Post B4jEXM0TC7M2Gefq1w by autolycos@beige.party
(DIR) More posts by autolycos@beige.party
(DIR) Post #B4iJ7pCuJHZZuoB4KG by futurebird@sauropods.win
2026-03-28T10:42:59Z
0 likes, 0 repeats
What is a math concept or theorem that you wish there were a better explanation of?It could be from arithmetic: Why is adding fractions so complicated?From grade-school algebra: Why does the teacher get so sad and angry if I just √(x²+y²)=x+yFrom the calculus: Why do I need to write dx with the integral?or beyond.
(DIR) Post #B4iK11xKc005NkDSmO by ligasser@social.epfl.ch
2026-03-28T10:52:50Z
0 likes, 0 repeats
@futurebird I'm working with elliptic curve cryptography for over ten years now, and I still have a major revelation at least once per year... The whole concept of fields, groups, curve arithmetic, bilinear curves - it's awesome, but I just barely understand some connections.
(DIR) Post #B4iK52pTJ6fHFTSCbA by jakobtougaard@mastodon.online
2026-03-28T10:53:36Z
0 likes, 0 repeats
@futurebird Last summer I read "Is math real?", by #eugeniacheng with great pleasure. A key takeaway for me is to ask "when", instead of "why". I
(DIR) Post #B4iKXPKRFS0Z8bSkwC by jakobtougaard@mastodon.online
2026-03-28T10:58:47Z
0 likes, 0 repeats
@futurebird Last summer I read "Is math real?", by #eugeniacheng with great pleasure. A key takeaway for me is to ask "when", instead of "why". The question becomes more open and interesting. E.g. "when is 1+1=2?" is open for discussing why one apple plus one apple is two apples, but one pile of sand plus one pile of sand is still only one pile of sand.
(DIR) Post #B4iKqYoGDq9q2pXpia by jmax@mastodon.social
2026-03-28T11:02:13Z
0 likes, 0 repeats
@futurebird Why e is special. I understand why, but I've never seen a good short explanation, nor do I have one.
(DIR) Post #B4iLVW0nWrodExwCGG by futurebird@sauropods.win
2026-03-28T11:09:40Z
0 likes, 0 repeats
@jmax Exponential grow is growth that explodes. This is because the rate that an exponential is growing is also increasing. How fast the exponential function is growing is larger when the function is larger. The bigger x the steeper it gets. This is true for all kinds of exponentials with different (positive >1) bases but if you want the function where the rate of growth is *exactly* the value of the function that's e^xThat's what I think of first but I don't think it's simple enough.
(DIR) Post #B4iLjI720IEYBbQEXg by jmax@mastodon.social
2026-03-28T11:12:09Z
0 likes, 0 repeats
@futurebird Yeah. It's not hard, but it's very resistant to extensive simplification.
(DIR) Post #B4iNdxgWj11lI9bxmC by suetanvil@freeradical.zone
2026-03-28T11:33:30Z
0 likes, 0 repeats
@futurebird I have a math degree and did not understand e until now.(tbf to me, I mostly studied computer-related stuff that doesn't use e.)
(DIR) Post #B4iNmhT8NwMTbZVv9M by suetanvil@freeradical.zone
2026-03-28T11:35:10Z
0 likes, 0 repeats
@futurebird @jmax I have a math degree and did not understand e until now.(tbf to me, I mostly studied computer-related stuff that doesn't use e.)
(DIR) Post #B4iUPjCyS3a81Bwr1k by PizzaDemon@mastodon.online
2026-03-28T12:49:26Z
0 likes, 0 repeats
@futurebird yeah, I echo a lot of above. I could answer test questions correctly about Euler's identity but I didn't *get* it in my bones. Always felt that if I could understand it that I could UNDERSTAND. https://en.wikipedia.org/wiki/Euler%27s_identity
(DIR) Post #B4iVpvXbJz4TkbeXKK by Meowthias@mastodon.world
2026-03-28T13:05:21Z
0 likes, 1 repeats
@futurebird I would like an explanation for why pi goes on forever. Is it evidence we are living in a simulation? Is it because if you trace the circumference of a circle with your finger you never reach a beginning or an end? Is it a message from the gods?
(DIR) Post #B4iXAXu2VCQGTRXmPg by cford@toot.thoughtworks.com
2026-03-28T13:20:16Z
0 likes, 0 repeats
@futurebird How a proof is both irrefutable and can have mistakes.
