursuit. You become restless, thinking that maybe the next one will be right for you. This makes you want to keep on searching, never being quite satisfied because there is a possibility that something else is better. That might be true or false. I can’t answer that. What I <em>can</em> say is that you can waste a lot of time searching for the “right” activity without ever doing anything. Therefore, my advice would be to find something you seem to like, and then dig deeper. Give yourself a few months in which you will focus exclusively on this pursuit. It doesn’t have to be forever, but you need to give yourself enough time to find joy in the repetitiveness. At <em>that</em> point, you can decide if you want to keep your search going.</p>
<p>Finding the work we love to do seems like the difficult part, but the real challenge is being brave enough to stop searching and say, “I’m going to try this and see what happens if I give it all of my focus.”</p>
Mon, 13 May 2019 00:00:00 +0000
https://jeremycote.me/being-happy-with-being-repetitive
https://jeremycote.me/being-happy-with-being-repetitivePeeling Back the Onion<p>No matter how much advanced mathematics you study, the great thing is that you rarely have to <em>accept</em> anything as-is. If you come across a procedure, technique, or result and you wonder how in the world it works, you can always retrace your steps and get back to the foundational reasons as to why it works. If you keep on asking “why”, you will eventually get back to your starting axioms. In between that and your starting point, you should be able to understand the concept as obvious<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>.</p>
<p>It’s in this sense that any mathematical concept is like an onion. You have the initial concept, which is something that may or may not seem obvious. You can then peel away the layers, moving away from the tools you built up and going back to the axioms in which you built them from. It can be a good exercise to do this for any given concept you’re studying. How far back can you go before there no more layers to peel?</p>
<p>The reason this is a good exercise is because it makes you think about your initial assumptions. What did you absolutely <em>have</em> to start with in order to build up your mathematics? School can give us the impression that mathematics is just <em>there</em>, a bunch of rules and techniques you can follow in order to solve problems. But these rules and techniques didn’t appear out of thin air. They were constructed by mathematicians. And if they were constructed, that begs the question: constructed out of what? Answer this question enough times, and eventually you will say that the materials were just <em>there</em>. That’s when you’ve hit the bedrock, your axioms. The thing that isn’t often noted is how infrequently one is at the bedrock. In fact, most mathematical concepts aren’t the “centre” of their onion. They are part of some outer layers, with the core concept hidden from view. It takes some effort to peel back the layers and uncover it. Furthermore, one can get so used to the outer layer that it seems like that’s <em>all</em> there is to it.</p>
<p>One example that comes to mind is that of the derivative. If we consider a function of one variable, the definition of the derivative uses limits when looking at the rate of change between two neighbouring points as that distance closes to zero. But if I gave you the function <em>f(x) = x<sup>2</sup></em> and asked for its derivative, I’m guessing you wouldn’t even <em>mention</em> limits. Instead, you would tell me that it’s <em>f’(x) = 2x</em> because of the power rule, and be done with it.</p>
<p>While that’s correct, where does this power rule come from? Do we have to assume it works all the time, or can we <em>show</em> that it comes from some deeper assumptions? In essence, is the power rule an outer or inner layer of the onion?</p>
<p>It turns out that the power rule comes as a consequence of applying the definition of the derivative (using limits) to these functions with powers. From there, you can derive the power rule. Once you’ve done that, you don’t need to play around with the definition of the derivative anymore. You’ve <em>shown</em> that functions with powers always behave in this certain way. In essence, you’ve found a way to sidestep using the limit definition of the derivative, since the derivative of these functions always works in the same way.</p>
<p>This shows that the power rule has some subtleties layered inside. It’s the result of using the full definition of the derivative, and this involves limits. Therefore, more interesting mathematics is nestled inside of the power rule. Furthermore, we haven’t even discussed what it means to have a limit. We just assumed it meant something obvious. But what does a limit mean, exactly? The definition of a limit involves inequalities, deltas, and epsilons, to the chagrin of many real analysis students. It’s not <em>difficult</em>, exactly. It’s just technical in nature, which means it can seem far-removed from what we might say a limit is when speaking.</p>
