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How the square root of 2 became a number
- bandrami 2y agoIt's funny that everybody remembers Pythagoras for the right-triangle theorem, but that wasn't what made him important at the time. The right-triangle square equality had been known empirically for centuries. What he proved that was so completely earth-shaking is that for some right triangles A, B, C there is no rational Q for which QA = C. That was what was so important about his proof.
- moioci 2y agoThat would seem to imply that for A =/= 0, C/A is irrational. This seems counterintuitive.
- asolove 2y agoWhat is counter-intuitive? In a triangle with sides A=1, B=1, then C=root(2), so C/A is irrational. That's what was so impactful about the discovery. Imagine not knowing about irrational numbers. You assume all numbers are just integers and fractional ratios between integers. It would be weird (terrifying?) that something as simple as a right triangle would require a whole category of numbers you can't express.
- Ekaros 2y agoFor some reason that feels so weird that it would be that late "discovery"... Once you define a square(sides same length) the length of diagonal is one of the first questions. And this being very weird number is something I believe someone must have thought about long before that point of time.
- asolove 2y agoThese are more or less the first people to think about geometry rigorously as an abstract system. Anyone previous would have just pointed to the hypotenuse and said “it’s that length right there” and not asked a further question.
- dontlikeyoueith 2y ago> more or less the first people to think about geometry rigorously as an abstract system. The first people whose thinking was preserved until the present. Which is still noteworthy, but a different thing.
- bandrami 2y agoOK but at the time it was literally an open research question: given two reals A, B is there always a rational Q such that QA=B? Number theory as such was still in its infancy but I think it's impressive that this was exactly the right question to ask and they understood how important it was.
- InitialLastName 2y agoA lot of early math was done using geometry tools rather than symbolic representation. If you are drawing a diagram for a building and you need a distance equal to the diagonal of a square, you set your compass to the two points and use that distance. No need to determine that it can't be represented by a comfortable multiple of the sides.
- moioci 2y agoMy bad. I was thinking A, B, and C were integers.
- hinkley 2y agoThe 3/4/5 triangle is rational. The unit right triangle is not. You’ve dropped the “some” from the parent.
- bandrami 2y agoFor "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1]) But before Pythagoras, it was still an open question if for any two reals A, B there might be a rational Q such that QA = B. Whereas we now know that for "most" reals there is no such Q, thanks to Pythagoras. 1: https://en.wikipedia.org/wiki/Pythagorean_triple https://en.wikipedia.org/wiki/Pythagorean_triple
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- seanhunter 2y agoI'm not sure you've got that right. We know almost nothing about the historical Pythagoras and none of his actual writings survived. We only know of him through the Pythagorean brotherhood and other people (eg Plato) who he influenced. While the Pythagoreans did come up with many things (eg the first rigorously documented scientific experiments) my understanding was that it is known for certain that they definitely were not responsible for Pythagoras’ theorem (or this rationality corolary you're talking about), and that the earliest formulation of it that is currently known is from Babylon where it's documented to do with sizing of farm plots[1] about a thousand years before Pythagoras. The proof of the irrationality of the square root of two was so terrifying to the Pythagoreans that the legend has it they threw the dude who produced it off a boat into the sea to drown because it was such a heresy (although I believe that is also known not to be true and the pythagoreans knew that root 2 was irrational). In that sense it's like Euler's number (first documented by Napier), Lambert's W function (Invented by Euler to solve a family of equations Lambert couldn't solve), Lagrange's notation for calculus (used by Lagrange yes but first also invented by Euler) etc etc. [1] https://www.researchgate.net/publication/222892801_Methods_and_traditions_of_Babylonian_mathematics_Plimpton_322_Pythagorean_Triples_and_the_Babylonian_Triangle_Parameter_Equations https://www.researchgate.net/publication/222892801_Methods_a...