(DIR) Post #B4iXI52hghpvHYBnFo by futurebird@sauropods.win
2026-03-28T13:21:42Z
0 likes, 1 repeats
@Meowthias Pi goes on forever because if you take the diameter of a circle and try to wrap it around the circle there is no simple ratio between these lengths. Now why isn't there a simple ratio? With a hexagon the diameter fits three times. So, why can't exactly three diameters make up the circumference of a circle?I'm thinking about how to answer this without just going "it's not Euclidian" which isn't a real explanation.Maybe someone else can help here.
(DIR) Post #B4iXNwDDQqvvdGtoXI by futurebird@sauropods.win
2026-03-28T13:22:46Z
0 likes, 0 repeats
@cford What proof are you thinking of that's like this. I tend to think a proof with "mistakes" is simply not a proof.
(DIR) Post #B4iXTFLPj5yePyGVUW by jtnystrom@genomic.social
2026-03-28T13:23:40Z
0 likes, 0 repeats
@futurebird what precisely constitutes proof? (I know to some degree now but remember that when we first encountered the idea in school, proofs weren’t defined, just illustrated by example.)
(DIR) Post #B4iXYT0iLLw0Fb4YD2 by Meowthias@mastodon.world
2026-03-28T13:24:38Z
0 likes, 0 repeats
@futurebird I'm a little nervous that if you explain it in a way that makes sense to my English major brain the universe might get unplugged.
(DIR) Post #B4iXh7BupvNPBYR45I by leadegroot@bne.social
2026-03-28T13:26:09Z
0 likes, 0 repeats
@futurebird @Meowthias my theory for a while now, has been that the value of pi is a result of the curvature of space - somewhere else pi might be a whole number
(DIR) Post #B4iXjhlvnhFrDaW5R2 by futurebird@sauropods.win
2026-03-28T13:26:42Z
0 likes, 1 repeats
@jtnystrom People will try to blow this up into something much more complex but a proof is simply a convincing and correct *deductive* argument. It's a series of sentences (logical statements such as "If A then B") that you string together to justify a more concise and useful statement. "The sum of the interior angles of parallel lines is 180"
(DIR) Post #B4iXpKWl8XUKrb6Ojw by Gustodon@mas.to
2026-03-28T13:27:41Z
0 likes, 0 repeats
@futurebird I'm sorry if this question is boring but I'm a simpleton.Can you "fool" pi with a circle that is distinctly a shape with 360 sides? I remember making clocks with LOGO and some of the circle discussions were interesting.
(DIR) Post #B4iYNW7YXSzkfLcCmm by Cheeseness@mastodon.social
2026-03-28T13:33:51Z
0 likes, 0 repeats
@futurebird @Meowthias I don't think I have much that can help, but I feel like it's important to note that a regular hexagon doesn't have a consistent "diameter" (distance between two opposing corners is not equal to the distance between two opposing sides)
(DIR) Post #B4iYPSzb8nWdXhp3gG by khleedril@cyberplace.social
2026-03-28T13:34:13Z
0 likes, 0 repeats
@futurebird @Meowthias To answer the part about running your finger around the circle: despite the fact that pi goes forever, it is clearly bounded above by 3.2 (for example; 3.15 is another bound), so if you move your finger 3.2 diameters around the circumference, you will have gotten back (and past) where you started, no infinities involved.
(DIR) Post #B4iYXrlmu530hkMszI by Phosphenes@mastodon.social
2026-03-28T13:32:17Z
0 likes, 0 repeats
@Gustodon @futurebird Is every regular polygon perimeter-to-radius ratio rational?If so, then could you show Pi is irrational by solving a polygon, adding another side, then solving it, and adding another side, so the student understands that with infinite sides, the fine adjustments go on forever?
(DIR) Post #B4iYXshZRFGhaxP23M by futurebird@sauropods.win
2026-03-28T13:35:45Z
0 likes, 0 repeats
@Phosphenes @Gustodon "Is every regular polygon perimeter-to-radius ratio rational?"Oh no no no. A triangle and a square will produce irrational ratios.But there are two kinds of irrational numbers. Some can be represented as roots. It makes sense that the root of a square would be the ratio of the diameter of a square to the perimeter... these are numbers that go on forever like pi.But pi is even more irrational than roots... it can't even be written using roots. It's "transcendental."