<p>The point here is that we are rarely standing on the axioms. If you’ve ever taken a class in linear algebra, you often spend a small bit of time in the first class talking about the properties of vector spaces and how elements in a vector space behave under the usual operations. You then probably jumped straight into discussions about linear independence, rank, solving matrix equations, and finding the eigenvalue for matrices. You left those initial properties behind without thinking about them too much. Well, <em>those</em> properties were the point at which you could say you were standing on the bedrock. As soon as you move on, you’re building on these initial axioms (plus any extra definitions such as scalar products and whatnot). Therefore, it seems a lot more likely that the mathematics you’re studying right now lies on the outer part of an onion, not the interior.</p>
<p>Of course, I’m not trying to get you to go back to the axioms every time you solve a problem! Frankly, that would be ridiculous. Imagine if you started doing proofs with limits every time you need to take a derivative. You would tire yourself out before you even got started on the problem. As such, it’s useful to have these rules and extra mathematics. It’s what mathematicians do. They try and build more structure on top of the underlying axioms.</p>
<p>The point I want you to reflect on is how most of mathematics is just a layer of an onion. You can often go in <em>both</em> directions, outwards to mathematics that builds on the concept you’re studying, and mathematics that was used to build <em>your</em> concept. The concept you’re looking at is almost certainly not fundamental.</p>
<p>It’s worth thinking about the next time you’re studying a concept in mathematics. What was used to build it? Is there anything interesting lying underneath the concept? My experience is that peeling back the layers of a mathematical onion is both a useful and enlightening activity.</p>
<p>The next time you want to learn something new, my suggestion is to take what you know and go <em>deeper</em>. Find the underlying mathematics that builds up to what you know. It will give you a greater appreciation for the way mathematics is constructed.</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Though, retracing the steps of a proof or technique can be a lot trickier than you initially bargained for. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Fri, 10 May 2019 00:00:00 +0000
https://jeremycote.me/peeling-back-the-onion
https://jeremycote.me/peeling-back-the-onionHow Many People Need To Watch?<p>We like being recognized for the work we do. This is even more relevant now, with the idea of documenting everything you do. (If it isn’t documented, did it happen?) We don’t want to do work unless there is some reward attached.</p>
<p>This is true for both the “regular” work we do as well as our side projects. Despite claiming to enjoy what we do, I have fundamental doubts about how many people would continue doing what they do if no one cared about their work.</p>
<p>When nobody cares about your work, it’s not easy to keep going. We wonder if our work is ever going to do anything. By definition, most of what we create is to cause a change in someone. Without an audience, it can seem pointless to continue.</p>
<p>However, the trouble is that having people look at our work is <em>not</em> enough to sustain us over the long run. It might provide a boost of motivation, but it’s not sustainable. It flickers out. Even worse, consider the beginning of any pursuit. You have this work that you’re doing, and nobody cares. You’re creating in a vacuum, so how do you continue?</p>
<p>Simple. You fall in love with the work.</p>
<p>It’s the only option that will last you a lifetime. If you can find an activity that you will do <em>even if</em> no one watches, you’re set for life. Suddenly, you’re free from the clutches of an audience, and you can get on with making the work that matters to <em>you</em>.</p>
<p>Will this mean you will build an audience later on? Of course not. Growing an audience is a difficult thing, and a lot of it is out of your control. However, if you’re already content with doing the work in a consistent manner, you’re setting yourself up to grow an audience.</p>
<p>There’s nothing wrong with having an audience and being motivated by it to create more. We all feel this when we have an audience. The question is: Would you do the work anyway? The sooner you can find the pursuit in which the answer is “yes”, the easier it will be to get an audience later.</p>
<p>Don’t start with chasing what everyone else enjoys. Start with what you like. After all, you’re the one on this creative journey. You get to decide what to do. Make it count.</p>
Mon, 06 May 2019 00:00:00 +0000
https://jeremycote.me/how-many-people-need-to-watch