- lordnacho 2y agoStigler's Law of Eponymy
- bandrami 2y agoThe documentatary evidence is fragmentary but there is significant evidence that a 5th-century BC Greek mathematician proved the incommensurability of the side of a square with its diagonal (it was apparently trivially known a century later since it appears in Plato's "Meno"). Whether that person was named Pythagoras or Hippasus or something else is really neither here nor there since he was pretty clearly part of the Pythagorean tradition that got associated with one name. The point in any case is that incommensurability as a concept was not widely accepted at the beginning of the 5th century BC and was widely accepted at the end, and the name "Pythagoras" gets attached to the mathematicians who discovered that. But like for that matter Plato's name wasn't "Plato"; that was a nickname his wrestling coach gave him.
- wolfi1 2y agoI thought the Pythagoreans threw one of their members over board for proving the root of 2 to be irrational
- ants_everywhere 2y agoThis is a myth, it's covered in https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus
- bandrami 2y agoThere's multiple stories about this; one of the better-attested is that for a time the brotherhood swore each other to secrecy (with threats of drowning) about it because it ran against the Parmenedian epistemology of the time.
- mehulashah 2y agoI never learned of these formal definitions in high school mathematics. Nor in the lower level college ones that I took. There’s a beauty to this perspective— irrational numbers are what rationals are not.
- bubblyworld 2y agoI think that's at the heart of mathematics - deceptively simple definitions that capture the essence of something.
- crdrost 2y agoYes, but it also messes with a lot of normal intuitions. Some examples: “Because rationals are dense —between any two rationals there are infinitely many other rationals—there are actually vastly more spaces between rational numbers, than rational numbers themselves. These spaces-between are the real numbers.” “Because every finite text document can be converted to UTF-8 and thus then an integer, it is only possible to describe 0% of the real numbers between 0 and 1 with text.” “Since most numbers are indescribable, there are (discontinuous) functions which have the value 2 for almost all numbers, but any number that you actually can describe and try to evaluate the function on, gives 1 and not 2.” You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls.
- tomrod 2y ago> You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls. Oh poppycock. We are the eldritch horror. We are the universe experiencing itself. Humans are space orcs, if Reddit is to be believed.
- markusde 2y ago> describe 0% of the rational numbers between 0 and 1 I think you mean irrational :)
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- m3kw9 2y agoHow do you prove irrational numbers doesn’t repeat down the line?
- IngoBlechschmid 2y agoThat's a great question, and the answer is by direct inspection that repeating digits cause the number to be rational. For instance, 0.123123123... is checked to be the same as 123/999, a fraction -- hence rational. Similarly, 0.abcdabcdabcd... is the same as abcd/9999. This works for repeating blocks of digits of any length.
- coldcache 2y agoTo add on to this, the question then can become why can’t the number start repeating after a certain point (e.g., 3.14133333333…). But then we can represent it as a sum of 3.141 and 0.000333333…, i.e., two rational numbers. Then we can construct a fraction that represents the number.
- empath75 2y agoand it can also be shown that all rational numbers repeat because there's only so many remainders possible for a given denominator (all the numbers from 0 to n-1), and as soon as you repeat a remainder, you necessarily have to repeat everything from the last occurrence of that remainder.
- BobaFloutist 2y agoOk, a different question: How do we know that several orders of magnitude past the digits we've calculated so far, Pi (or e, or 2^1/2, or any irrational number) doesn't start repeating (or end), and turn out to be rational. If an irrational number has to have infinite digits without repeating (or stopping), and we can't calculate infinite digits, how can we ever know that a number is actually irrational? What if it's just an absurdly specific rational number, and we just haven't gotten to the end?
