Rational or not? This basic math question took decades to answer

https://www.quantamagazine.org/rational-or-not-this-basic-math-question-took-decades-to-answer-20250108/

by nsoonhui • 2 years ago

173 132 2 years ago

tejohnso• 2 years ago

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?

treyd2• 2 years ago

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• 2 years ago

Constructed is the wrong word. sqrt(2) is constructible and irrational

erehweb• 2 years ago

I think you may be thinking of transcendental numbers, which are a subset of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number

CassianAI2• 2 years ago

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.

wat100005• 2 years ago

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?

crabbone10• 2 years ago

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-kt2• 2 years ago

Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc.

Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.

tsimionescu3• 2 years ago

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 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.

sunshowers6• 2 years ago

These things are useless for centuries until they suddenly underpin all of modern society.

octachron2• 2 years ago

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• 2 years ago

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.

yen2232• 2 years ago

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• 2 years ago

When asked where his formulas came from, he claimed, “They grow in my garden.”

This makes me think of Ramanujan's notebooks. And based on my limited interaction with professional mathematicians, I think there is something to this - some hidden brain circuitry whereby mathematicians can access mathematical truths in some way based on their "beauty", without going through anything resembling rigorous intermediate steps. The metaphor that comes to my sci-fi-fed mind is that something in their brains allows them to "travel via hyperspace".

And this then makes me think of GenAI - recent progress has been quite interesting, with models like o1 and o3 at times making silly mistakes, and at other times making incredible leaps - could it be that AI's are able to access this "garden" too? Or does there remain something that we humans have access to, while AIs do not?

ysofunny• 2 years ago

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• 2 years ago

Terrence Tao wrote a nice blog post which captures this idea (post-rigorous phase)

https://terrytao.wordpress.com/career-advice/theres-more-to-...

sdwr• 2 years ago

I appreciate that he cites corroborating sources for his ideas, it makes the whole thing feel well-rounded

heisenzombie• 2 years ago

Also see David Bessis:

https://www.quantamagazine.org/mathematical-thinking-isnt-wh...

anthk• 2 years ago

Leaves and branches/roots in nature are fractal.

fermigier• 2 years ago

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).

thaumasiotes7• 2 years ago

When mathematicians do succeed in proving a number’s irrationality, the core of their proof usually relies on one basic property of rational numbers: They don’t like to come near each other.

This property of the minimum distance between two rational numbers is what the ruler function* relies on to be continuous at all irrational numbers while being discontinuous at all rationals.

* When x is irrational, f(x) = 0; otherwise, when p and q are integers, f(p/q) = gcd(p,q)/q. Note that this leaves f(0) undefined, which is fine for the result of being discontinuous at rationals. You could define f(0) to be any value other than 0. The function is traditionally defined over the open interval (0, 1), which avoids the issue.

DerekL• 2 years ago

Actually, f(0) is well-defined. If q is positive, then gcd(0,q) = q, so f(0) = 1.

thaumasiotes7• 2 years ago

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_992• 2 years ago

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.

sunshowers6• 2 years ago

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 :)

c0redump2• 2 years ago

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• 2 years ago

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• 2 years ago

Same for pi^pi^pi^pi

https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz

tshaddox• 2 years ago

Out of curiosity, why would it be mind-blowing if either of them were a rational number?

charlieyu1• 2 years ago

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

cbm-vic-20• 2 years ago

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 ?

aidenn02• 2 years ago

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.

AlotOfReading3• 2 years ago

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.

LPisGood4• 2 years ago

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.

umanwizard2• 2 years ago

pi and -pi are both irrational and their sum is zero.

ryandv• 2 years ago

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.

yen2232• 2 years ago

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.

vrighter• 2 years ago

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• 2 years ago

I could believe pi*e rational but pi+e being rational would blow my mind.

sunshowers6• 2 years ago

I would be shocked if either of them were proven to be rational.

chongli• 2 years ago

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• 2 years ago

Well, e^pi - pi = 20, is rational.

dvt• 2 years ago

I'm kind of in the opposite camp. If Schanuel's conjecture is true, then e^iπ = 0 would be the only non-trivial relation between e, π, and i over the complex numbers. And the fact that we already found it seems unlikely.

paulddraper• 2 years ago

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_dshiv6• 2 years ago

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.

DoctorOetker4• 2 years ago

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_dshiv6• 2 years ago

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.

Sniffnoy2• 2 years ago

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_dshiv6• 2 years ago

It’s because you can’t create square root of two with a fraction. Pythagoreans were all about fractions (such as their musical tuning).

dhosek3• 2 years ago

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).

jncfhnb5• 2 years ago

If you pick a point along the number line at random, it’s almost guaranteed to be irrational.

I’m having a hard time grasping this one. Feels like the coastline paradox on a straight line of a known length.

Are irrational numbers even on a number line? Isn’t it definitionally impossible to pick it as a “point along the line”?

IanKerr2• 2 years ago

Are irrational numbers even on a number line?

Yes, e is between 2 and 3 and Pi is between 3 and 4. There are geometrical lengths corresponding to each number.

Isn’t it definitionally impossible to pick it as a “point along the line”?

No, it's mathematically possible to have a random process which picks a random real between 0 and n, with equal probability. Imagine it akin to throwing a dart at a line and picking the point it lands on as the number. Since there are only countably many rationals and uncountably many irrationals (i.e. not just infinitely more, but so many that you could never pair off the rationals with the irrationals, there are just too many) on any such length of the real line, chances are the number you end up with is overwhelmingly likely to be irrational.

wat100005• 2 years ago

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.

jncfhnb5• 2 years ago

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.

greekanalyst• 2 years ago

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!

dhosek3• 2 years ago

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• 2 years ago

If you want a probability flavoured statement, you will need to appeal to measure theory.

justinpombrio• 2 years ago

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.

edanm• 2 years ago

That's a really cool well of "visualizing" this.

zyklu5• 2 years ago

Here is Frank Calegari's excellent talk on this work:

https://www.youtube.com/watch?v=znBdPEyDScY

Rational or not? This basic math question took decades to answer

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