(DIR) Post #B4iYdt4oxOu0Un8qwK by skotchygut@social.seattle.wa.us
2026-03-28T13:36:42Z
0 likes, 0 repeats
@futurebird @Meowthias well polygons are made of straight line segments
(DIR) Post #B4iZ0iqb9it7QjNUMS by Jestbill@mastodon.world
2026-03-28T13:40:57Z
0 likes, 0 repeats
@futurebird @Phosphenes @Gustodon So, somehow adding more sides transitions in the limit from roots to transcentants?Doesn't sound like a subject that can be "answered" simply.
(DIR) Post #B4iZCkq1a1gn4yqZ2O by SkylarkDuquesne@mas.to
2026-03-28T13:43:06Z
0 likes, 0 repeats
@futurebird I identified with Brad in "Close Encounters of the Third Kind" so much when I was 9.
(DIR) Post #B4iZE2maLeTN1Tqwme by IngaLovinde@embracing.space
2026-03-28T13:43:16Z
0 likes, 0 repeats
@futurebird idk what's so complicated about adding fractions? Or substraction them even. E.g. 49/14-25/10 = (49-25)/(14+10), easy
(DIR) Post #B4iZJgVK5kd1fIT6TA by futurebird@sauropods.win
2026-03-28T13:44:25Z
0 likes, 0 repeats
@IngaLovinde **tortured whimpering**stooooop
(DIR) Post #B4iZbitPwwCKHmL8F6 by SeanPLynch@mastodon.social
2026-03-28T13:47:38Z
0 likes, 0 repeats
@futurebird @Meowthias Think about the sponges you were posting about a few days ago ...If they were intelligent they wouldn't use base 10 because they don't have 10 digits (fingers).Sponges might develop some way of counting quantities that wasn't based on distinct numbers, but was more fluid and could handle irrational division.We are trapped in our 'digital' world by our own biology!
(DIR) Post #B4iZm8fEzGhR6sfvpg by futurebird@sauropods.win
2026-03-28T13:49:33Z
0 likes, 1 repeats
@SeanPLynch @Meowthias Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
(DIR) Post #B4ia1CsfamPMX08qR6 by fay@lingo.lol
2026-03-28T13:52:14Z
0 likes, 0 repeats
@futurebird @Meowthias a first, usually non satisfying answer: if you pick a number uniformly between 3 and 4 (which is easy to show that's where pi lives), the probability of landing on a rational number (or even an algebric irrational like sqrt(11) is 0), so for pi to be irrational was very likely. And now I'm trying to think of a more satisfying answer before looking up what others said :)
(DIR) Post #B4iaC05DVux81e9oNE by futurebird@sauropods.win
2026-03-28T13:54:13Z
0 likes, 1 repeats
@SeanPLynch @Meowthias It's like the lengths come from two incompatible lego sets. There's no ratio to make them perfectly even. But if you don't care about "perfect" 22 diameters will match up almost perfectly with 7 circumferences.
(DIR) Post #B4iaHfzN8XQhRS878K by futurebird@sauropods.win
2026-03-28T13:55:15Z
0 likes, 0 repeats
@fay @Meowthias This makes sense but we know circles are important and not just "random" so I think that's why this fails to feel like it really explains it.
(DIR) Post #B4iaYO3n34XKlAAGqu by SeanPLynch@mastodon.social
2026-03-28T13:58:14Z
0 likes, 0 repeats
@futurebird @Meowthias Yes, that's why I mentioned sponges.You'd want something that isn't going to count in distinct digits.Like 10 for us, 8 for an octopus, maybe 6 for an insect?You'd want something with no digits.
(DIR) Post #B4ianFjoGyOHIBuKiO by futurebird@sauropods.win
2026-03-28T14:00:48Z
0 likes, 0 repeats
@agturcz @IngaLovinde 'good'
(DIR) Post #B4iau62whfdBGK7bPM by independentpen@mas.to
2026-03-28T14:02:10Z
0 likes, 0 repeats
@futurebird @SeanPLynch @Meowthias how does a mathematician know such a thing?
(DIR) Post #B4ibmmnFzGxDTdrSF6 by fay@lingo.lol
2026-03-28T14:12:03Z
0 likes, 0 repeats
@futurebird best i (and apparently anyone at this point) can do is https://crypto.stanford.edu/pbc/notes/contfrac/cheat.html and continuous fraction expansions, which might be derived from pure geometry, but probably not in a way that makes it intuitive :(@Meowthias
(DIR) Post #B4ic030ozuDrXkWtVY by rallias@hax.social
2026-03-28T14:14:23Z
0 likes, 0 repeats
@futurebird @Meowthias so, the short answer is, the more sides to an even-sided regular polygon that you have, the closer and closer you reach to a limit of the ratio between the distance between two oppos and corners and sum of side lengths. A circle is functionally an infinitely sided regular polygon. And so, with an infinitely sided regular polygon, the ratio of the distance between two opposing corners and the sum of the length of the sides happens to be that limit. That limit happens to be pi.
