[HN Gopher] Rational or not? This basic math question took decad...
___________________________________________________________________
Rational or not? This basic math question took decades to answer
Author : nsoonhui
Score : 160 points
Date : 2025-01-09 13:01 UTC (1 days 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.
| crabbone wrote:
| No. 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 wrote:
| I feel like you mean basic arithmetic, not basic algebra.
| crabbone wrote:
| Algebra 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
|
| 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 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
| crabbone wrote:
| Yeah... the article has a lot of "simplifications" that
| the reader has to just kind of trust the author on... and
| those are meant for people with not a lot of mathematical
| sophistication. All this talk about "shrink fast enough"
| and why it's important is just some intense handwaving
| w/o any actual explanation.
|
| To be fair, it's kind of upsetting, but maybe there's no
| way to help it... some mathematical proofs can be "dumbed
| down" to the point that people with very little
| background can understand them. The proof of sqrt(2)
| being irrational might be one of those. But, what's given
| in the article feels like either the author didn't really
| understand the subject, or she couldn't explain what she
| understood in simple terms. But, it's really rare that
| there's such an easy to understand proof or concept. So,
| I don't blame her.
| 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
| crabbone wrote:
| No. I mean computable functions:
| https://en.wikipedia.org/wiki/Computable_function
| sunshowers wrote:
| Computable numbers are those that are described by
| computable functions. Irrationals like pi can be
| described by computable functions that take a precision
| as input.
| LPisGood wrote:
| "God created the natural numbers, all else is the work of
| man."
| tsimionescu wrote:
| A lot of this is very much wrong. To the extent that your
| square is close to perfect and its sides are actually 1m,
| it's diagonal is just as truly sqrt(2)m long. And a circle
| that you draw correctly enough with a radius of 1m will have
| its circumference actually equal to 2pi.
|
| There is nothing more special about sqrt(2) than about 1, nor
| about a perfectly 90deg angle versus a perfect circle. All of
| our drawings and constructions are approximations, but that
| doesn't make them naturally be integers or rationals any more
| than they are irrational.
|
| In other words, it would be just as accurate to say that the
| sides of the physical square are not 1, but x*pi for a pretty
| small x (that is, they are ever so slightly curved) as it
| would be to say that the circle's circumference isn't really
| 2*pi, it's actually some rational number, because the square
| is actually some very very many sided polyhedron.
|
| Even if we look at this from a purely physical perspective,
| elementary particles travel in perfectly straight lines and
| radiate in perfect circles in our models. And the directions
| of movement after a collision are not quantized, they can be
| arbitrary angles (just as space is not quantized, and in fact
| not even quantizable, in QM). And if you tried to look at a
| physical object and count the atoms to determine its length,
| you'd quickly find that it doesn't even have a constant
| number of atoms or a constant length, so in fact the least
| real concept is "an object of x meters in length", regardless
| of whether x is natural, rational, or irrational.
| crabbone wrote:
| Well, no, no physical square, no matter how precisely its
| sides are 1m long has an irrational-lenght diagonal. That
| is simply impossible in the physical universe because the
| universe is discrete at this level. Irrational numbers are
| impossible in discrete context. The whole point they were
| invented is to capture the idea of continuous functions or
| continuous number line. But this is a "slight of hand", a
| definition that's made for convenience of solving useful
| problems, but isn't based in the physical reality. In a
| very similar way to how sqrt(-1) is not a thing in a
| physical reality, but it's useful to work with problems
| that can be described using complex numbers.
|
| > There is nothing more special about sqrt(2) than about 1
|
| That fact that you _don 't understand_ what's special about
| it doesn't mean there isn't. It's a very different thing
| though.
|
| > All of our drawings and constructions are approximations
|
| Drawings: yes. Constructions: no.
|
| > but that doesn't make them naturally be integers or
| rationals any more than they are irrational.
