Episode 274: Recreational Math, Calculators & the Quadratic Formula

0:00Ben:I write most of the AP most of the code I write is C. It's highly generic. It's mostly templates. And so it's I'm writing a lot of function templates and I'm writing a lot of sort of thinking a lot about the concepts, right? So the arguments have to fulfill. And so it does lead to surprising things sometimes.

0:21Conor:For example, honestly, probably one of the best days of my education from like kindergarten to grade 12 was the day I learned the quadratic formula. Welcome to ADSP the podcast, episode 274, recorded on February 4th, 2026. My name is Connor. Today with my co-host Ben, we chat about recreational math, calculators, the quadratic formula, and more.

1:03Ben:It is the case that oftentimes things are taught suboptimally or not to a sufficient level where kids or learners understand what, you know, it's quite common that for example, when you when you learn mathematics as a child, right? We start out, we learn arithmetic, right? We learn and then we learn geometry and trig and calculus. And it's not until you get to that lesson which finally your teacher explains how trigonometry and calculus link up, and maybe Euler's identity is on the board, and and complex numbers come into the equation.

1:42Ben:And that's the finally, maybe the first time you realize, okay, these things that I thought were all strange, distinct techniques that I had to learn by rote. No, there's something underlying here. There's a language that links these things together. And at that point, it's not a point you often get to in learning something, right? So there are ways that things are taught badly, and in particular, mathematics gets taught badly. One of my favorite kind of minor examples of that is like when you learn binomials, polynomials, and you learn how to multiply them together, right?

2:19Ben:And invariably you learn something like FOIL first outside, inside last. Did you learn that? I actually didn't, but I I think it is taught, right? So, you know, when you have like 3x plus 5 times 4x plus 7 or something, right? And you're doing just binomial multiplication. Well, that if but and for years, that kind of thing to me, I I just did it. I knew how it worked. I I wasn't actually taught FOIL, but I was taught the technique. But it was only years later, you know, when I was late teens, whenever, later on in my mathematic education, when I realized that that's the exact same algorithm as I had learned in grade school for multiplying like two-digit numbers together.

3:03Ben:It's just that in one case, x is ten, right, and it's hidden. But if you write out your polynomials with your, you know, units, an X's column, an X squared's column, that's exactly the same as writing a regular decimal number with units, tens, and hundreds. And then you can do the exact same, it's the exact same multiplication algorithm for polynomials as it is you learned at age six. But but realizing that comes way later if you're if if that's not elucidated in your education.

3:36Conor:I was gonna say, if you even get that far, I don't think I've ever had that insight. Interesting. So I couldn't be upset by that because I didn't realize I'd been I'd been not taught the generic version of it.

3:49Ben:Well, you know, like Feynman said, teaching physics is the process of telling telling people fewer and fewer lies. Is that is that just a quote of his? Uh I I bat butchered the quote, but um but yes, that that idea. It's like we start out, we learn, you know, Newton and then we learn that, oh actually, you know, Einstein modified Newton's equations and that that kind of idea. Simplif early on, massive simplification. As you progress, less and less simplification, more and more fidelity.

4:24Conor:So I yeah, I guess I'm trying to what what am I trying to do in my head? Is that that I mean, how do I feel about the fact that sometimes they're teaching you, you know, something that is omitting whatever the full truth of what you should know. I guess that's fine. I mean, I imagine that they've done some kind of study that shows that that's the best way to teach things. You can't throw everything at everyone all at once.

4:47Ben:I mean, I that is the hopeful outlook that they've done studies showing that's the best way to teach things. Perhaps the more realistic outlook is that's the way it's just been taught for a while. Yeah.

4:59Conor:Yeah, I don't I don't convention, tradition. I guess I yeah. I d I don't know. I definitely I haven't been moved. Your devil's advocateness has I don't think like I I I guess I I'm willing to not concede, but I am I'm aware that, you know. But I guess the strongest form of the argument is that, you know, Python has won for a reason. And for people like former me that says, you know, oh it's just it's not a good language for X, Y, and Z. It's like you're missing some truth there.