(DIR) Post #B4id5u8kzpIPidb8SG by evan@cosocial.ca
2026-03-28T14:01:30Z
0 likes, 0 repeats
@Meowthias @futurebird this isn't easy or intuitive! The key property is that pi can't be represented as a fraction or ratio, a/b. If it could, its decimal representation would eventually stop (a = all the digits, b = 10^number of digits). But it can't, so they don't.
(DIR) Post #B4id5uqiMOSRuyAEyG by evan@cosocial.ca
2026-03-28T14:03:20Z
0 likes, 0 repeats
@Meowthias @futurebird why is pi irrational, that is, can't be represented as a fraction? That was not clear for a long time. People kept doing rational approximations, and they weren't exactly pi. So they started guessing that it might be irrational.
(DIR) Post #B4id5vbrX6AiHCDtSa by evan@cosocial.ca
2026-03-28T14:06:59Z
0 likes, 0 repeats
@Meowthias @futurebird the first proof came in 1764 from Johann Lambert. He showed that if a number were non-zero and rational, its tangent was irrational. Because we know that the tangent of pi/4 is 1, then pi/4 can't be rational, so pi can't be rational. The first part is kind of hard, though!
(DIR) Post #B4id5wIkxcU0QEI9Jo by evan@cosocial.ca
2026-03-28T14:07:40Z
0 likes, 0 repeats
@Meowthias @futurebird I've never seen an intuitive or visual proof that pi is irrational.
(DIR) Post #B4id5x3u8KCGmSLno8 by evan@cosocial.ca
2026-03-28T14:16:33Z
0 likes, 0 repeats
@Meowthias @futurebird an aside: we watched the film "Train Dreams" last night. There's one scene where the couple are discussing whether a puppy or a baby of the same age is smarter. And they come up with some pretty convincing theories about it, based on evidence they'd seen with their own eyes -- how independent a puppy can be after weaning, how dependent a baby is even when it can walk and talk.
(DIR) Post #B4id5xnzMz3n5Nubdg by evan@cosocial.ca
2026-03-28T14:20:25Z
0 likes, 0 repeats
@Meowthias @futurebird it made me think about how science has crossed from rational examination and experimentation with our normal everyday sense experiences to extremely specialized equipment and methodologies. The question of whether puppies or babies have greater intelligence would be answered very differently in 2026 than in 1920, the setting of the film.
(DIR) Post #B4id5yHlaGsaZjqO3M by evan@cosocial.ca
2026-03-28T14:23:25Z
0 likes, 0 repeats
@Meowthias @futurebird I bring it up because of this question of pi's irrationality. I did physics as an undergraduate, which requires a lot of math, and I can kind of follow along with some of the proofs in this article. But they're definitely not gut level, and I don't come away with an intuitive sense of *why*.https://en.wikipedia.org/wiki/Proof_that_pi_is_irrational
(DIR) Post #B4id5z0msstMpMuLE8 by evan@cosocial.ca
2026-03-28T14:25:16Z
0 likes, 0 repeats
@Meowthias @futurebird maybe part of the tradeoff of getting to know these facts is having specialists who dig very deeply into an area, such that they can tell us what they learned, but they can't exactly communicate why it's true. And we can't just chat about it over the campfire.
(DIR) Post #B4id5zppp5j1Ngn6nI by futurebird@sauropods.win
2026-03-28T14:26:18Z
0 likes, 0 repeats
@evan @Meowthias "And we can't just chat about it over the campfire."I always take this as a challenge. "watch me cook!"