|
| Here you've ventured into the territory you have no idea
| about... I'm sorry. You sound more like some LLM-generated
| gibberish here than anything a human with any expertise on
| the subject would write. Of course some things are
| naturally integers. We've invented integers to capture
| those things (in the physical universe). Similarly,
| rationals. There's nothing in the physical universe that's
| irrational in the same way how it can be an integer or a
| rational. Irrational numbers don't describe quantities or
| passage of time or forces acting on physical objects or the
| speed etc. because all those things are made up of small
| indivisible parts, and there's always a finite computable
| answer to how big something is, how long a process would
| take, how strong is the force applied to an object etc.
|
| Irrational numbers are a mathematical device to deal with
| different kinds of problems. Similar to how generating
| functions use "+" to mean a completely different thing from
| how it's used in algebra, or how it's used in regular
| languages, so are irrational called "numbers". But they
| aren't the same kind of thing as integers or rationals. To
| be honest, it would've been better not to call them
| "numbers" at all, to avoid this kind of confusion, but
| mathematics has a lot of old and bad terminology that's
| used due to tradition.
|
| > elementary particles travel in perfectly straight lines
| and radiate in perfect circles in our models
|
| The root of your problem in understanding this is: _our
| models_. Particles don 't radiate in perfect circles in
| reality. Physical reality is discrete and cannot create
| perfect circles. You can imagine, however, a perfect circle
| and use it to a great effect to estimate the result of some
| physical process. But, if you truly measure the effect, you
| will never have an irrational number. There's no physical
| process of measuring anything that will end up with an
| irrational answer. That's simply impossible.
| tsimionescu wrote:
| To the best of our knowledge, physical reality has both
| discrete quantities and continuous quantities. Space and
| time are continuous, for example. Energy, charge, mass -
| all discrete. And this is not a question of just models -
| we've tried to create models where time or space are
| discrete, and they don't work. They fail to predict
| reality correctly.
|
| And you're still wrong about the nature of irrational
| numbers. They are not a model for computational
| irreducibility. They basically separate different types
| of quantities that are not directly relatable as ratios
| of one another. The circumference of a circle (or any
| non-trivial ellipse, for that matter) is a fundamentally
| different quantity than the length of a straight line:
| this is what Pi being irrational tells us.
|
| And yet, you can do arithmetic with circle lengths just
| as much as you can with sides of squares. You can add
| them up, divide them, everything. If for whatever reason
| we decided to define 1m as the circumference of a
| particular circle, and base our geometry around circles
| rather than straight lines, we'd consider circle lengths
| to be integer/rational numbers, and then line lengths
| would be irrational (the radius of a circle with
| circumference 1 would be 1/(2pi), an irrational number).
| Similarly, if we decided that 1m is defined as the
| diagonal length of a particular square, we'd say that the
| lengths of that square have irrational length, and then
| you'd claim that no true square has lengths of exactly
| sqrt(2)/2.
|
| As I said, it's just as correct to think of any physical
| circle/ellipse as a many-sided polyhedron as it is to
| think of any physical straight line as a segment of the
| circumference of a circle/ellipse with a really huge
| radius (focal length).
|
| In fact, it's more correct physically to think of
| apparently straight lines as curved than to think of
| curves as composed of polygons, as straight paths are
| very rare in physics, any kind of bias will tend to
| induce a slight curvature. And atoms are more circular in
| nature than they are "boxy".
|
| Ultimately we can only really measure ratios of things,
| and we know some things are not an exact ratio of another
| thing when we measure them precisely enough. Which of the
| quantities you consider to be represented by a rational
| number and which you consider to be irrational is purely
| a choice of definitions.
| crabbone wrote:
| > And yet, you can do arithmetic with circle lengths just
| as much as you can with sides of squares.
