Rational or not? This basic math question took decades to answer
149 comments
·January 9, 2025FartyMcFarter
rokob
I could believe pi*e rational but pi+e being rational would blow my mind.
sunshowers
I would be shocked if either of them were proven to be rational.
dvt
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.
seanhunter
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^(3i/2pi)/i=1
e^(5i/2pi)/i=-1
e^(i 2n pi) = 1 for all n in Z ...
etc
programjames
Well, e^pi - pi = 20, is rational.
nimih
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).
toth
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)
hollerith
It is not exactly 20.
c0redump
Wow, you just made my day with this! What a fantastic result! Beautiful.
Edit: looks like I swallowed the bait, hook like and sinker
chongli
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.
gosub100
Same for pi^pi^pi^pi
cbm-vic-20
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 ?
AlotOfReading
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
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.
umanwizard
> Take any irrational a where 1/a is also irrational.
In other words: any irrational at all
pfdietz
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.
aidenn0
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.
umanwizard
pi and -pi are both irrational and their sum is zero.
ryandv
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
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
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
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.
tshaddox
Out of curiosity, why would it be mind-blowing if either of them were a rational number?
charlieyu1
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
> 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
Someone
> 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.
pishpash
e and pi are highly related, both pop out of periodic phenomenon.
paulddraper
> 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.
fermigier
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).
falcor84
> 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?
PhilipRoman
Terrence Tao wrote a nice blog post which captures this idea (post-rigorous phase)
https://terrytao.wordpress.com/career-advice/theres-more-to-...
sdwr
I appreciate that he cites corroborating sources for his ideas, it makes the whole thing feel well-rounded
heisenzombie
Also see David Bessis:
https://www.quantamagazine.org/mathematical-thinking-isnt-wh...
ysofunny
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"
anthk
Leaves and branches/roots in nature are fractal.
thaumasiotes
> 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
Actually, f(0) is well-defined. If q is positive, then gcd(0,q) = q, so f(0) = 1.
thaumasiotes
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.
greekanalyst
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!
tejohnso
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?
yen223
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.
sunshowers
These things are useless for centuries until they suddenly underpin all of modern society.
treyd
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.
erehweb
I think you may be thinking of transcendental numbers, which are a subset of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number
treyd
Ah I'm mixing up my terms, you're right.
nh23423fefe
Constructed is the wrong word. sqrt(2) is constructible and irrational
crabbone
OP said constructed from finite number of symbols from basic algebra. There's no finite construction of sqrt(2) using addition and multiplication.
jjtheblunt
what's the right word?
octachron
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).
CassianAI
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
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
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.
alpple
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.
crabbone
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 :)
tsimionescu
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.
crabbone
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.
tim-kt
> 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
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.)
LPisGood
“God created the natural numbers, all else is the work of man.”
zyklu5
Here is Frank Calegari's excellent talk on this work:
dhosek
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.
justinpombrio
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
That's a really cool well of "visualizing" this.
sorokod
If you want a probability flavoured statement, you will need to appeal to measure theory.
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hn_throwaway_99
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
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
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)
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dr_dshiv
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
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
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.
wat10000
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.)
AlotOfReading
They probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. Regardless, they were a weird religious cult. They could have just regarded numbers that didn't produce rational numbers as unnatural and not something that was going to occur in the actual functioning of the world.
DoctorOetker
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?
Sniffnoy
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
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
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
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).
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throw848484
[flagged]
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