https://jeremycote.me/how-many-people-need-to-watchDo I Have What It Takes?<p>This is a question that students encounter over and over throughout their education. It crops up when deciding what classes to take, what projects to embark on, and what programs to study. It is a natural question, because we don’t like embarrassing ourselves. Therefore, we want to avoid pursuits that are <em>too</em> difficult if possible.</p>
<p>As a physics and mathematics student, I can only give you my perspective from this small corner of life. How can you tell if you have what it takes to study mathematics and physics?</p>
<p>I don’t want this to turn into a celebration of every person’s ability to do mathematics and physics if they give enough effort. While this may be true, it isn’t exactly helpful. That is avoiding the question, so I want to address it a bit differently.</p>
<h2 id="work-work-and-more-work">Work, work, and more work</h2>
<p>This is the first thing you should be prepared for. Studying mathematics and physics <em>requires</em> a lot of work. You might get by in the beginning by understanding the “trick” to a problem, but from what I have seen, this doesn’t last forever. At some point, you will encounter problems that confuse you and are not solved in five seconds. When this happens, are you willing to put in the work to understand?</p>
<p>On a practical level, I spend many hours each week doing homework. This is not studying for pure enjoyment. It is studying in order to finish my homework. This often means digging through textbooks and various resources to figure out the essence of a problem. It’s not always fun, and sometimes I spend hours labouring over a specific example that I just cannot seem to get. Perhaps I am slow, but I think it is safe to say that you will encounter your own version of this throughout your studies.</p>
<p>I am not saying the work is always boring. I enjoy a lot of the aspects of solving a problem, and gaining a better understanding of a concept is satisfying. I just think it is worth pointing out that this does not come for free. You <em>will</em> work a lot in a mathematics or physics program, and there is not much you can do to get around that.</p>
<h2 id="being-brilliant">Being brilliant</h2>
<p>Connected to the previous section is the idea of needing to be very smart in order to fare well in these programs. I do think some are better than others, but it’s rare to find someone with zero ability in the subjects. Why is there an initial imbalance? That is probably a question that sociologists can answer better than I can, but I would say that your upbringing can prepare you better or worse for any program. I don’t know if there are any genetic tendencies for people to be better at mathematics or physics, but I do think it’s clear that someone may seem more naturally “brilliant” than others.</p>
<p>Does that mean you need to have an easy time in order to study physics or mathematics? Not at all. In fact, I would say that most of the students I have met do not fall in this category. I would definitely include myself in this category.</p>
<p>Being smart does not automatically mean you are ready to study physics and mathematics. I would put a lot more emphasis on the willingness to do work, because <em>everyone</em> will hit a roadblock. Some people might hit a roadblock later on than others, but each person has their own difficulties. The real question is how you will respond when you encounter a roadblock. Will you quit, or will you persevere?</p>
<h2 id="memorization-and-first-principles">Memorization and first principles</h2>
<p>Throughout our primary and secondary education, the lesson that is reinforced over and over again is that memorization works. We hear teachers and others bemoaning the fact that memorization is all students do, but the simple truth is that it <em>does</em> work.</p>
<p>At least, to a certain extent. When you go further and further into mathematics and physics, you learn that memorization is only part of the picture. The emphasis now is on first principles thinking.</p>
<p>What this means is that, instead of remembering how everything fits together, you start from some bedrock foundation, and build from there. In terms of classical physics, this often begins with Newton’s second law. From that simple equation, you can derive a bunch of consequences that depend on the system. The point is that you no longer worry about the end result. Instead, the emphasis is on understanding the foundations, and building from there.</p>
<p>If you are someone who gets by through memorization (and don’t worry, we all are, to some extent), this shift in mindset will take some getting used to. However, I think it is important to be aware of this, because it is quite different from early education.</p>
<hr />
<p>My goal with this piece is to remove the fear of not being “smart” enough to study physics or mathematics. If I had to isolate one quality that needs to be built, it is the willingness to work hard. If you can do that (and even find joy in it), you will do fine. You might not be the best researcher who ever lived, but you <em>will</em> be able to understand more than you once did. Therefore, if you enjoy physics and mathematics, I would recommend giving it a shot. With the the right mindset, I think many people are capable of studying these subjects.</p>