|
| No, you can't. There's no way to add or to multiply two
| numbers that don't have a finite expansion in some basis
| beyond just writing it as a sum outside of a very few
| special cases where there's a round-about way of finding
| the answer. I.e. if you try to do pi + pi, well, you may
| get a 2pi, if you pray hard enough and your faith is
| strong enough, but really, there's no proof that even
| that is true. You just choose to believe that it will
| check out somehow. But, even if you get a 2pi, it's still
| not an answer you want because to figure out what 2pi is,
| you still need to add a pi to a pi, so, you are back to
| square one.
|
| > As I said, it's just as correct to think of any
| physical circle/ellipse as a many-sided polyhedron
|
| Because you prayed hard enough and it was revealed to you
| in a dream? Based on what do you believe this?
| com2kid wrote:
| Take a circle with a radius of 1cm. Unroll its perimeter
| and declare that length is 1 of new unit called a squelk.
|
| You can measure things and build things in squelks just
| fine, but if you try to take something that is 100
| squelks long and measure it in centimeters you will get
| an irrational number of centimeters because there is no
| rational conversation from squelks to centimeters.
|
| A given length can be irrational in one unit of measure
| but not in another.
|
| Of course we do have limits of precision in the real
| world, so in reality nothing lines up quite right.
| tsimionescu wrote:
| > I.e. if you try to do pi + pi, well, you may get a 2pi,
| if you pray hard enough and your faith is strong enough,
| but really, there's no proof that even that is true.
|
| Sure there is. pi + pi = 2pi <=> (pi + pi) / pi = 2pi /
| pi <=> pi/pi + pi/pi = 2 <=> 1 + 1 = 2, which we know is
| true. QED.
|
| This is in fact exactly what I'm saying about the circle
| and its radius. We can't get rid of the irrationality
| when calculating the ratio between the circumference and
| the radius of a circle. But it's arbitrary which one we
| call rational and which we call irrational: a circle with
| a rational circumference will have an irrational radius,
| and vice versa.
|
| > Because you prayed hard enough and it was revealed to
| you in a dream? Based on what do you believe this?
|
| I don't know what exactly you are responding to here.
|
| It's _your_ claim that in the real physical world all
| "circles" have a rational circumference (perimeter),
| which is equivalent to saying that the "circle" is really
| a very very many-sided polyhedron (since only a
| polyhedron can have a rational perimeter if the sides are
| of a rational length and all angles are constructible). I
| don't need to pray (?!?) to see this.
|
| And if you were responding to my full quote, that this
| comparison is equivalent to saying that all physical
| "squares" are in fact rounded-corner ovoid shapes (and so
| their actual side lengths are some multiple of pi, or at
| least some other irrational number that we don't even
| have a name for) then that follows from the observation
| above, that you can arbitrarily decide to call the
| circumference of a circle "2 pi" or the radius "1/2pi".
|
| It also follows from how trajectories work in physics -
| if a particle is moving in a straight line and then some
| force starts acting on it in some direction other than
| directly in front or behind, its trajectory will become
| circular, not go at a straight angle. So if an electron
| in a perfectly isolated environment would follow a
| perfectly straight line, an electron in a real
| environment where there are electrical fields everywhere
| will follow a line that's curvy all around. In contrast,
| it's in fact impossible to create a trajectory for an
| electron that has any kind of angles, even in an ideally
| isolated environment - it's impossible for a physical
| object to turn on the spot like an ideal angle.
|
| So, again, curves (and their associated irrational
| numbers) are in fact closer to physical reality, we just
| chose to approximate them using straight lines because
| its easier.
|
| And as a final thought, related to the reality of the
| continuum. In all of the models that we have of physics
| that actually work, if I fire two particles away from
| each other arbitrarily in space, the distance between
| them will cover every real number in some interval
| [minDist, maxDist]. And any model that requires a minimum
| unit of distance to exist (so that the distances would be
| minDist + n*FundamentalMinimum, with n = 1, 2, 3...)
