Rational or not? This basic math question took decades to answer
quantamagazine.org
quantamagazine.org
It would be mind-blowing if either of them were rational numbers, yet it's very hard to prove either way.
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^(3i/2pi)/i=1
e^(5i/2pi)/i=-1
e^(i 2n pi) = 1 for all n in Z ...
etc
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)
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.
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.
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.
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.
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`.
Edit: looks like I swallowed the bait, hook like and sinker
Even moving from addition and multiplication to exponentials won’t save you: there are irrational numbers to irrational powers that are raational.
In other words: any irrational at all
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.
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.
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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
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.
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
∫(−∞ 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.
And for a given variance, gaussian distributions are exactly the maximal entropy distribution.
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.
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.
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).
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?
https://terrytao.wordpress.com/career-advice/theres-more-to-...
https://www.quantamagazine.org/mathematical-thinking-isnt-wh...
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"
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.
This same problem will occur everywhere negative, though. I wasn't thinking about it; I was just being sloppy.
Thank you Quanta and thank you science writers all over the world for making science more accessible!
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?
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.
> 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.
Transcendental functions s.a. sqrt() are to algebra like the trolley problem is to physics: deliberately excluded from the domain of discourse.
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.)
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).
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.
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.
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 :)
There is nothing more special about sqrt(2) than about 1, nor about a perfectly 90° 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.
> 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.
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.
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?
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.
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).
Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.
To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.)
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.
> 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
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.
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1. This result follows from the fact that |ℚ|/|ℝ| = 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.
If you like these, you may also like “Simplifying Complexity” (no relation to Quanta)
“ 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.
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.
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.
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.)
What do you see as the obvious thing that goes from n*n = 2 to “you can’t make n with a fraction”?
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.
> 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?
“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”
You took initially took issue with the following snippet from the HN article:
> “ 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.”
So I reference a Wikipedia page stating ancient / quasicontemporary sources describe such an event, but those ancient sources didn't attribute it to Hippasus by name. In the centuries after the supposed event authors start attributing this to Hippasus, which is under historical contention, since we don't find this attribution in writings from the same era as the event.
The HN article (just check your own snippet) describe the person as "a member of their school", which is in line with the hermeneutic interpretation of ancient texts (who said what and when).
Perhaps some day newfound libraries will be uncovered, or burnt scrolls deciphered, possibly finding contemporary (in the same lifetime) attributions of this discovery of the irrationality of sqrt(2) to Hippapus.
Basically: oldest sources already describe the events without naming the "offender" that was killed. Currently the oldest sources attributing this "offense" to Hippapus are lifetimes separated from the supposed events, casting doubt on the identity of the victim.
(These days the irrationality of sqrt(2) is obvious due to unique prime factorization, but the ancient Greeks didn't have that concept!)
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).
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”?
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.
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?
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.
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.
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.
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.
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.
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.
These are both uniform. But the outcome is different
As you originally pointed out, a physical dart can't hit a single point on a number line. It will hit an infinite number of them simultaneously. This is true whether you're worrying about rationals or reals.
But if you have a dart so sharp that its tip is zero-dimensional, one that can hit a single point on a real line, and you throw it at a composite of the rationals from [0,9] and the reals from [9,10], it will have a 10% chance of hitting an irrational number (within [9,10]), and it will have a 90% chance of missing the line entirely, striking one of the holes in the rational interval [0,9]. The chance of hitting a rational number will not improve from 0.
Do you have a model of uniform selection in mind, or do you find that it's easier to say the words without assigning them any particular meaning?
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?
Not a 0% chance because there happens to be an uncountable infinity number of options in [9,10)
I'm not sure what I was thinking of; 3.3333... is obviously ten thirds.