Fri, 03 May 2019 00:00:00 +0000
https://jeremycote.me/do-i-have-what-it-takes
https://jeremycote.me/do-i-have-what-it-takesAnalogies in Mathematics<p>Learning mathematics is an additive process. What I mean by this is that new mathematics often builds on what came before. Learning mathematics isn’t exactly a linear journey, but it’s a good enough rough approximation.</p>
<p>In order to go from one concept to the next, it’s useful to offer analogies to explain what is happening. For example, when you first learn to count, you learn about the natural numbers. These are positive integers that grow in size by one each time. It might be difficult to understand at first, but with a bit of practice, most people become good at doing this. However, teachers then throw you a curve ball with <em>negative</em> numbers. What in the world does this mean? It seems ridiculous at first. For the counting numbers, you were able to <em>point</em> to these quantities. They made sense and were clear. Negative numbers, on the other hand, don’t seem to be as tangible.</p>
<p>But then your teacher explains how negative numbers can be thought of as moving across the familiar number line in the opposite direction (usually to the left). Just like the number “after” zero is one, the number “before” (to the left of) zero is -1. All you get with negative numbers is a way to go past zero to the left when you’re doing things like addition and subtraction.</p>
<p>The analogy here is very straightforward: negative numbers act just like positive numbers, but they lie on the <em>other</em> side of the familiar number line. If you have a good understanding of the number line, extending to to include negative numbers becomes easier.</p>
<p>What I want to highlight here is that I <em>didn’t</em> mention anything about additive inverses, which is the technical way to introduce negative numbers in structures known as <em>rings</em>. I didn’t mention that, not because I’m trying to deceive a student, but because I know that it’s not helpful. What <em>is</em> helpful is an analogy that takes a student from a place they are familiar with to one that they aren’t.</p>
<p>Here’s another, more advanced example. If you’ve taken a course in linear algebra before, you know that matrices are a big part of the subject. You learn that matrices can be added, multiplied and are useful for solving many simultaneous equations. However, have you ever thought of what it means to take the <em>exponential</em> of a matrix? In other words, if you have a square matrix <em>M</em>, what is <em>e<sup>M</sup></em>?</p>
<p>It turns out that the way to go about this is to use the fact that the exponential function can be written as a power series, which is just a polynomial that is infinite in length. What’s nice about this is that we know how to deal with squaring, cubing, or raising a matrix to any power. Just do a bunch of matrix multiplication! Barring some technical details, this is how you can define the exponential of a matrix. Here, the analogy is that the exponential of a matrix behaves similar to taking the exponential of a real number, but the way to <em>compute</em> it requires a power series.</p>
<p>Mathematics is littered with examples like this. They pop up all over the place, since we often look for generalizations of ideas we know. The familiar factorial operation <em>n! = n(n-1)…(2)(1)</em> can also be written in a more complicated looking form using integrals and exponential functions. This new form behaves just like the usual factorial when <em>n</em> is an integer, but it turns out that the integral isn’t limited to integer inputs. Instead, one can use <em>any</em> real number. That means you can calculate interesting expressions such as <em>π!</em>, even though we aren’t using the usual form for the factorial.</p>
<hr />
<p>There are advantages and disadvantages to using analogies while generalizing ideas in mathematics. The upside is that it gives you a foothold to getting to a new concept. Without an analogy, learning something new can be quite jarring. You end up wondering what in the world this new idea has to do with everything else you learned, and the experience tends to be one of confusion. I’ve written about this multiple times before, because I find that it’s critical to give yourself footholds while learning. When I don’t understand what’s happening at all (I have nothing to latch onto), it’s difficult for me to follow and the experience is often frustrating. Therefore, I try to always find some sort of analogy to help me through the initial stage of learning.</p>
