It would be mind-blowing if either of them were rational numbers, yet it's very hard to prove either way.
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)
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.
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.
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"
It is a "joke:" "a thing that someone says to cause amusement or laughter."
"Formula:" "a concise way of expressing information symbolically."
"Garden:" "a small piece of ground used to grow vegetables, fruit, herbs, or flowers."
Similar to:
> When asked where his HTML/CSS/JavaScript came from, he claimed, "It grows on trees."
This type of joke is called "Absurdity:" The humor is amplified by the absurdity of imagining abstract things like formulas or programming code as physical objects that could grow in nature.
> my sci-fi-fed mind is that something in their brains allows them to "travel via hyperspace".
What is a "hyperspace?" Are you talking about hyperbolic geometry?
> "and threw paper airplanes"
Were the paper airplanes also traveling via hyperspace?
The definitions of the words formula and garden also seem out of place because they don't add anything to what you were saying -- at least I don't see any analogies that would make the joke funnier.
[1] Like this.
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.
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 :)
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
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.
“ 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.)
> 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?
(These days the irrationality of sqrt(2) is obvious due to unique prime factorization, but the ancient Greeks didn't have that concept!)
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.