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- ordu 2y agoAccording to Van der Warden "Science awakening"[1] Ancient Greeks treated numbers as some kind of "dirty" (Real) model of a platonic Ideal of a quantity. Numbers were invented by filthy traders and accountants while wise philosophers used geometry to reason about quantities. This attitude to numbers can be felt even now when we are taught to solve straightedge and compass construction problems. Greeks had no issues dealing with square root of 2 with geometry or "geometric algebra" how Van der Warden names it. [1] https://archive.org/details/scienceawakening0000waer https://archive.org/details/scienceawakening0000waer
- vinnyvichy 2y agoHipparchus must have been the greatest accountant of ancient times https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number#History https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparch... https://en.wikipedia.org/wiki/Hipparchus https://en.wikipedia.org/wiki/Hipparchus
- diffxx 2y agoI happen to be reading Poincare's Science and Hypothesis right now and he introduced a way of defining the square root of two that I found enlightening. In less articulate form: consider two sets, one of which contains all numbers whose square is less than two and one of which contains all numbers whose square is greater than two. The square root of two is then the symbolic name for the element that divides those two sets. Poincare says it better in the book though.
- __MatrixMan__ 2y agoI think you're describing a https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut (edit: oh, I see the article makes that clear. Well here's a link anyway)
- postoplust 2y agoThat's also the definition given in the article, attributed to Richard Dedekind.
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- andoando 2y agoYou can do the same with all numbers, and you get the construction of the surreal numbers.
- tombert 2y agoIrrational numbers have become a constant nuisance for me in Isabelle. I really wish that the Greeks' initial hypothesis that everything could be expressed in rationals was actually correct.
- andrewla 2y agoI'm with Dedekind on this one -- Cantor's work, by and large, was hot garbage and led and continues to power some of the most naval-gazing mathematics ever invented. Dedekind had a great idea that at its heart was a constructive notion of what a "number" was in terms of our ability to approximate it. That's the core of intuitionism, and after a long dark interval has finally come back into prominence in modern mathematics.
- sandworm101 2y ago>> constructive notion of what a "number" was in terms of our ability to approximate it. That's the key. The problem isn't whether the number exists or not. It does exist as a point on the number line. The issue is our inability to describe its position using our number system. Adopt a different numbering system, a different language for describing locations on the number line, and one can avoid the debate altogether.
- Tainnor 2y agoIt's really hard to see the history of the sciences especially in the last couple of centuries as anything but a resounding success of modern mathematics. Whatever qualms some people may have about classical mathematics, nobody has shown it to entail a contradiction, nor has any practical result obtained in physics, engineering or anywhere else been shown to be erroneous for mathematical reasons (modelling errors, of course, happen all the time, no matter what mathematics you use). All the issues such as Banach-Tarski disappear once you apply mathematics to real-world things. Meanwhile, classical mathematics remains insanely practical. People like you, who call Cantor's work "hot garbage", are giving constructive mathematics, which by itself can be a very useful additional way of doing maths, a bad name. Cantor's diagonal argument, for example, doesn't just disappear in a constructive framework.
- andrewla 2y agoI would argue that all the success of mathematics in modern times has been the result of constructive branches of mathematics. Where has anything useful been achieved from a non-constructive premise? Physicists and engineers are notoriously imprecise with their use of mathematics -- "all functions are integrable" and "all matrices are invertible", etc., and that's where all the real-world uses of mathematics have yielded results. I would love to hear examples where non-constructive techniques have yielded anything of interest. There have been places, for sure, where mathematicians working in those spaces have emerged with real and interesting work (Turing and von Neumann come to mind) but their work ultimately fits well within the bounds of intuitionism.
- abtinf 2y agoWhy is it better to invent a weird new class of numbers (irrationals) rather than just identify that there is something wrong with how we think about this issue? Put another way, why don’t we reject out-of-hand the notion that sqrt(2) cannot be calculated, given that right isosceles triangles do exist in reality and their hypotenuse has a definite length? Put yet another way, why not just say sqrt(2) equals 1.41 (or however much precision you need) + some infinitesimal amount?