5:28Conor:That being said, when it comes to like I mean, PedMass is the one that is topical. Like, I don't it'd be very I'd be curious if a listener out there has a argument that is able to convince me, but it just seems to me that that is it's just awful. And maybe this is the whole Chesterson's face fence, you know. I don't actually know why did they create this hierarchy? Is there a reason?

5:51Ben:Maybe I should go learn the reason first instead of just I mean I I'm guessing, but I think I think the reason is now in history. I I think you'll find that it wasn't it wasn't purposefully created. It evolved from how mathematicians talked to each other hundreds of years ago, you know, and and how they wrote down their equations and how they made them understood. There are some cases where we can point to this was the first usage of a particular mathematical symbol, for sure, or a particular way of writing things, you know, like you know, New Newton's Newton's way of explaining calculus was uh considered inferior to Leibniz because you know we the more symbolic uh ways tend to carry the terser meanings that are more useful as you become more proficient.

6:41Ben:That's one thing we can say in general, right? You know, for most of that, uh particularly for PEMDAS, I I I I think it just comes about from from history, from from what's been useful in history, right?

6:56Conor:It's it's also it's even uh it's so beautiful. It's in in this conversation, I just realized, because in your acronym, and I thought it in my head, I just didn't say anything. You said something, my dear Aunt Sally, like you said MDAS. Yeah.

7:09Ben:PEM P Yeah, M and D get swapped around, yeah.

7:12Conor:And so I was taught PEDMAS, but you've been using an acronym that is PEMDAS.

7:17Ben:But the reality is that, you know, quite you know, addition and subtraction can be either way around. Multiplication and division can often be either way around. It's like reducing it to an acronym belies the underlying ideas. That the convention is more important than the acronym, is what that tells me, right?

7:33Conor:Right, right. I guess that's true. Yeah. My my critique was going to be that like, you know, you can just have different acronyms. It's a flaw of the hierarchy of operations that we apply. But I guess it's that's not actually the concrete algorithm, is there's a set of operations you do, then another set, then another set. The order within that set doesn't matter.

7:50Ben:And all these, you know, a lot of times you see these I hesitate to call them memes, but they go around the internet saying, you know, what what's the answer to this? And it relies on applying some some version of that. But oftentimes, whenever one of those comes across my feet, I look at it and I think, well, that, you know, it's it the question on its face is ridiculous because nobody who wanted no mathematician, nobody who wanted to be clear about conveying the intent of their equation would write that in the first place.

8:20Ben:Right? No, nobody would write that. So the fact that it's some kind of puzzle that's supposed to be you're good at mathematics if you get this, no. That's ridiculous.

8:29Conor:See, that's it's so funny that like because in my head there's a little voice screaming right now that's going, like, isn't it a clear flaw of the articulation of that math equation that the no one would write it that way? Is like, that's because ped mass is terrible. Like you would communicate it with brackets or parentheses to like make your intent clear. And it's like, you don't need any of that for APL. You know, it's like uh and this isn't an argument for APL.

8:54Conor:This is an argument against the order of evaluation that is necessitated by PedMass and Botmas or whatever.

9:01Ben:Well, there's particularly the particular sort of poor candidate here is division, right? Because what invariably, whenever you see this online, it's like the divide symbol, you know, the the hyphen with two dots on either side. And I don't know, maybe maybe it's just me, but I don't know anyone who's used that symbol since grade school. Like anyone who actually manipulates numbers like that and and equations, they just they just write a fraction.

9:31Ben:Or, you know, they'll use the other dimension. You know, computers, in some sense, for a while at least reduced us to kind of one-dimensional equations, and that was also a problem. Yeah, it's true. Right? You've just got the line. But but in reality, nobody does that when they actually write things on paper, or when they want to even, you know, typeset a paper for mathematics, of course. Yeah, yeah, yeah.

9:52Conor:I mean, that's true. I mean, on my favorite calculator of all time, the I believe. It's been years since I've had one of these calculators. A TI30XS multiview. I believe. It has the ability to Okay, you're wrong.

10:07Ben:You're wrong. This is the best calculator of all time.