| doesn't work with special relativity, that says that
| lengths contract in the direction of movement (because if
| two particles are at a distance of FundamentalMinimum as
| measured by one observer, they will be at a distance of
| gamma*FundamentalMinimum to another observer moving at
| some speed relative to the first one, with gamma < 1,
| thus breaking the assumption that all lengths are >
| FundamentalMinimum).
| 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.
| yen223 wrote:
| The 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.
| 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"
| 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
| heisenzombie wrote:
| Also see David Bessis:
|
| https://www.quantamagazine.org/mathematical-thinking-isnt-
| wh...
| anthk wrote:
| Leaves and branches/roots in nature are fractal.
| 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 :)
| c0redump wrote:
| If 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 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.
| Someone wrote:
| > because these two numbers are not supposed to be related
|
| Says who? They're not known to be related in that way, but
| it's not like nature set out to prevent such a thing, or
| that large parts of mathematics would break down if it
| happened to be the case.
| 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.
| dhosek wrote:
| To put the first equation more formally, we know that Q is
| closed under addition1, so given k[?]R\Q, l[?]Q then if
| k+l=m[?]Q, then m-l=m+(-l)[?]Q, but m-l=k which is not in Q
| so k+l[?]Q.
|
| [?]
|
| 1. For p,q[?]Q, let p=a/b, q=c/d, a,b,c,d[?]Z, then
| p+q=(ad+bc)/bd, but the products and sums of integers are
| integers, so p+q[?]Q
| vrighter wrote:
| x - (x - floor(x)) == x truncated to an integer
|
| when x is an irrational number > 1:
|
| "x - floor(x)" is just the fractional part of x, so it's an
| irrational number which is not equal to x.
|
| Subtracting the fractional part from the original leaves only
| the integer part, which is obviously rational.
| 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)
| mvdtnz wrote:
| You didn't know that one because it's a lie. He's telling
| lies.
| toth wrote:
| Charitably it was a joke, as was my quip about
| `e^(sqrt(67) pi)`. It is a funnier joke without a
| disclaimer at the end, but unlike GP I couldn't bring
| myself to leave one out and potentially mislead some
| people...
|
| What I meant was that I didn't know that `e^pi - pi` is
| another transcendental expression that is very close an
| integer. You might think this is just an uninteresting
| coincidence but there's some interesting mathematics
| around such "almost integers". Wikipedia has a quick
| overview [1]. I didn't realize it before, but they have
| GP's example and also the awesome `e + pi + e pi + e^pi +
| pi^e ~= 60`.
|
| [1] https://en.wikipedia.org/wiki/Almost_integer
| isaacfrond wrote:
| The number you are looking for is e^(sqrt(163) pi).
| According to Wikipedia:
|
| _In a 1975 April Fool article in Scientific American
| magazine,[8] "Mathematical Games" columnist Martin
| Gardner made the hoax claim that the number was in fact
| an integer, and that the Indian mathematical genius
| Srinivasa Ramanujan had predicted it - hence its name._
|
| It is not an integer of course.
| toth wrote:
| Actually `e^(sqrt(n) pi)` is very close to being an
| integer for a couple of different `n`s, including 67 and
| 163. For 163 it's much closer to an integer, but for 67
| you get something you can easily check in double
| precision floats is close to an integer, so I thought it
| worked better as a joke answer :)
|
| FYI, the reason you get these almost integers is related
| to the `n`s being Heegner numbers, see
| https://en.wikipedia.org/wiki/Heegner_number.
| Someone wrote:
| > The number you are looking for is e^(sqrt(163) pi)
| [...] It is not an integer of course.
|
| Of course? I'm not aware that we have some theorem other
| than "we computed it to lots of decimals, and it isn't an
| integer" from which that follows.
| less_less wrote:
| It's not really "of course", and I don't think we have
| such a theorem in general. But in this case, I believe
| the fact that it's not an integer follows from the same
| theorem that says it's very close to an integer. See eg
| https://math.stackexchange.com/questions/4544/why-is-e-
| pi-sq...