<p>That being said, analogies don’t automatically make you a better learner. In fact, analogies can be harmful if you don’t approach them with the right mindset. I’ve found that when we are very comfortable with a concept, we end up creating our own analogies. These are simplified versions of what is going on, and they make a lot of sense to us. The crucial thing to realize though is that part of the reason they make sense to you is <em>because</em> you have the background. Sure, your analogy might be short and sweet, but if it only makes sense to those who know the subject inside and out, it won’t exactly help others learn. As such, when I hear an expert on a subject give an analogy, I don’t outright accept it. I approach it with caution, since I don’t want to be led astray.</p>
<p>This is an important point to reiterate: analogies can and will lead you astray if you follow them too literally. Analogies are there to be helpful and to give you a foothold to a subject. They don’t <em>replace</em> the subject. I think of it as similar to trying to understand the message of a book by looking at chapter summaries. Sure, you can get the essence of the book in a much shorter text, but there are bound to be nuances that you miss. That’s just the nature of compression. In the same way, analogies in mathematics often compress the technical details. This is helpful for initial understanding, but it doesn’t replace the fact that you need to do the difficult work of learning the subject.</p>
<p>If there’s one idea I want you to take away from this essay, it’s that analogies in mathematics are very helpful, but they don’t <em>replace</em> the rest of learning. Almost any concept in mathematics can be explained with an analogy, but there’s always more to that explanation. Therefore, use those analogies to your advantage as a starting point, but don’t rely on them too much. Doing so will likely lead you down the path of misconceptions, which gets worse and worse as time goes by and you hold on to these incorrect ideas.</p>
<p>At the end of the day, your goal is to learn some new mathematics. Analogies can bring you part of the way, but you have to be willing to do the rest of the work.</p>
Mon, 29 Apr 2019 00:00:00 +0000
https://jeremycote.me/analogies-in-mathematics
https://jeremycote.me/analogies-in-mathematicsRegurgitating<p>When I sit down to take a final exam, I don’t think about how to make my answer as perfect as it could be. I don’t waste time making it as clear and concise as possible. After all, that’s not the goal I have when I write my exam. Instead, I’m looking to answer all the questions as best I can in the allotted time. If I finish early, <em>then</em> I’ll go back and look to make things nicer. The first priority is always getting to the answer, though.</p>
<p>I think this mindset can be defended, since a final exam requires a lot of effort all bunched up into a short amount of time. You’re not necessarily rushed for the entire duration, but you do have to remain focused. Therefore, I try to just get to the answers as quickly as possible.</p>
<p>This implies that I’m not thinking about my answers or the content of the questions too much. That’s fair. In other words, I’ve found that final exams tend to be a process of giving the teacher what they’re looking for. By the time you sit down for the final exam, you should have a rough idea of what kind of questions will be asked. In my experience, I often go on autopilot during final exams, since I recognize the patterns needed to solve problems.</p>
<p>If you’re someone that values deep learning over grades and memorization, you’re probably cringing from the previous sentence. Honestly, I do too. I wish I didn’t have to do that, but it’s the way one gets through exams. When you have three hours to answer a bunch of questions and you don’t have a lot of time to reflect on each one, there’s not much else you can do. I think it’s unfortunate, but I also won’t punish myself for having these standards. Instead, I make sure that I only employ this kind of work in exams.</p>
<p>Of course, regurgitating information is something that can happen all the time during the process of learning. I find myself thinking about this when using certain terminology in my classes. If I don’t have a clear conception of what these terms mean, I can get stuck with saying them because I know they mean <em>something</em>, yet I can’t explain them.</p>
<p>It’s so easy to get caught up in the terminology of a concept. Furthermore, it’s easy to let yourself just regurgitate information throughout your whole education, giving the teacher what they want to hear. The result is good grades, but another byproduct is little “real” learning. When all you do is mirror what the teacher says, how much do you really understand?</p>
<p>Think about your own education. Do you just focus on parroting back the information that was given to you? If so, perhaps it’s worth reflecting on what you’re getting out of your education. I don’t deny that it can be useful to employ this strategy in certain situations (such as exams where you need to answer a bunch of questions quickly), but <em>please</em> don’t let it take over your education. Frankly, it’s the lazy way out. While you may benefit in the short term, the long term rewards won’t be as great.</p>
