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Rational or not? This basic math question took decades to answer
- throowi59594o 2y ago[flagged]
- tejohnso 2y agoWhy the interest in whether a number is irrational or not? Is it just a researcher's fun pastime or does it tell us something useful? From the article: Even though the numbers that feature in mathematics research are, by definition, not random, mathematicians believe most of them should be irrational too. So is there some kind of validation happening where we are meant to be suspicious of numbers that aren't irrational?
- treyd 2y agoThe interesting thing about irrational numbers is that they can't be constructed from a finite number of symbols from basic algebra. This is especially interesting when they have relationships with other irrational numbers, like the unexpected relationship between pi and e (and i) demonstrated in Euler's formula.
- nh23423fefe 2y agoConstructed is the wrong word. sqrt(2) is constructible and irrational
- jjtheblunt 2y agowhat's the right word?
- crabbone 2y agoOP said constructed from finite number of symbols from basic algebra. There's no finite construction of sqrt(2) using addition and multiplication.
- aidenn0 2y agoSurely exponentiation is basic algebra? And sqrt(2) is trivially constructable from exponentiation.
- crabbone 2y agoNo. exponentiation and square root are not algebraic operations. Only addition and multiplication are. That's kind of the whole point of this theory / subfield of mathematics. The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)
- LPisGood 2y agoI feel like you mean basic arithmetic, not basic algebra.
- crabbone 2y agoAlgebra is a very well defined mathematical term: > an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear" from: https://en.wikipedia.org/wiki/Algebra_over_a_field https://en.wikipedia.org/wiki/Algebra_over_a_field Historically though, the word "algebra" was used more broadly, and the further into the past you go, the more vague this term becomes. But, today, if you ask a mathematician, the definition above is how they would immediately understand algebra, and other kinds of algebras would need a qualification, eg. "linear algebra" or "abstract algebra" etc. Another way to look at this is to say that various subfields of mathematics that are called "algebra" are studies of particular kinds of algebra (from the first definition). And so they will still have all the same elements: a set (with some restrictions on it), a multiplication and addition. It could be surprising that so few basic elements give rise to such a rich field, but that's how math is... In a way, the elements you work with act more as constraints rather than extra dimensions. So, theories with very few basic elements tend to capture more stuff and be richer in terms of theorems than theories with more basic elements.
- erehweb 2y agoI think you may be thinking of transcendental numbers, which are a subset of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number https://en.wikipedia.org/wiki/Transcendental_number
- treyd 2y agoAh I'm mixing up my terms, you're right.
- CassianAI 2y agoPractically there's uses in areas like cryptography and simulation where Pseudo-random number generators (PRNGs) are used. If the numbers aren't irrational then there may be flaws in the assumptions being used. Beyond direct application, knowing a number is irrational can be a form of validation for theoretical modelling. If a number arising in a model turns out to be rational, it could mean an unexpected simplicity or symmetry, which is worth exploring further. Conversely, irrationality is often expected in complex systems and may confirm the soundness of a mathematical construct or physical model. I guess a good example of that is the relationship of light spectra and Planks constant.
- wat10000 2y agoHow could it be relevant to cryptography/simulation? Short of a symbolic algebra system, all numbers on a computer are rational. Pi is irrational but M_PI is rational. How can a PRNG be based on an assumption of irrationality when it has no access to irrational numbers?
- AlotOfReading 2y agoYou can make PRNGs based on approximations to (disjunctive) irrational numbers, and the irrationality of the number being approximated is important to its quality. I'm not aware of any widespread real-world PRNGs constructed this way because they're less efficient than traditional PRNGs. It's mostly a mathematical trick to be used in proofs and thought experiments. I suspect they're referring to the more common practice of taking the first N digits of a well known number like Pi or e that happens to be irrational as a magic constant of known provenance. 1245678 is another common one though, which obviously isn't irrational.
- CassianAI 2y agoAgreed re approximating and use of constants. Admittedly, I haven’t looked into PRNGs much since my numerical analysis college days! On simulation I did once do some work with Monte Carlo using quasi-random sequences (e.g Sobol), which can provide better coverage than pure randomness for certain problems.