|
| Basically e^(sqrt(163)*pi) is the leading term in a
| Laurent series for an integer, and the other (non-
| integer) terms are really small but not zero.
| 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.
| c0redump wrote:
| Wow, you just made my day with this! What a fantastic
| result! Beautiful.
|
| Edit: looks like I swallowed the bait, hook like and sinker
| dvt wrote:
| I'm kind of in the opposite camp. If Schanuel's conjecture is
| true, then e^ip = 0 would be the _only_ non-trivial relation
| between e, p, and i over the complex numbers. And the fact
| that we already found it seems unlikely.
| seanhunter wrote:
| you mean e^(i pi)=-1, which is known as Euler's identity
| and is a specific case of Euler's formula
|
| e^(i theta) = cos theta + i sin theta
|
| That formula gives infinitely many trivial relationships
| like this due to the symmetry of the unit circle
|
| e^(i 2 pi) = 1
|
| e^(3 _i /2_pi)/i=1
|
| e^(5 _i /2_pi)/i=-1
|
| e^(i 2n pi) = 1 for all n in Z ...
|
| etc
| botanical76 wrote:
| Thanks for ringing some bells. It's been a long time
| since I used that equation.
| paulddraper wrote:
| > although it's proven that at least one of them is irrational
|
| And not particular to e and pi. More generally, at least one of
| a+b and a*b must be irrational, if an and b are transcendental.
| 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.
| thaumasiotes 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.
|
| No, but the form they used was that they thought about
| lines in a plane that had no particular scale other than
| whatever you might assign to them. "Plugging in 1" for
| the sides of a right triangle in that model just means
| that you look at an isosceles right triangle. It will
| always be obvious.
| 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?
| dr_dshiv wrote:
| It's not about conclusively debunking. There is no
| evidence for it at all! They say he drowned for his
| impiety at publishing the dodecahedron (stated first by
| Iamblichus, 700 years later).
|
| "The discovery of irrational numbers is said to have been
| shocking to the Pythagoreans, and Hippasus is supposed to
| have drowned at sea, apparently as a punishment from the
| gods for divulging this and crediting it to himself
| instead of Pythagoras which was the norm in Pythagorean
| society. However, the few ancient sources who describe
| this story either do not mention Hippasus by name (e.g.
| Pappus)[4] or alternatively tell that Hippasus drowned
| because he revealed how to construct a dodecahedron
| inside a sphere.[5] _The discovery of irrationality is
| not specifically ascribed to Hippasus by any ancient
| writer_ "
| 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.)
| dr_dshiv wrote:
| Because you can't make it with a fraction.
| DoctorOetker wrote:
| Ah yes, thanks for reminding us that the Pythagoras
| proved the square root of 2 was irrational by ...
| circular reasoning. /s
| 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!)
| dr_dshiv wrote:
| It's because you can't create square root of two with a
| fraction. Pythagoreans were all about fractions (such as
| their musical tuning).
| Sniffnoy wrote:
| That's not an argument, that's a restatement of the
| problem; that's what being irrational means. The problem is
| to _prove_ that you can 't create sqrt(2) with a fraction.
| How, exactly, would this statement have been obvious to the
| Pythagoreans? Can you give me an argument for this that
| they would have found obvious? One that uses the
| Pythagorean theorem even, perhaps, since you brought that
| up? Remember, no using concepts they wouldn't have had like
| prime factorization!
| dhosek wrote:
| So many circular arguments. The Pythagorean theorem tells you
| that [?]2 exists, but not that it's irrational.
|
| As for the story, it's apocryphal, not ahistorical, but even
| so, it was too good of a story not to tell my students when I
| taught Math for Liberal Arts Majors (the version I'd heard was
| that the proof was presented while the Pythagoreans were on a
| boat and they were so offended by the idea that [?]2 is
| irrational, they threw the guy off the boat. I would guess that
| of all the things I said in lectures for that class, this is
| the one that my students would be most likely to remember).