10:10Conor:Ben was reaching for something. I was like, is he about to pull out the TI30XS multiview? But no, it's a Casio. This is the Casio FX 115D. It's how okay, well now we're this episode. This episode started in like Toronto, Canada and ended up in like Tanzania. Why is this the best uh calculator?

10:33Ben:Well, it's it is sort of peak non-graphical calculator, in my opinion. So it I've had this calculator for 35 years. It's still working, it's solar powered. It still works fine. What makes it very good in particular, the thing that I use it for, and sometimes still use it for, less so these days, but it has sort of one touch binary decimal hex conversion.

11:00Conor:Oh, that is really good. I don't think the 30xs multiview has that. In fact, I'm sure if your calculator is pre-graphing calculator, it's got an order of magnitude more functionality. I'm just looking at the key set. The main thing that was nice about the 30xs was for at least the actual exams, because it was an actually an SOA approved calculator, it had the, you know, it could show you square root or like a fraction. And a lot of times their multiple choice questions actually have them in the in the pure form or whatever, not the rational uh floating point form.

11:35Conor:And you could also and you could store stuff in variables as well, which I imagine yours, if yours is pre-graphing calculator, I'm sure they have that functionality.

11:43Ben:That so that's another thing of like progress in mathematical education, right? Sometime when you're around, oh, I don't know, whether it might be eleven, twelve, there's a point where you need to learn when you're solving an equation, you know, if the answer is, let's say, route three, just write route three. Do not write the decimal. Oh yeah. Right? The answer is route three, that's fine.

12:08Conor:Yeah. Yeah, that's that's very true. I don't know what point I went. And that that also just made me think that the the lies that you we were told in our education, uh you know I d I don't know why I did this, but every summer we would drive either north or south. You know, I lived in the middle of nowhere in British Columbia, a city called Prince George, and so we'd always drive to visit my grandparents. One was in Terrace and one was in uh Vancouver.

12:31Conor:And each one is eight hours and one's ten hours. And sometimes I would just take math, I'd take a sheet of like loose leaf and like write some random math equation with X's and things, and then I would just solve it. It was just something to pass the time. And every once in a while you end up with X squared. I'm like, you know, like, oh, I put I put a bad combination, like an incorrect combination at the top, because it was before like grade eight when they taught you the quadratic formula, and then I'm sitting in class one day and then teach you the quadratic formula, and I was like, You're telling me this whole time there was a formula for solving this, and I thought I was doing something wrong.

13:03Conor:What the hell? And I guess that's just uh an example of not having gotten to that point yet.

13:09Ben:Yeah. The quadratic formula to me always feels like cheating. I don't know if it feels like it, but like how because it feels like it feels impure, right? If I have to break out the quadratic formula to solve an equation, it means usually it means that I'm looking for some real answer. Like I'm looking for a decimal, you know, a real number. Uh like that that to begin with is somehow impure. Maybe. But it feels like, you know, I don't know, breaking out the quadratic equation, it's fine, but it feels like you know, it feels like taking sugar in your tea.

13:47Conor:Really? So you're not a big you're not a big fan of the quad. Honestly, probably one of the best days of my education from like kindergarten to grade 12 was the day I learned the quadratic formula.

13:59Ben:I would I would much rather I mean it's expedient and it's fine and it's good, but but aesthetically, I would much rather solve a quadratic equation by by inspection or by completing the square or some some other method rather than breaking out just having to fall feels like a fallback, right? It feels like I couldn't solve this by any other method, so I had to use the quadratic equation. I don't know, maybe it's just me.

14:24Conor:Yeah, interesting. I've never f I've never f I mean I guess it was the my path to the quadratic formula. I I had run into all these equations that I had mangled together myself, and and then this was the and I guess I guess I didn't have graph paper. I wasn't drawing these curves and then you know circling points and estimating.

14:43Ben:When I was in grade school, when uh what what age was I? Thir let's say 13, 12, 13, something like that. So second year of secondary school, I think it was. I was lucky enough to have a recreational mathematics class on the syllabus once a week, one period. Recreational mathematics. David Beckwith was the teacher, if if if you're out there, thank you very much. But uh it was a great class, and what made it great was just that it was it was a class where the students firstly we got to collaborate on problems, right?