- crabbone 2y agoRational numbers are a lot more useful than irrational. Eg. everything that happens in digital computers is rational. If you need a measuring tool, the scale is going to be rational. Irrational numbers, in practice, cause lack of precision. So, for example, if you draw a square 1m x 1m, its diagonal isn't sqrt(2)m. It's some rational number because that square is made of some discrete elements that you can count, and so is its diagonal. But, upfront, you won't be able to tell what exactly that number is going to be. Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. So, again, from a practical point of view, and especially in fields that like to measure things or build precise things, you want numbers to be rational, and, preferably with "small" denominators. On the other hand, irrational numbers give rise to all sorts of bizarre properties because they aren't usually considered as a point on a number line, but more of a process that describes some interesting behavior, sequences, infinite sums, recurrences etc. So, in practical terms, you aren't interested in the number itself, but rather in the process through which it is obtained. * * * Also, worth noting that there's a larger group that includes rationals, the algebraic numbers, which also includes some irrational numbers (eg. sqrt(2) is algebraic, but not rational). Algebraic numbers are numbers that can be expressed as roots of quadratic or higher (but finite) power equations. These, perhaps, capture more of the "useful" numbers that we operate on in everyday life in terms of measuring or counting things. And the practical use of these numbers is that they can be "compactly" written / stored, so it's easy to operate on them and they have all kinds of desirable mathematical properties like all kinds of closures etc. Algebraic numbers are also useful because any computable function has a polynomial that coincides with it at every point. Which means that with these numbers you can, in principle, model every algorithm imaginable. That seems pretty valuable :)
- tim-kt 2y ago> Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.
- sunshowers 2y agoThese things are useless for centuries until they suddenly underpin all of modern society.
- octachron 2y agoMathematicians are more interested in the gap in our proof framework. Like stated in the articles, many "interesting" constants appearing in mathematics feels like obviously irrational. However, proofs that they are irrational have been eluding mathematicians for centuries. This contrast is seen as a sign that we may be just missing the right mathematical insights. And if we find this insight, we might be able to adapt it to unlock other open problems in mathematics (or computer science?). This is one of these cases where the path (the new proof framework) is expected to be much more interesting than the initial destination (the fact that yes the Euler constant is irrational, of course).
- alpple 2y agoImagine you're a character in a Lord of the Rings novel. And math is the imaginary landscape that you are going on an adventure through. When a number is irrational, it is not easy to work with compared to natural numbers. So they're marking the map as a hard pass in the mountain ridge that is on your journey, assuming you want to explore that world.
- yen223 2y agoThe interesting thing to me is how poorly-understood irrationality is. Despite it being relatively common knowledge nowadays that pi is irrational, we've only proven that pi is irrational in the past 300 years or so. And the proof is not simple (at least to me) As the article states, we didn't have a general purpose "plug this number in and it'll spit out whether the number is rational or not" formula. The irrationality proofs that we do have tend to bespoke to the structure of the number itself. That's why this research is exciting.
- throw848484 2y ago[flagged]
- falcor84 2y agoWhat are you referring to? Is there anything specific that happened in Marseilles?
- throw848484 2y ago[flagged]
- tokinonagare 2y agoMaybe the insecurity of the city since he mentions the evening? Hardly somethings specific to mathematicians tho. It's seems like a weird AI-generated rant.
- throw848484 2y ago[flagged]
- falcor84 2y ago> When asked where his formulas came from, he claimed, “They grow in my garden.” This makes me think of Ramanujan's notebooks. And based on my limited interaction with professional mathematicians, I think there is something to this - some hidden brain circuitry whereby mathematicians can access mathematical truths in some way based on their "beauty", without going through anything resembling rigorous intermediate steps. The metaphor that comes to my sci-fi-fed mind is that something in their brains allows them to "travel via hyperspace". And this then makes me think of GenAI - recent progress has been quite interesting, with models like o1 and o3 at times making silly mistakes, and at other times making incredible leaps - could it be that AI's are able to access this "garden" too? Or does there remain something that we humans have access to, while AIs do not?
- ysofunny 2y agoit's like learning the letters of the alphabet permits one to see meaning behind their glyphs, namely the words; and then through reading text you perceive stories and so on when somebody learns enough letters of "the mathematical alphabet of concepts" one begins to perceive a sort of "meaning", the mathematical realm i.e. the "garden"
- PhilipRoman 2y agoTerrence Tao wrote a nice blog post which captures this idea (post-rigorous phase) https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...
- sdwr 2y agoI appreciate that he cites corroborating sources for his ideas, it makes the whole thing feel well-rounded
- heisenzombie 2y agoAlso see David Bessis: https://www.quantamagazine.org/mathematical-thinking-isnt-what-you-think-it-is-20241118/ https://www.quantamagazine.org/mathematical-thinking-isnt-wh...