| 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.
| alexashka wrote:
| > 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?
| josephcsible wrote:
| > Do math people not feel the need to explain themselves
| when they state things that defy common sense everyone
| except math people agree upon? Is that part of thinking
| you're 'smart'?
|
| It's a pretty basic thing covered in undergrad prob/stats
| classes. We don't re-explain it every time we use it for
| the same reason computer scientists don't re-explain the
| halting problem every time it comes up.
| BalinKing wrote:
| > 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.
|
| This is only true if there are only a finite number of
| things in your pocket, though... I think an analogy is
| how we always have 1/n>0 for any finite (positive) number
| n--and yet, 1/infinity=0.
|
| For something more precise, I think the corresponding
| Wikipedia page (FWIW) is
| https://en.wikipedia.org/wiki/Almost_never.
| jncfhnb wrote:
| I'm struggling to understand how a thrown dart could land on
| an irrational number. It seems definitionally that any
| physically realized outcome must pertain to a rational number
| because it is impossible to physically measure one at any
| level of precision.
|
| It is possible to write a random process that returns 5 or pi
| with 50/50 odds so this isn't a very compelling argument that
| it's possible. I don't feel the semantics of picking a random
| point along a number line is gg solved just by appealing to
| the existence of uncountably infinite irrationals.
|
| By most people's definitions of random points along the
| number line, including the dart throw, it seems to me the
| probability of getting an irrational is 0.
|
| Invoking the number of possible outcomes has bad feeling
| implications. For example if your set is 1 2 3 pi 4, then the
| probability of getting an outcome in [3,4) is higher than
| [2,3) and that seems like it's breaking the intuition of what
| the line represents. Like as a stupid example say we only
| include the irrational numbers between 9 and 10 and pick a
| random point between 1 and 10. If the random method uniformly
| sampled a point along the line by distance we would suggest a
| 90% chance of getting a rational number <= 9 and a 10% chance
| of getting an irrational number above 9.
|
| But if we sample by naive odds you'd probably claim there's a
| near 100% chance of getting an irrational number above 9
| because there's an uncountable infinity up there.
|
| That seems dumb.
| thaumasiotes wrote:
| > It seems definitionally that any physically realized
| outcome must pertain to a rational number because it is
| impossible to physically measure one at any level of
| precision.
|
| Sure, that's correct, but it isn't what people are talking
| about here.
|
| > By most people's definitions of random points along the
| number line, including the dart throw, it seems to me the
| probability of getting an irrational is 0.
|
| That depends on the number line you're using. You can say
| that irrationals don't exist and you won't lose anything.
| But if your number line includes the reals, then the
| rationals form 0% of it.
|
| > Invoking the number of possible outcomes has bad feeling
| implications.
|
| That isn't how this is measured. You don't want to compare
| a count to an area. For probability, you need to compare
| like with like. A number line is one-dimensional, so we
| consider one-dimensional areas, or "lengths".
|
| The interval from 0 to 50 has length 50. How much of that
| length is occupied by rationals, and how much by
| irrationals?
|
| Each value is a point with no length. So, to measure the
| rationals, we assign to each rational point an interval
| that contains it. We will estimate the total length
| occupied by the rational numbers within the interval as
| being no greater than the total length of the intervals we
| put around each one.
|
| Since there are only countably many rationals, we can use
| an infinite series with a finite sum to restrict our total-
| length-of-intervals to a finite amount. (Rational number
| one gets an interval 3 units wide. Rational number two gets
| one 0.3 units wide. Number three gets one 0.03 units wide.