<p>For myself, I keep this in mind with every assignment I do. I make sure to give my best effort on each one. Not because the assignments are worth a lot (they aren’t), but because they are where I get to really understand the concepts in a class. In order to let myself go on autopilot for a final exam, I make sure to spend a lot of time thinking about the assignments. This way, I get the results I want from my classes, while avoiding the temptation of regurgitating information without thinking about it further.</p>
<hr />
<p>I know it’s difficult to apply yourself to every class as a student. Often, we have to juggle multiple classes, the associated homework, sometimes a job, and hopefully find some free time within that. As such, the temptation to regurgitate is high. For example, I know many students do their homework with a copy of the solutions beside them. While I think there’s a time and place for such resources, it isn’t during the time you do your homework. It’s just too easy to start looking at the solutions and filling in the blanks. This is another example of regurgitation, since you only care about finishing the assignment. If you get through a whole assignment without pausing to think at least once, I don’t think you’ve considered the nuances of the concepts as much as you could have.</p>
<p>The lesson here is simple: regurgitation is efficient, but it comes at a cost. There are benefits to being stuck on a question in an assignment. For one thing, you remember the experience more, which means you’re unlikely to be stuck in the same position again. Furthermore, you teach yourself that a bit of confusion isn’t detrimental to your problem-solving skills. You <em>can</em> work through them, and doing so gives you confidence for later on. If I immediately flipped to the back of the book each time I had difficulty, I might finish assignments quick, but I wouldn’t be able to do them on my own. Moreover, I wouldn’t build the habit of figuring out where I went wrong and how I can fix it. I think the latter is much more important than we give it credit for.</p>
<p>The result of regurgitation is a lot of short term gains, but few longer lasting ones.</p>
<p>And aren’t long term gains what we’re looking for in an education, in the end?</p>
Fri, 26 Apr 2019 00:00:00 +0000
https://jeremycote.me/regurgitating
https://jeremycote.me/regurgitatingPicking Yourself<p>I like school. That’s probably clear from reading my blog. I enjoy learning about science and mathematics, and the school system is one that I’ve learned to navigate with ease. Sure, I sometimes have complaints and suggestions for improvements, but on the whole, I enjoy going to school.</p>
<p>That being said, there is one part of school that I think doesn’t prepare us well for life outside of education. It’s about learning how to pick yourself.</p>
<p>What does this mean? Well, think about an activity you enjoy doing, or want to do in the future. Are you doing it right now? If not, what’s holding you back? Many of us in school are hoping that at some point, a person will look at our performance in school and choose us. In the meantime, we just need to work hard and do well at school, because everything else will be taken care of. It’s as if we believe doing our work in school will somehow make people decide to choose us, and then we can finally do the work we want.</p>
<p>Here’s a radical idea though: what if you just started now?</p>
<p>All by yourself, without the blessing of others. What if right now, today, you decided to finally start that project you wanted to do? Instead of waiting to be picked, you decide to do the work the work that’s important to you.</p>
<p>I first heard this idea from writer Seth Godin on his <a href="seths.blog">blog</a> (so don’t give me any credit for it, I didn’t come up with it). He writes a lot about doing the work that matters to you and not being afraid to start. When I read this idea, I realized that this has big implications for how we conduct ourselves in school.</p>
<p>Think about it. We go to school, study in order to get good grades, and then hope that we will get the right internship, or be accepted into a prestigious school, or get to work on a big science experiment. Throughout our education, the lesson is clear: work hard, and wait to get picked. The system of applying for grants, scholarships, and schools reinforces this message over and over. <em>Work hard, give us proof, and then wait to be picked.</em></p>
<p>This was an important realization for me, because a lot of the things I want to do with my life aren’t things that I need to wait to be picked. Instead, I can choose to take initiative and start.</p>
<p>My particular interests are in writing here on my blog about science and mathematics, drawing my webcomic <a href="jeremycote.me/handwaving">Handwaving</a>, and teaching/researching. Those are my main interests, and the great thing is that I don’t need to be picked to do these. I’ve written on my blog now for over two years, and I’ve been drawing consistently for my webcomic for nearly a year. I didn’t wait until someone told me I was good enough to write or draw ideas about science and mathematics. I <em>chose</em> to start on my own. Even for teaching, there’s nothing stopping me from teaching. Sure, I might not be a college professor at the moment, but that doesn’t stop me from taking other opportunities such as with tutoring or giving presentations. I can still do this. I can pick myself.</p>