15:19Ben:Which is which is great, which normally you don't get to do because you know the structure of education means you have to be tested and you have to make sure that you individually you know it. No, in in in the recreational mathematics class, the teacher would throw out a problem and solicit ideas from the class, right? And then and then over the course of two weeks, three weeks, the class would get towards a solution to the problem and solve sub problems along the way, maybe split into groups.

15:48Ben:It was that kind of a class. Really, really good. I remember it to this day.

15:52Conor:That sounds very I would have died for a class like that. Well, not die. I guess that's a bit I would have Well, I was about to say killed, but I I also wouldn't have killed all these figure speeches.

16:04Ben:An example problem would be he'd come into class and he'd say, you know, how many squares are there on a chessboard? And someone would put their hand up and say, of course, there are 64. And he'd say, Okay, so you're counting the one by one squares. And then that might never at that point everyone's mind expands a little bit and they're like, oh my gosh. And and so then we go away, we work on that problem, we come back with the answer. And then he says, Well, what if the chessboard had a length of side length of n?

16:30Ben:Can you can you figure that out? We go away, we do that, we and he's like, Okay, what about now if it's rectangular? How many rectangles are there, not just squares? So that was the kind of way that the class proceeded, right? It was that kind of open-ended problem where you just keep adding generalizations, or maybe you keep adding constraints. And over time, you know, everyone's mind gets broadened. What grade was this? Second year of secondary school.

16:58Ben:So I'm gonna find it hard to answer your question because I don't know how grade numbers go, and the school I went to did not use conventional grade numbers. But it would be when I was 12 turning 13.

17:09Conor:Okay, so it's for I mean, at least for Canada. I don't even know. I assume America and other parts, but we call that either junior or high school. But it's like you've got three or four years left in your Yeah, yeah.

17:21Ben:It's pre-high school, for pre-US high school for sure. Free GCSE in in the UK.

17:28Conor:Recreational math. I mean recreational we've talked about so many things now, you know. My complaints about a lot of stuff. I mean, what is the I mean the what did I write? What was my exact thing that I wrote in the email?

17:39Ben:It was algorithmic Algorithmic truth, we've mentioned, follow-up on inverse plus scan, arraybox.dev, which which you pulled up. So yeah. And then finally recreational math. Uh the other thing about recreational math is it's it, you know, I grew up geeking out on reading those kind of books, which is like, you know, amuse your friends and confound your enemies kind of things. You know, and it's when you have kids, it's great because you can, you know, blow their mind in all sorts of different ways with these with these silly little number problems, you know.

18:10Ben:It's like the thing about, you know, take any three-digit number, reverse it, take it away from itself, or take do the do the subtraction, then reverse and do the subtraction again, and you always I think that's the algorithm, and you always get 1,089. Like it's just it's a way and and and it look on the face of it, it looks amazing. It's like you always get you guessed my number? How did you do that? Well, you always get that. And the the the reason why you always get that is a property, a structural property of the base 10 and the and the place value system, right?

18:44Ben:Just the way it works out. Things like that. It's things like knowing, you know, knowing the decimal expansion of one seventh and that kind of cyclical for number, right? Yeah, yeah, yeah. So like I went out to karaoke and there were seven of us, and the bill at the end of the evening was two hundred dollars. And someone says, How much is that per person? And I say, Well, it's 2857. And the hostess is reaching for a calculus, she's like, How do you know that so fast?

19:11Ben:Like it's two sevenths. 200 divided by seven.

19:16Conor:Yeah, it's funny because I I always memorized the like to four digits, one seventh. And then for for like years didn't. I think I tweeted it out the day that I realized it. I was like, just realized today because I was messing around an APL with like formatting, I don't know, seventh fractions that they all had the same character set. And I was like, who knew? And I thought I was like discovering something cool. And then everyone on Twitter was like, Yeah, here's the Wikipedia pade on cyclic fractions.

19:41Conor:I was I was just like, oh, there's other like you know, I I obviously you know about one ninth and one third and all those ones, and and one eleven. Sure, sure. But one seventh is to what is it, six digits? One one four two nine something something. Yeah. So that one's a lot harder to just like observe, right?