- neeleshs 2y agoWe probably have not discovered the unified theory of numbers yet, and these are all patches to the current system
- function_seven 2y ago> some infinitesimal amount? What does that even mean? If we're rejecting the notion of an irrational, then the statement, "some infinitesimal amount" might as well be "some gorkly boggleboop". Sure, we can approximate to whatever precision is required for building a wall or calculating an orbit, but math itself would be hobbled by trying to make discoveries with the handicap of only allowing rationals. > given that right isosceles triangles do exist in reality and their hypotenuse has a definite length? I might be agreeing with you in a sideways manner, but right isosceles triangles don't exist in reality. Nor do any of the simple shapes like squares and circles. We have physical things that approximate those ideal shapes, but even the most precise triangle will not have a perfect right or 45 deg angle. Nor will the real-world hypotenuse be precisely sqrt(2). These physical items are made of a countable amount of molecules each of which is in some quantized state. Hell, the length of each side of the most perfect triangle we can make will be in constant flux. So for practical everyday purposes, sure. We can't work directly with irrationals, and there's no need to. But for making new discoveries in math, we must work out how to deal with "weird new" classes of numbers, like 0, or the negatives, or the complex, etc. Each one of those classes of numbers has survived because it has proven useful. If you can identify the "something wrong with how we think about this issue", you would probably win a big old prize for that :)
- xigoi 2y agoHow would that help you? Being able to reason about irrational numbers is useful.
- autoexec 2y ago> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. Let me try it! If I make a square with sides 1 inch long, then measure the diagonal with a tape measure I get... 1 and 6/16ths! Only whole numbers and the ratios between them involved there, so I guess that's all the Greeks needed after all.
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- xigoi 2y agoYou can’t do math by measuring real-world objects.
- autoexec 2y ago> You can’t do math by measuring real-world objects. Don't we all start out doing math by counting/measuring real-world objects? It was all "Sally has x apples" and rulers when I started school but some kids do math using other tools like cubes and cuisenaire rods which are also used to do math through measuring/counting real world objects.
- xigoi 2y agoMathematics is a tool that can be used to describe the real world, and mathematical discoveries are often motivated by practital uses. However, mathematical structures are not directly influenced by the real world; they can just model it.
- autoexec 2y ago> Mathematics is a tool that can be used to describe the real world, And that's what the article says the Greeks believed they could do using whole numbers and fractions. In the case of a real world square with sides of length 1 it seems that you can get away with describing the length of its diagonal in those terms which made it seem odd that it was what caused them to abandon their belief/aspiration. Maybe the author just described their dilemma poorly/strangely. I'm not at all suggesting that math has to be limited to the real world or that irrational numbers don't have their place. We're certainly better off with them.
- paulpauper 2y agolol finally a quantamag math article where I can follow the math
- Razengan 2y agoSomething I always love to ponder: Would aliens who perceive reality vastly differently than humans come up with different number systems? For example, for the longest time we thought of numbers as 1 dimensional and refused to consider 2-dimensional numbers. Even know we try to shunt them off to the side as much as possible ("complex", "imaginary") even though they model reality more closely. Might a hypothetical alien race begin with 2D numbers? or something entirely different?
- yen223 2y agoI have pondered if a hypothetical race of liquid or gaseous aliens living in a fluid world invented maths that were not rooted in counting numbers, what would that look like.
- tim333 2y agoOr less hypothetically with AIs with no knowledge of physical space, I wonder how they'd do with deducing mathematics. There are non counting approaches to mathematics such as set theory or geometry.
- Razengan 2y agoOr consider a radially symmetric organism with sensory organs all around it, so it has no concept (need) for "left" and "right", or "forward" and "back" etc. How would they define angles, coordinates etcetera?
- andoando 2y agoUltimately mathematics is describing space-time. You can definitely think to start with the intuition of having 2D numbers, but to define that you'd have to define what 1D number is anyway.
- tim333 2y agoYou can use it to describe space-time but there's much more to mathematics.
- nico 2y agoInteresting timing for this video that talks about bias in stem history (for example how naming discoveries/inventions is done in different cultures) https://www.tiktok.com/t/ZTNLLvDYm/ https://www.tiktok.com/t/ZTNLLvDYm/