(DIR) Post #B4idDPLxLV9IYELk5Q by evan@cosocial.ca
2026-03-28T14:27:42Z
0 likes, 0 repeats
@futurebird @Meowthias do it! I hope you can. 🙏🏼
(DIR) Post #B4idFp2eo4q4noTRpY by jenesuispasgoth@pouet.chapril.org
2026-03-28T14:28:30Z
0 likes, 0 repeats
@futurebird that leads some people to saying that logic (first order and higher order) are not part of maths, but is the language that allows maths to be done. :-) @jtnystrom
(DIR) Post #B4idSBEs6Xkl7W51DU by dvandal@infosec.exchange
2026-03-28T14:13:19Z
0 likes, 0 repeats
@SeanPLynch @futurebird @Meowthias I think there is a fundamental misunderstanding of what an irrational number is going in here. Because regardless of the base that is being used, or the counting system at play, you can’t tweak how you count to make the irrational numbers suddenly rational.The “ratio” in rational is about how the number can be described as a ratio of two other integers. To be irrational means that it “cannot be expressed as a ratio between two integers”Whatever base you use does not get around this. Using a base that is fractional doesn’t change the fundamental definition of “expressed as a ratio between two integers” either, it just means that it is incredibly difficult to do math because you have to express things in complicated addition and subtraction chains to represent a whole integer.
(DIR) Post #B4idSCjkXE2xlZXBz6 by SeanPLynch@mastodon.social
2026-03-28T14:27:36Z
0 likes, 0 repeats
@dvandal @futurebird @Meowthias Yes, that's why I first mentioned sponges.We'd need something without distinct digits to develop a 'math' not based on distinct set of counting numbers. A non-real number system. Something more fluid. It's not a matter of choosing a different base. Even choosing pi as your base won't help.I like our math, and its unreasonable effectiveness...https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf
(DIR) Post #B4idSDseHcl9JShoUi by futurebird@sauropods.win
2026-03-28T14:30:43Z
0 likes, 0 repeats
@SeanPLynch @dvandal @Meowthias Pi is defined as a ratio and the irrationality is a property of the ratio. I'm having trouble knowing how you could somehow have pi and it didn't have that property. You could have some other notion of calculating where this didn't come up... but then you'd never define pi. You see, to me, that pi is irrational is so intrinsic to what it is in a Euclidean space that I don't think it'd be "pi" anymore if it didn't have that property.
(DIR) Post #B4iddQS5slTVUMcv5s by futurebird@sauropods.win
2026-03-28T14:32:46Z
0 likes, 1 repeats
@seachaint @rallias @Meowthias That explains why it *could* go on forever. It explains why it's possible to have an irrational number that isn't a nice ratio of integers... but it doesn't show that whatever process you use to estimate pi won't at some point down the line just start repeating. You can define 1/3 as an infinite process too.
(DIR) Post #B4idfgb8p5ti8xw7dI by TobyBartels@mathstodon.xyz
2026-03-28T14:33:11Z
0 likes, 0 repeats
@futurebird @MeowthiasIt's easier to reason about the area of a flat surface than the length of a curve, so instead of the circumference of a circle of diameter 1, let's look at the area of a circle (technically a disc) of radius 1.If you truncate π at any digit, say to 3.141, then it's possible to construct a polygon (even a regular polygon inscribed in the circle) whose area is greater than 3.141 even though it fits entirely within the disc. If instead you truncate and round up, say to 3.142, then it's possible to construct a polygon (even a regular polygon circumscribed around the disc) whose area is less than 3.142 even though it entirely contains the disc. Therefore the area of the disc is between all of those rounded-down quantities and all of those rounded-up quantities, which ultimately is what it means to say that the area is given by this infinite sequence of digits.Of course, this is only more true if you take a terminating decimal that isn't even an approximation of π; you can use the same polygon as you would for the rounded-down or rounded-up approximation (as appropriate) with the same number of digits. It's because every terminating decimal gives us an area that is either too small or too large that π cannot be equal to any of them.But don't ask me to prove this; it's actually very hard prove it! Archimedes used all of his power just to find the areas of 96-sided polygons and demonstrate that π is between 223⁄71 and 22⁄7, which in decimals only gets you as far as 3.14+. Lambert proved it in 1761 using a continued-fraction expansion of the tangent function, which doesn't have much to do with the areas of anything.…
(DIR) Post #B4idqS7oyPMGRTM8fo by darkling@mstdn.social
2026-03-28T14:35:05Z
0 likes, 0 repeats
@futurebird @jtnystrom It's fairly important to note that there are always some (usually fairly simple) assumptions down at the bottom of everything. Like a+b is the same as b+a.You can't dispose of or prove those assumptions. (Well, you can, but always by making others that you derive the original ones from).
(DIR) Post #B4ieFFGTvOXnnQEG3s by SeanPLynch@mastodon.social
2026-03-28T13:54:30Z
0 likes, 0 repeats
@futurebird @Meowthias Using base 6 (ants?), or base 2 (binary), or base 16 (hexadecimal) doesn't help the pi issue because you still get an irrational ratio.The distinct digits of any rational number set will always produce an irrational pi.So maybe something that is more fluid in its own biology would develop a math where pi would not go on forever.