- fermigier 2y agoWhoa, good to know that Henri Cohen was involved in this story. He is the co-creator of PARI/GP, the algorithmic number theoretic C library that I used for my thesis (https://pari.math.u-bordeaux.fr/ https://pari.math.u-bordeaux.fr/) as well as four books in Springer's Graduate Texts in Mathematics (GTM 138, 193, 239 and 240 - most mathematicians achieve fame with just one book in this series).
- throw848484 2y ago[flagged]
- thaumasiotes 2y ago> When mathematicians do succeed in proving a number’s irrationality, the core of their proof usually relies on one basic property of rational numbers: They don’t like to come near each other. This property of the minimum distance between two rational numbers is what the ruler function* relies on to be continuous at all irrational numbers while being discontinuous at all rationals. * When x is irrational, f(x) = 0; otherwise, when p and q are integers, f(p/q) = gcd(p,q)/q. Note that this leaves f(0) undefined, which is fine for the result of being discontinuous at rationals. You could define f(0) to be any value other than 0. The function is traditionally defined over the open interval (0, 1), which avoids the issue.
- DerekL 2y agoActually, f(0) is well-defined. If q is positive, then gcd(0,q) = q, so f(0) = 1.
- thaumasiotes 2y agoAnd when q is negative? Don't we have 1 = 3/3 = f(0/3) = f(0/-3) = 3/-3 = -1? This same problem will occur everywhere negative, though. I wasn't thinking about it; I was just being sloppy.
- deleted 2y ago[deleted]
- pro14 2y ago[flagged]
- tim-kt 2y agoIt's difficult to read this comment. Why are you writing like this?
- pro14 2y agoCan you please help me understand where you are having difficulty reading the comment? I am genuinely interested. It seems I need to improve my writing skills.
- tim-kt 2y agoTo sum it up in one sentence: You don't need to explain words that I can look up in a dictionary, unless you are commenting on that explanation. I know what the word "joke" means but you are forcing me to either read the definition or look for where you continue your comment. It's also slightly demeaning since I can read this as "you don't know what the word joke is, let me explain it to you". You do this again for the words formula, garden and absurdity. If you really need to give extra details to a word that you are using, you can do it in brackets (like this) or with a footnote [1]. The definitions of the words formula and garden also seem out of place because they don't add anything to what you were saying -- at least I don't see any analogies that would make the joke funnier. [1] Like this.
- hn_throwaway_99 2y agoPerhaps tangential, but as a non mathematician, I'm very impressed by the writing in Quanta Magazine. It's very understandable to me as a layperson without being too "dumbed down". There was an article in Quanta about the continuum hypothesis that hit the HN front page yesterday that I also thought was very well written and clear. So kudos to the authors, as explaining complicated topics in understandable language is a tough skill.
- deleted 2y ago[deleted]
- sunshowers 2y agoQuanta tends to be quite good. From the intro, I was wondering if they'd define zeta(3) or if they'd just leave it as some mysterious mathematical object. But they did define zeta(3) thankfully :)
- c0redump 2y agoIf you’re a podcast person, I highly recommend Quantas podcasts “The Joy of x” and “The Joy of y”, they are both excellent. If you like these, you may also like “Simplifying Complexity” (no relation to Quanta)
- FartyMcFarter 2y agoI find it quite interesting that pi+e and pi*e are not proven to be irrational (although it's proven that at least one of them is irrational [1]). It would be mind-blowing if either of them were rational numbers, yet it's very hard to prove either way. [1] https://math.stackexchange.com/a/159353 https://math.stackexchange.com/a/159353
- gosub100 2y agoSame for pi^pi^pi^pi https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz
- tshaddox 2y agoOut of curiosity, why would it be mind-blowing if either of them were a rational number?
- charlieyu1 2y agoIf pi+e=a/b then you can write one as a/b minus the other Which is pretty insane because these two numbers are not supposed to be related
- hn_throwaway_99 2y ago> Which is pretty insane because these two numbers are not supposed to be related Not really, there is Euler's identity: https://en.m.wikipedia.org/wiki/Euler%27s_identity https://en.m.wikipedia.org/wiki/Euler%27s_identity
- sunshowers 2y agoThat one's "just" a special case of how complex numbers happen to work. I think the really cool relationship between e and pi is the fact that the Gaussian integral acts as a fixpoint/attractor when sampling and summing data from any distribution (this is the central limit theorem): ∫(−∞ to ∞) e^(-x²) dx = √π I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.