19:59Ben:It's and it's cyclic. And if you multiply of course, if you multiply by seven, you get nine nine nine nine nine nine. Because the you know, because it's repeating, the extra one seventh is cut of of that part is cut off, I think. So yeah, it makes you look amazing to your friends when the when you when you just have, you know, when numbers are when you Who was it who said like maybe it was Feynman, maybe it was someone else, but it's like or it was said of Romanagend that he ever every every number was his friend.

20:27Ben:Right. So, you know, when you when you when you think of numbers not just as just the number that you've learned by road, but of uh uh with it with all of its web of of interactions with other numbers, like those kind of things. Then that that kind of thing is what recreational mathematics opens your mind to, I think. Yes.

20:48Conor:Yeah. What is it? What's the name of that movie that uh The Man Who Knew Infinity? Yeah, yeah, yeah. With Dev Patel playing Roman Ajahn. Yeah, there's a scene in that movie. I thought for sure I had included that clip in my sick P CPP con talk from like years ago, where he gets out of the taxi and says, Oh, what an unremarkable number. What are you talking about?

21:11G. H. Hardy (Clip):Should have known for his number. What was that? More than a dull one. 1729.

21:17Ramanujan (Clip):No hardy. It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.

21:25Conor:Oh, that yes, taxi cab numbers. Yeah, taxicab numbers. Or 1729, which is like a nice little Easter egg because there's tons of folks. I think you, STL, Stefan Lavay, he he sneaks those in. Yeah, stuff like that. I don't I don't remember. I I feel like maybe when I was super young I used to have the birthday tricks where it's like take your birthday and do something to it, and then I can reverse engineer your birthday with some simple trick, but I don't fully remember if uh if I actually like had any of the I don't I spend too much time on combinatory logic land.

21:53Conor:And that is that's a reason why I love uh, you know, because I mean one of the I don't know if it is a a through line. Through this conversation, because we've been full of the show. Yeah. Was it Papushkin dolls, the Russian dolls, or whatever? Matroshka dolls. M Matroshka dolls. You know, I've we're so many tangents deep on some of these that we might have lost sight of the algorithmic truth and the recreational algorithms and math. Is that that is why I love combinatory logic so much, is because, you know, in my on my my quest for algorithmic truth in, you know, criticizing adjacent difference and discovering unplus scan and all this stuff, you know, sometimes it feels like you have really found like the underlying generic.

22:38Conor:And that is, you know, the kind of the pursuit of generic programming as well, is what is the most generic form of this algorithm? You know, there's a connection between generic programming and and algorithms and whatnot. And combinatory logic, it's like the purest form of anything that I've come across because it's just like composition patterns, you know, like it's so obvious when you're applying in some pattern, you know, a unary operation and a binary operation.

23:04Conor:It's like, well, clearly this is there's no debating that this is the insert, you know, B1 combinator or psi combinator or whatever. They've got fancy names, which is a whole other thing. It's like, well, that's the convention. Well, it's a terrible convention. And I think most people would agree it's a terrible convention, because it doesn't, it doesn't mean anything. And it's like, whereas at least everybody knows PedMass. Anyways, there's something here, but it's just like it's the it's the pursuit of this stuff that's like undeniably beautiful, and there's like nothing below it.

23:31Ben:And to bring it back to APIs, you know, to bring it back to an application here rather than pure mathematics, uh it's the same the same kind of feeling, right? When you write a really nice piece of code, when you get an API correct, I write most of the API most of the code I write is C ⁇ . It's highly generic. It's mostly templates. And so it's I'm writing a lot of function templates, and I'm writing a lot of sort of thinking a lot about the concepts, right?

24:02Ben:The arguments have to fulfill. And so it does lead to surprising things sometimes. For example, I have an API where where it does a myO and there's a read function and a write function. Okay. Now, now the surprising thing is that so okay, so what does the write function take? Well, being very generic, it takes a write specification, right? It takes it takes and the specification is uh data plus where to write it. Okay? Just just staying very high level here.