(DIR) Post #B4ieFGA8aT40a2GhoO by darkling@mstdn.social
2026-03-28T14:03:02Z
0 likes, 0 repeats
@SeanPLynch @futurebird @Meowthias In that case, though, the description of the base would go on for ever.
(DIR) Post #B4ieFGwLhDd0zYpCxU by SeanPLynch@mastodon.social
2026-03-28T14:09:29Z
0 likes, 0 repeats
@darkling @futurebird @Meowthias Yeah some kind of fractional base. Maybe a tree, or a fern, with its fractal body design, would develop some kind of weirdly based counting system that could work.Transforming to base 10, would still give irrational pi.Great band name, irrational pi.
(DIR) Post #B4ieFHhUrvLHLmsrRo by darkling@mstdn.social
2026-03-28T14:25:08Z
0 likes, 0 repeats
@SeanPLynch @futurebird @Meowthias Unless some component of that fractional base is itself related to pi (by a rational multipler), you're still going to end up with an infinite-length description of pi.If you go for a multi-component base with non-transcendental components (say, the first digit is base 5, the second digit is base 3/2, the third is base sqrt(13), ...), then you'd still not be able to describe pi in a finite number of digits, even if your base has an infinite description.
(DIR) Post #B4ieFIddNlqYG65I48 by SeanPLynch@mastodon.social
2026-03-28T14:37:14Z
0 likes, 0 repeats
@darkling @futurebird @Meowthias What if, when our ancestors started to count, they decided...Hey that middle finger is 'one' thing. The ones to the right and left of it are each almost 'one' thing.That little finger is 2/3 of a thing.The thumb is the same length as the little one, but it's fatter, so we'll call that 3/4 of a thing...What would math be like?
(DIR) Post #B4ieFJ9tRpePs9B3Lc by futurebird@sauropods.win
2026-03-28T14:39:32Z
0 likes, 0 repeats
@SeanPLynch @darkling @Meowthias You could have a math where this thing that vexes us "why is pi irrational" isn't an important question I think. But if you dug deep it'd still be there.
(DIR) Post #B4ieYtoHP3IH2xPRqK by cford@toot.thoughtworks.com
2026-03-28T14:43:04Z
0 likes, 0 repeats
@futurebird I've scraped together enough maths education to realise that, but it still feels intuitive to me that we might say something like "X thought they had proved the theorum, but it turns out they made a mistake so that the thing they published that everyone thought was a proof actually wasn't", especially when the common way of explaining what a proof is is "a watertight argument".
(DIR) Post #B4iei7MkN78thCvXLE by futurebird@sauropods.win
2026-03-28T14:44:51Z
0 likes, 0 repeats
@cford When that has happened in math we call it a massive error in peer review. And it's generally been VERY rare compared to other areas of study.
(DIR) Post #B4ieozs0dhGsTWpWDI by cford@toot.thoughtworks.com
2026-03-28T14:44:34Z
0 likes, 0 repeats
@futurebird For example, Wiles' first proof of Fermat's last theorem. You could argue that it's not actually a proof, but I don't think you had trouble understanding what I meant when using the word.
(DIR) Post #B4iep12KIp7O5ofGvw by futurebird@sauropods.win
2026-03-28T14:46:01Z
0 likes, 0 repeats
@cford I would say it was not a proof since it didn't prove it? IDK maybe I've bought the math orthodoxy too much.
(DIR) Post #B4if4KWc7wDysZ8MOe by cford@toot.thoughtworks.com
2026-03-28T14:48:50Z
0 likes, 0 repeats
@futurebird I agree with your definition, but it's my impression that "How can a proof have mistakes?" is a source of confusion to non-mathematicians. And telling them they've made a category error doesn't seem to help. 🙂
(DIR) Post #B4ifOUjFtn44Jkf6e0 by fay@lingo.lol
2026-03-28T14:52:26Z
0 likes, 0 repeats
@futurebird it took me many years of teaching to get an answer I like to the first question: adding fractions is very simple, what's complicated is finding a given number as a fraction
(DIR) Post #B4ifOcdAWIwupBsMGe by fay@lingo.lol
2026-03-28T14:52:27Z
0 likes, 0 repeats
@futurebird like if i give my student a quarter of an apple and a third of an apple, it's very intuitive that they now have 1/3+1/4 of an apple and that this is a completely fine way of writing that number. And once that's intuitive we can move on to the hard task of finding how to cut an apple in pieces of equal sized so that a whole number of such pieces is the same amount of apple as that first 1/3+1/4
(DIR) Post #B4igYJXsQwrETyaOdk by evan@cosocial.ca
2026-03-28T14:38:25Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias this is extremely important and it's how Archimedes did the first approximations of pi! If you take a hexagon whose corner-to-corner length is the same as the diameter of a circle, it will fit inside the circle. It's "inscribed". Its perimeter is smaller than the circumference of the circle.If you take another hexagon whose side-to-side length is the same as the diameter of the circle, it's entirely outside the circle. Exscribed! Its perimeter is bigger.