<p>Of course, I want to be clear that this doesn’t work for <em>all</em> interests. If you want to be a miner, you’re going to need access to a mine. There’s no way around that. But more and more, the kinds of interests that people have are ones that require creativity, but whose investments in terms of cost are minimal. This is particularly true for a lot of science-related activities, such as science communication. We have better access to tools than ever before. If you want to do science communication, all it takes is for you to pick yourself.</p>
<p>It’s such a simple thing, yet I can imagine the immediate retorts.</p>
<p>“But no one is going to hire me to do this work I love!”</p>
<p>“Okay, I want to do science communication, but no one will listen to me.”</p>
<p>These are valid concerns, but they are tangential to my point. Picking yourself isn’t a guarantee that you will be hired by the organization of your dreams or that you will have an audience. Chances are you won’t have anything, at least not at first. But the key is that this doesn’t matter! What’s important is to <em>start</em>. Everything starts from there. When I started writing on my blog, I had nobody reading my work. Two years later and over a hundred thousand words later, I <em>still</em> have almost no one reading my work. Does this mean I haven’t made any progress? Not at all. First, I’ve improved my writing skills just by showing up every day to write. Plus, my work is still there, waiting to be read. It’s not going anywhere. I’m slowly building up my portfolio of work that <em>proves</em> how serious I am about writing and explaining scientific and mathematical ideas. As such, when someone shows up to my blog and sees the huge backlog, they will be more likely to stick with me. I’ve shown that I’m here for the long haul, even if no one is reading quite yet.</p>
<p>I want this to sink in. I’ve shown up week after week for about two years and I have no “results” to show for it. I don’t have a huge following that reads my words, but that’s not what picking yourself means. Picking yourself is about saying, “This work is important enough to me that I will do it <em>even if</em> no one else sees it.” My strategy for this blog is simple. I will keep writing until pure stubbornness sees me through. And then, once I’ve gotten to the point where people read my work, it won’t be a fluke. It will be <em>because</em> I chose myself many years ago, without anyone else.</p>
<p>This is my message to you. If there’s something you want to do, <em>particularly</em> if it has to do with spreading your interest in science and mathematics, please don’t wait for someone else to pick you. I know this is what you’re used to, because it was what I was used to through school. But the truth is that waiting for someone to pick you is a losing strategy. Some may get lucky and be picked, but most won’t be. Instead, my suggestion to you is to think about something you’ve wanted to do for a long time, and just start. Don’t overthink it. Start with the equipment and resources you have. More than anything, don’t expect big results. People won’t care about your work, at least not at first. That’s not part of the deal when you pick yourself. The deal is to decide that this work is important to <em>you</em>, and that you will do it no matter what.</p>
<hr />
<p>I started thinking about this in the context of where I am in my education. I’m finishing my undergraduate degree and looking to go to graduate school. I can’t help but see a lot of my education as jumping through hoops just to get the chance of doing research. There’s a lot of work involved just to get the opportunity of being picked. I’ve been wondering if there’s another way, or at least a different road I can travel for my other interests (such as writing). This is why I started this blog, and it’s why I chose myself instead of waiting for someone else to do it. I’m still travelling down the academic road (it’s what I want to do), but I’m also making choosing myself in these other areas.</p>
<p>Picking yourself sounds scary, because it means you’re committing yourself to something. You’re taking a stand and saying, “This is important to me.” However, the truth is that it’s so liberating. Instead of worrying about others choosing you, the choice is on you. Yes, this means you won’t suddenly jump to stardom, but the slow burn is likely more sustainable anyway.</p>
<p>Don’t wait for someone else to pick you. It’s not going to happen, and it will just be an exercise in frustration. Take the initiative to pick yourself. It’s worth it.</p>
Mon, 22 Apr 2019 00:00:00 +0000
https://jeremycote.me/picking-yourself
https://jeremycote.me/picking-yourself