[HN Gopher] Rational or Not? This Basic Math Question Took Decad...
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Rational or Not? This Basic Math Question Took Decades to Answer
Author : nsoonhui
Score : 92 points
Date : 2025-01-09 13:01 UTC (9 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| tejohnso wrote:
| Why 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 wrote:
| The 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 wrote:
| Constructed is the wrong word. sqrt(2) is constructible and
| irrational
| jjtheblunt wrote:
| what's the right word?
| crabbone wrote:
| OP said constructed from _finite_ number of symbols from
| basic algebra. There 's no finite construction of sqrt(2)
| using addition and multiplication.
| aidenn0 wrote:
| Surely exponentiation is basic algebra? And sqrt(2) is
| trivially constructable from exponentiation.
| LPisGood wrote:
| I feel like you mean basic arithmetic, not basic algebra.
| erehweb wrote:
| I think you may be thinking of transcendental numbers, which
| are a subset of irrational numbers.
| https://en.wikipedia.org/wiki/Transcendental_number
| treyd wrote:
| Ah I'm mixing up my terms, you're right.
| CassianAI wrote:
| Practically 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 wrote:
| How 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 wrote:
| You 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 wrote:
| Agreed 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 wrote:
| Rational 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 wrote:
| > 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.
| crabbone wrote:
| I think it's kind of obvious what my definition of
| existence could be from the answer above: if it's possible
| to count up to that number in finite time, that number
| exists. By counting I mean a physical process that requires
| discrete non-zero intervals between counts. And you don't
| have to count in integers, you can count in fractions, not
| necessarily equal at each step: the only requirement is
| that the element used for counting exists (in terms of this
| definition) and that you are able to accomplish counting in
| finite time.
|
| To me, this pretty much captures what people understand the
| numbers to be used for outside of college math (so no
| transfinite, cardinals etc.)
| tim-kt wrote:
| That sounds plausible. I think that definition is
| equivalent to a number being rational.
|
| > To me, this pretty much captures what people understand
| the numbers to be used for outside of college math (so no
| transfinite, cardinals etc.)
|
| I'm in my fourth year of mathematics right now. I guess
| I'm not in the target group of articles such as these :P
| sunshowers wrote:
| Do you mean the computable numbers? (there's an algorithm
| to compute them to arbitrary precision)
|
| The irrational numbers used outside of college math, like
| pi or e or sqrt(2), are computable, though almost all are
| not.
|
| You can do a lot of productive math using just computable
| numbers since they form a real closed field [1]. I
| believe they're a little harder to work with though.
|
| [1] https://en.wikipedia.org/wiki/Real_closed_field
| LPisGood wrote:
| "God created the natural numbers, all else is the work of
| man."
| sunshowers wrote:
| These things are useless for centuries until they suddenly
| underpin all of modern society.
| octachron wrote:
| Mathematicians 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 wrote:
| Imagine 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.
| falcor84 wrote:
| > 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 wrote:
| it'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"
| pro14 wrote:
| > When asked where his formulas came from, he claimed, "They
| grow in my garden."
|
| It is a "joke:" "a thing that someone says to cause amusement
| or laughter."
|
| "Formula:" "a concise way of expressing information
| symbolically."
|
| "Garden:" "a small piece of ground used to grow vegetables,
| fruit, herbs, or flowers."
|
| Similar to:
|
| > When asked where his HTML/CSS/JavaScript came from, he
| claimed, "It grows on trees."
|
| This type of joke is called "Absurdity:" The humor is amplified
| by the absurdity of imagining abstract things like formulas or
| programming code as physical objects that could grow in nature.
|
| > my sci-fi-fed mind is that something in their brains allows
| them to "travel via hyperspace".
|
| What is a "hyperspace?" Are you talking about hyperbolic
| geometry?
|
| > "and threw paper airplanes"
|
| Were the paper airplanes also traveling via hyperspace?
| tim-kt wrote:
| It's difficult to read this comment. Why are you writing like
| this?
| pro14 wrote:
| Can 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 wrote:
| To 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.
| PhilipRoman wrote:
| Terrence Tao wrote a nice blog post which captures this idea
| (post-rigorous phase)
|
| https://terrytao.wordpress.com/career-advice/theres-more-to-...
| sdwr wrote:
| I appreciate that he cites corroborating sources for his
| ideas, it makes the whole thing feel well-rounded
| fermigier wrote:
| Whoa, 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/) 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).
| thaumasiotes wrote:
| > 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 wrote:
| Actually, f(0) is well-defined. If q is positive, then gcd(0,q)
| = q, so f(0) = 1.
| thaumasiotes wrote:
| And 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.
| hn_throwaway_99 wrote:
| Perhaps 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.
| sunshowers wrote:
| Quanta 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 :)
| FartyMcFarter wrote:
| I 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
| gosub100 wrote:
| Same for pi^pi^pi^pi
|
| https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz
| tshaddox wrote:
| Out of curiosity, why would it be mind-blowing if either of
| them were a rational number?
| charlieyu1 wrote:
| If 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 wrote:
| > 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
| sunshowers wrote:
| That 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 = [?]p
|
| 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.
| octachron wrote:
| This is only true for distributions with finite variance
| (and the edge case of distribution with slowly growing
| infinite variance).
|
| And for a given variance, gaussian distributions are
| exactly the maximal entropy distribution.