(DIR) Post #B4igYKvJJJCEkwYcnw by evan@cosocial.ca
2026-03-28T14:40:29Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias so the circumference is between the perimeter of the inside and outside hexagons.
(DIR) Post #B4igYLtvfvgZmwv2I4 by evan@cosocial.ca
2026-03-28T14:42:13Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias the neat thing about hexagons is that they can be made up of six equilateral triangles. So, the inside hexagon (six sides) has a perimeter that's six times the radius, or 3 times the diameter. So, the circumference is more than three times the diameter.
(DIR) Post #B4igYN1lUHY1HXao8u by evan@cosocial.ca
2026-03-28T14:52:58Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias for the outside hexagon, figuring out the perimeter is a little harder. But you can use the Pythagorean Theorem on this equilateral triangle. The distance from the centre to the edge is 1/2 the diameter of the circle (by definition). That makes a triangle that splits the equilateral triangle in two. If we say the edge length is x, we have a right triangle with one side x, one side d/2, and one side x/2. We know (x/2)^2 + (d/2)^2 = x^2 so x^2 = d^2/3
(DIR) Post #B4igYNjMsAQTSlzd6e by evan@cosocial.ca
2026-03-28T14:56:21Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias that means x = d/sqrt 3. Since the perimeter is 6x, it's 6d/sqrt 3, or 2 times the sqrt of 3. That's around 3.4.So, we know pi is between 3 and 3.4.
(DIR) Post #B4igYOVE0EhtrCNqhU by evan@cosocial.ca
2026-03-28T14:56:58Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias Archimedes did this with polygons up to 96 sides (!), and came up with a pretty good approximation of pi.
(DIR) Post #B4igYPLKsUOISolSvQ by futurebird@sauropods.win
2026-03-28T15:05:10Z
0 likes, 0 repeats
@evan @Cheeseness @Meowthias Was he doing his calculations in Roman numerals? I feel like I used to know this...
(DIR) Post #B4igYegRqHBw2dl2Po by evan@cosocial.ca
2026-03-28T15:03:39Z
0 likes, 0 repeats
@Cheeseness @futurebird @Meowthias I think the math is a lot easier if I you use an inscribed hexagon and an exscribed square, but I wanted to use your insight first.
(DIR) Post #B4ihbfMzqZmWzcevEu by futurebird@sauropods.win
2026-03-28T15:17:17Z
0 likes, 0 repeats
@xarvos @evan @Cheeseness @Meowthias Did the Greek numerals have place value?
(DIR) Post #B4iiPAYSgtUztucuh6 by rallias@hax.social
2026-03-28T15:26:03Z
0 likes, 0 repeats
@futurebird @seachaint @Meowthias because in each intermediate step that you take, say from six sides to eight sides, you change that ratio ever so slightly closer to the limit. That pattern doesn't just exist from six size to eight sides to 10 sides, that pattern continues on as you go from a billion to a billion and two sides, from a trillion to a trillion and two sides, from a quintillion to a quintillion and two sides, to whatever arbitrarily large even number you think of to whatever arbitrarily large even number plus two sides. And thus, any intermediate repeating digit set is going to be wiped out by adding two more sides in perpetuity.
(DIR) Post #B4ijLb4GH544vPMS7U by evan@cosocial.ca
2026-03-28T15:36:46Z
0 likes, 0 repeats
@futurebird @Cheeseness @Meowthias I don't know! But Greek mathematicians often didn't write things out. They did some of their proofs geometrically. They had these giant compasses, about a meter long, and they would draw out their proofs in the dirt on the ground. Supposedly Archimedes died with his compass in his hand. When they did write out proofs, they were verbal and logical -- not equations. Euclid's proofs are a good example.