- dr_dshiv 2y agoWhy do they always gotta throw the Pythagoreans under the bus? “ Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the ratio of two whole numbers. They were shocked when a member of their school proved that the square root of 2 is not. Legend has it that as punishment, the offender was drowned.” Not only is this story ahistorical, it is obviously wrong if you have developed the Pythagorean theorem.
- deleted 2y ago[deleted]
- DoctorOetker 2y agoA person who is born rich can proclaim it is easy to be rich. A person who has been educated with intellectual richess, for example having been shown the proof of irrationality of sqrt(2), can similarily think this observation is obvious. The Pythagoreans were a semi-secretive cult. It is not because you know a theorem that you automatically know all future proofs that apply this theorem as a step. https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus We don't know if it happened or didn't happen.
- dr_dshiv 2y agoOh stop. If you have the theorem how would you not test it with sides = 1. Of course we know it didn’t happen. The ancient stories of Hippasus don’t have anything to do with this libel. As is conveniently mentioned in the Wikipedia article you posted. The Pythagoreans were absolutely incredible — and yet this is the only story people throw around. It’s just laziness.
- AlotOfReading 2y agoThey probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. Regardless, they were a weird religious cult. They could have just regarded numbers that didn't produce rational numbers as unnatural and not something that was going to occur in the actual functioning of the world.
- jncfhnb 2y ago> If you pick a point along the number line at random, it’s almost guaranteed to be irrational. I’m having a hard time grasping this one. Feels like the coastline paradox on a straight line of a known length. Are irrational numbers even on a number line? Isn’t it definitionally impossible to pick it as a “point along the line”?
- IanKerr 2y ago>Are irrational numbers even on a number line? Yes, e is between 2 and 3 and Pi is between 3 and 4. There are geometrical lengths corresponding to each number. >Isn’t it definitionally impossible to pick it as a “point along the line”? No, it's mathematically possible to have a random process which picks a random real between 0 and n, with equal probability. Imagine it akin to throwing a dart at a line and picking the point it lands on as the number. Since there are only countably many rationals and uncountably many irrationals (i.e. not just infinitely more, but so many that you could never pair off the rationals with the irrationals, there are just too many) on any such length of the real line, chances are the number you end up with is overwhelmingly likely to be irrational.
- wat10000 2y agoAnd it’s not “overwhelmingly likely” as in there’s a 99% chance or whatever. If you choose a random point on the line, the probability of choosing a rational is zero.
- IanKerr 2y agoYep, exactly. I glossed over that detail a bit because explaining how a meagre set has a truly zero probability of being picked, while technically still being a possible result of a random process, is a bit messy to wrap your head around colloquially.
- alexashka 2y ago> If you choose a random point on the line, the probability of choosing a rational is zero. Wat? If a thing is in my pocket, there's an above zero probability of me picking it when I randomly take a thing out of my pocket. What are math people doing that's different?
- greekanalyst 2y agoI want to take a moment and appreciate how important science writing is for the lot of us who might be curious enough to love these kinds of breakthroughs but not technical enough to fully understand them. Thank you Quanta and thank you science writers all over the world for making science more accessible!
- dhosek 2y agoA chance to bring up one of my favorite quotes from grad school: “the probability of any given number being rational is 0.”¹ ⸻ 1. This result follows from the fact that |ℚ|/|ℝ| = 0.
- sorokod 2y agoIf you want a probability flavoured statement, you will need to appeal to measure theory.
- justinpombrio 2y agoStating that more precisely, if you pick a real number uniformly from the range [0, 1), the probability that it's rational is 0. One way to see this is to imagine a procedure for picking the number: - Start with "0." - Roll a D10 and append the digit. - Repeat an infinite number of times. - In the unlikely event that you wrote down a number that's not in standard format, like "0.1499999..." (which should instead be written "0.15"), toss it out and start again. The digits of every rational number eventually repeat forever. For example, 1/7 is "0. 142857 142857 ...". So what's the probability that your sequence of rolls settles on a pattern and then repeats it forever, without once deviating in an infinite number of rolls? Pretty clearly zero.
- edanm 2y agoThat's a really cool well of "visualizing" this.
- zyklu5 2y agoHere is Frank Calegari's excellent talk on this work: https://www.youtube.com/watch?v=znBdPEyDScY https://www.youtube.com/watch?v=znBdPEyDScY