24:34Ben:What does the read function take? Well, again, it takes a read specification, which is where to read from, right? Because it's going to produce the data. So what does the read function return? Well, you want to be able to write what you've just read. So the read function actually returns a write specification. It it takes in a read spec, which is locations from where to read, and it produces those locations tied together with the data that it read.

25:00Ben:Right? And that is a write spec. So what this means is that if you include the read header, you got you because it returns the write specification, it needs the definition of it. Right. So so somewhat surprisingly, the write specification definition is entailed by the reading, right? And and not actually by the writing. Because because what the right function takes is not a concrete write specification, but something that fulfills a concept. It doesn't actually need the definition of a concrete thing.

25:35Ben:Right? Because when you when you get to that level of genericity, these are the kind of surprising effects you you discover.

25:42Conor:There's an asymmetry there.

25:44Ben:Yeah. Like you would think that in order to you would think that, you know, I just write specification goes along with the write function. But that's not actually true. It goes along with the read function because that's what's returned, and therefore you need to know a concrete thing to return. Just a small observation I had lately.

26:01Conor:Yeah, yeah. Well, I mean, those are the those are the things you end up thinking about when you when you're going for a walk or well, I don't know if this is stuff that you're thinking about when you're going for a walk, but I spend an inordinate an inordinate amount of time. Sometimes I'll be listening to a podcast, and then if even if it's it is an interesting podcast, if there's something interesting that I was thinking about or designing, I have to turn the podcast off and just like clearly my brain wants to think about whatever that problem is.

26:26Conor:Is there a better way to Yeah.

26:27Ben:When your mind is quiet. I have a talk about that. I'm not sure it's very well grounded in in fact, but it is an experiential talk I gave at a local meetup a while back. About this idea of bringing both sides of your mind to work on a problem. You know, it the talk was called The Blacksmith and the Gardener.

26:47Conor:Oh, yes, yeah.

26:48Ben:These are two personas that we use in writing code. Like the blacksmith is forging code, trying to get the machine to bend to our will. The gardener is the more subconscious side of our brain, which is also working on the problem if we're working on the meaty problem over the course of some time, right? This is this is why stepping away from the machine is a good debugger. This is why going for a walk or this is why you realize the answer to the problem in the shower.

27:11Ben:Yeah. Right? It's a cliche. But this idea that how can we think about both sides of our minds working on a problem and it's the blacksmith and the gardener. Yeah.

27:23Conor:I mean, should we call it there? I mean, there's there's some other things we could chat about, but we might leave them. Yeah, yeah. Maybe next time. Oh yeah, well, there's a bunch of stuff we can talk about, but one of the things that didn't end up, because we ended up talking about stuff. Was that insights that you've come across from this kind of either recreational math or recreational algorithms or this was Stepanov's experience, right?

27:49Ben:He told an anecdote about how he was he was ill, he had flu or something, like he was ill in bed. He was sort of, you know, in that of consciousness kind of ill, kind of ill where you want to be just asleep the whole day. And he had this fever dream, I guess, where he saw, you know, basically what we now know as the fold algorithm in his head or accumulate, and he and he understood how this was not just a mechanical thing happening in the computer, but an implementation of this mathematical idea, right?

28:20Ben:And in that moment he saw how computer programs could be he saw how generic programming could work, basically.

28:29Conor:Yeah. Well, I've never had a fever dream due to an illness, but one can hope that we'll be enlightened the way Stepanov was. I'm not sure. Have you ever had any sim similar uh experiences like that?

28:40Ben:Luckily I don't get very sick very often. But most of my good ideas, you know, my output cue for for for picking up the answers to problems, if you like, is not necessarily in the shower or on a walk, but it's it's early, early in the morning when I wake up before the alarm and I'm just dozing, and my my mind has been dreaming about the problem or something, and then I realize how to solve it. That's where I find my inspirations.

29:08Conor:Be sure to check these show notes either in your podcast app or at adspthepodcast.com for links to anything we mentioned in today's episode, as well as a link to a GitHub discussion where you can leave thoughts, comments, and questions. Thanks for listening. We hope you enjoyed and have a great day. I am the anti Bryce. Um