(DIR) Post #B4ijYibLZlbI3S5r3w by MichaelPorter@ottawa.place
2026-03-28T15:39:08Z
0 likes, 0 repeats
@futurebird @SeanPLynch @Meowthias π = 1 in base pi…I'll see myself out 😄
(DIR) Post #B4il6AdQT7dSVea4vY by javierg@mstdn.social
2026-03-28T15:56:23Z
0 likes, 0 repeats
@futurebird @SeanPLynch @Meowthias I think it's not about the circle being "unsquarable". A square has a ratio of sqrt(2), which is as irrational as pi. (Although less trascendental, I guess)
(DIR) Post #B4ilqLSmXdGXPicVJw by quizzicus@mastodon.online
2026-03-28T16:04:36Z
0 likes, 0 repeats
@futurebird @seachaint @rallias @Meowthias I feel like there's a sub-proof here that any repeating decimal is non-repeating in some base.
(DIR) Post #B4inDdJ2T5VdMojOIC by burnitdown@beige.party
2026-03-28T16:20:08Z
0 likes, 0 repeats
@futurebird pretty much everything. i gave up even trying in 10th grade math class cause it was just more of the same "if you don't understand, it's all your fault and you will get big angry red zeros on all of the tests you fail because you don't understand and have given up bothering with it".
(DIR) Post #B4iqZ1F2yGJZXtklCy by clf@mastodon.bsd.cafe
2026-03-28T16:57:36Z
0 likes, 0 repeats
@futurebird I've had very good teachers, so I don't think that I needed more explanation than what I received for most conceptsI do wish they had spent more time about the multidimensionality of complex numbers
(DIR) Post #B4iqogXNiUvpxwnc5w by michael_w_busch@mastodon.online
2026-03-28T17:00:21Z
0 likes, 0 repeats
@futurebird @seachaint @rallias @Meowthias To elaborate:That π is transcendental (and thus both non-repeating and not expressible in closed form algebraically) follows from Euler's identity: e^iπ = -1 and that e is transcendental.Proving e ins transcendental takes a few pages of playing around with series and integrals: https://en.wikipedia.org/wiki/Transcendental_number#A_proof_that_e_is_transcendental .
(DIR) Post #B4jDjmvapMWxHdl3I0 by autolycos@beige.party
2026-03-28T21:17:17Z
0 likes, 0 repeats
@futurebird hamiltonian math and matrix mathGood Lord those things make my brain hurt
(DIR) Post #B4jDrmcFs3mCdIWc0e by futurebird@sauropods.win
2026-03-28T21:18:47Z
0 likes, 0 repeats
@autolycos IDK about that hamiltonian, but matrices aren't so bad. Once you just accept them as linear multivariable functions... though I assume you are talking about something deeper than that.
(DIR) Post #B4jEXM0TC7M2Gefq1w by autolycos@beige.party
2026-03-28T21:26:16Z
0 likes, 0 repeats
@futurebirdI was introduced to it doing NMR stuff and I had just done intro calculus and being Greek would be an improvement. So obtuse to me https://en.wikipedia.org/wiki/Hamiltonian_system
(DIR) Post #B4jFV0sur0C28OaXJ2 by futurebird@sauropods.win
2026-03-28T21:37:04Z
0 likes, 0 repeats
@autolycos You need calc 1, calc 2 and probably multivariable (sometimes called calc 3) before you mess with this. This is differential equations. This is when you write equations about rates of change and then solve them for functions or rates. eg. "There is a function whose rate of change is equal to ten times it's value, who am I?"It's not awful, but into to calc isn't enough at all.
(DIR) Post #B4jFg87GJA3nkPbDpA by autolycos@beige.party
2026-03-28T21:39:03Z
0 likes, 0 repeats
@futurebird I was definitely too clever by half in a unique education environment
(DIR) Post #B4jNDrxNatRp9g0od6 by futurebird@sauropods.win
2026-03-28T23:03:36Z
0 likes, 0 repeats
@Space_Burger_Steve Do you program with "for" loops ever? If not no biggie, but if you do? That's what those are. They are "for loops" in math notation.
(DIR) Post #B4jNK9YN5DAtT3JLnM by futurebird@sauropods.win
2026-03-28T23:04:44Z
0 likes, 0 repeats
@Space_Burger_Steve It occurs to me you may have been vexed by the analysis of infinite sums, which is another matter.
(DIR) Post #B4jTMfXrhzggzbeoka by ghosttie@mastodon.gamedev.place
2026-03-29T00:12:23Z
0 likes, 0 repeats
@futurebird quaternions, matrices