| pishpash wrote:
| e and pi are highly related, both pop out of periodic
| phenomenon.
| cbm-vic-20 wrote:
| Is the result of the addition or multiplication of an
| irrational number with any other real number not equal to it
| (and non-zero in the case of multiplication) always irrational?
| ex: pi + e, pi * e, but also sqrt(2) - 1 or sqrt(3) * 2.54 ?
| aidenn0 wrote:
| Definitely not; consider the formula for calculating the log
| of any base given only the natural logarithm. That can result
| e.g. in two irrational numbers, the ratio of which are
| integers.
| AlotOfReading wrote:
| No, sqrt(5)*sqrt(16*5)=20. More trivially, there's always a
| number y such that z = x*y for a given irrational x. You can
| give similar examples for all the other basic operations.
| LPisGood wrote:
| Take any irrational a where 1/a is also irrational. Then a *
| 1/a = 1.
|
| Even moving from addition and multiplication to exponentials
| won't save you: there are irrational numbers to irrational
| powers that are raational.
| pfdietz wrote:
| The nonconstructive proof of that is simple and fun: either
| sqrt(2)^sqrt(2) or (sqrt(2)^sqrt(2))^sqrt(2) is just such
| an example.
| umanwizard wrote:
| > Take any irrational a where 1/a is also irrational.
|
| In other words: any irrational at all
| umanwizard wrote:
| pi and -pi are both irrational and their sum is zero.
| ryandv wrote:
| There is an important distinction to be made here. Examples
| in this thread show cases of irrational numbers multiplied by
| or added to other irrational numbers producing real numbers,
| but in the special case of a _rational_ number added to or
| multiplied by an irrational number, the result is always
| irrational.
|
| Otherwise, supposing for instance that (n/m)x is rational for
| integers n, m, both non-zero, and irrational x, we can
| express (n/m)x as a ratio of two integers p, q, q non-zero:
| (n/m)x = p/q if and only if x = (mp)/(qn). Since integers are
| closed under multiplication, x is rational, against
| supposition; thus by contradiction (n/m)x is irrational for
| any rational r = (n/m), with integers n, m both non-zero.
| Similarly for the case of addition.
| yen223 wrote:
| irrational number + rational number = irrational number [1]
|
| irrational number + irrational number could be rational or
| irrational.
|
| 5 - sqrt(2) is irrational
|
| sqrt(2) is irrational
|
| Add them up you get 5, which is rational
|
| [1] If it were rational, you will be able to construct a
| rational representation of the irrational number using this
| equation.
| rokob wrote:
| I could believe pi*e rational but pi+e being rational would
| blow my mind.
| sunshowers wrote:
| I would be shocked if either of them were proven to be
| rational.
| chongli wrote:
| I mean you just have to get to the point where all of the
| trailing decimal places (bits) form a repeating pattern with
| finite period. But since there are infinitely many such
| patterns it becomes extremely hard to rule out without some
| mechanism of proof.
| programjames wrote:
| Well, e^pi - pi = 20, is rational.
| hollerith wrote:
| It is not exactly 20.
| toth wrote:
| Very nice, didn't know about that one!
|
| In a similar vein, Ramanujan famously proved that
| e^(sqrt(67) pi) is an integer.
|
| And obviously exp(i pi) is an integer as well, but that's
| less fun.
|
| (Note: only one of the above claims is correct)
| nimih wrote:
| Do you have a citation for the rationality of e^pi - pi? I
| couldn't find anything alluding to anything close to that
| after some cursory googling, and, indeed, the OEIS sequence
| of the value's decimal expansion[1] doesn't have notes or
| references to such a fact (which you'd perhaps expect for a
| rational number, as it would eventually be repeating).
|
| [1] https://oeis.org/A018938
| chowells wrote:
| https://xkcd.com/217/
| nimih wrote:
| Is the joke here that if you lie to people (on the
| Internet or otherwise), they'll take it at face value for
| a little bit and then decide you're either a moron or an
| asshole once they realize their mistake?
| chowells wrote:
| Nah. I'd read it as there being an expectation that the
| audience already knew the joke and they were playing the
| favorites.
| dr_dshiv wrote:
| Why 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.
| DoctorOetker wrote:
| A 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
|
| We don't know if it happened or didn't happen.
| dr_dshiv wrote:
| Oh 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 wrote:
| They 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.
| DoctorOetker wrote:
| I am certainly open to the idea that
|
| > 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.
|
| I reread it BEFORE posting my initial comment.
|
| Can you point me to where ON THE WIKIPEDIA PAGE this story
| was conclusively debunked?
| wat10000 wrote:
| So you test it with sides = 1. Result: hypotenuse is some
| number n where n*n = 2.
|
| So far so good. How does this lead you to the obvious
| conclusion that n is irrational?
|
| (I'm familiar with the standard proof that it is, but
| that's not something that just naturally falls out of
| this.)
| Sniffnoy wrote:
| How is this story obviously wrong if you've developed the
| Pythagorean theorem? The Pythagorean theorem has nothing to do
| with rationality. If you think the irrationality of sqrt(2)
| follows easily from the Pythagorean theorem, then by all means,
| please demonstrate!
|
| (These days the irrationality of sqrt(2) is obvious due to
| unique prime factorization, but the ancient Greeks didn't have
| that concept!)
| jncfhnb wrote:
| > 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 wrote:
| >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 wrote:
| And 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 wrote:
| Yep, 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.
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