| What do all these intervals add up to? Four thirds.) We can
| scale those intervals however we like. We will scale them
| down. If our first set of intervals had total length 20, we
| can multiply them all by 1/400 and now they'll have total
| length 1/20. The limit of this process is a total length of
| zero, which is our _upper_ bound on how much of the length
| of our interval is occupied by rational numbers.
|
| Since zero is also a _lower_ bound on any length, we know
| that the total length of the interval occupied by rational
| numbers is exactly equal to 0. It is then easy to calculate
| the probability that a randomly chosen value from this
| interval will be rational: it is 0 (the amount of length
| occupied by rationals) over 50 (the total amount of
| length).
|
| > Like as a stupid example say we only include the
| irrational numbers between 9 and 10 and pick a random point
| between 1 and 10. If the random method uniformly sampled a
| point along the line by distance we would suggest a 90%
| chance of getting a rational number <= 9 and a 10% chance
| of getting an irrational number above 9.
|
| This seems to be just you being confused over the concept
| of a uniform distribution.
| lll-o-lll wrote:
| Throw infinity into the mix, get an irrational answer.
| jncfhnb wrote:
| > This seems to be just you being confused over the
| concept of a uniform distribution.
|
| Try and follow the example again.
|
| The distribution is all rationals 1-9 and all numbers
| 9-10.
|
| Sampling uniformly such that each distance is equally
| likely across the line gives at least a 90% chance of
| choosing a rational.
|
| Sampling uniform by elements of the set gives a 0% chance
| of choosing a rational.
|
| The problem with the latter is that even though you're
| claiming to be randomly sampling the _line_ you are never
| going to sample the first 90% of the line length because
| you are instead sampling the _distribution of set
| elements_.
|
| You are NOT more likely to throw a dart that lands in 9+
| just because you have magically introduced an infinitely
| tense series of irrationals in that range.
| thaumasiotes wrote:
| > The problem with the latter is that even though you're
| claiming to be randomly sampling the _line_ you are never
| going to sample the first 90% of the line length because
| you are instead sampling the _distribution of set
| elements_.
|
| This is all in your head. Who are you responding to?
| Where did your three claims ("sampling uniformly by
| distance from 0" / "sampling uniformly by element count"
| / "randomly sampling the line") come from? What does
| "sampling uniformly by distance" mean? Uniform sampling
| is done by count for discrete sets and by area for
| continua. You have yet to mention a discrete set.
| jncfhnb wrote:
| It is the difference between picking a random point along
| a line and picking a random number from a set. A dart
| throw will not land in the range of [9,10) more often
| than [1,9) simply because we are considering irrationals
| in the former.
|
| These are both uniform. But the outcome is different
| humodz wrote:
| > Sampling uniformly such that each distance is equally
| likely across the line gives at least a 90% chance of
| choosing a rational.
|
| Let's say the numbers are targets on the line. Your
| distribution implies the range 1-9 is less dense with
| targets than the range 9-10. Doesn't that mean you're
| less than 90% likely to hit something between 1-9?
|
| > You are NOT more likely to throw a dart that lands in
| 9+ just because you have magically introduced an
| infinitely tense series of irrationals in that range.
|
| If we turn this around, by forbidding a bunch of values
| in the 1-9 range from being hit, then won't the
| probabilities get skewed towards the 9-10 range?
| jncfhnb wrote:
| No, because a dart throw is not a uniform draw from set
| elements. Is a uniform draw of length which the inclusion
| of irrational numbers does not affect. You are 90% likely
| to throw something in the first 90% of the line. It
| doesn't matter if we say we will round anything in [1,2)
| to 1. There's a ten percent chance of falling in that
| range.
|
| Not a 0% chance because there happens to be an
| uncountable infinity number of options in [9,10)
| greekanalyst wrote:
| I 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 wrote:
| A 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 wrote:
| If you want a probability flavoured statement, you will need to
| appeal to measure theory.
| justinpombrio wrote:
| Stating 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.
___________________________________________________________________
(page generated 2025-01-10 23:02 UTC)