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

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

Do you have a citation for the rationality of e^pi - pi? I couldn't find anything alluding to anything close to that after some cursory googling, and, indeed, the OEIS sequence of the value's decimal expansion[1] doesn't have notes or references to such a fact (which you'd perhaps expect for a rational number, as it would eventually be repeating).

[1] https://oeis.org/A018938

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)

It is not exactly 20.

Wow, you just made my day with this! What a fantastic result! Beautiful.

Edit: looks like I swallowed the bait, hook like and sinker

> Take any irrational a where 1/a is also irrational.

In other words: any irrational at all

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.

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

> 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

> 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.

e and pi are highly related, both pop out of periodic phenomenon.

Ah I'm mixing up my terms, you're right.

OP said constructed from finite number of symbols from basic algebra. There's no finite construction of sqrt(2) using addition and multiplication.

what's the right word?

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.

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.

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.)

“God created the natural numbers, all else is the work of man.”

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.)

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.

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?

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!

To sum it up in one sentence: You don't need to explain words that I can look up in a dictionary, unless you are commenting on that explanation. I know what the word "joke" means but you are forcing me to either read the definition or look for where you continue your comment. It's also slightly demeaning since I can read this as "you don't know what the word joke is, let me explain it to you". You do this again for the words formula, garden and absurdity. If you really need to give extra details to a word that you are using, you can do it in brackets (like this) or with a footnote [1].

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.

[flagged]

Yep, exactly. I glossed over that detail a bit because explaining how a meagre set has a truly zero probability of being picked, while technically still being a possible result of a random process, is a bit messy to wrap your head around colloquially.

> If you choose a random point on the line, the probability of choosing a rational is zero.

Wat?

If a thing is in my pocket, there's an above zero probability of me picking it when I randomly take a thing out of my pocket.

What are math people doing that's different?

> It seems definitionally that any physically realized outcome must pertain to a rational number because it is impossible to physically measure one at any level of precision.

Sure, that's correct, but it isn't what people are talking about here.

> By most people’s definitions of random points along the number line, including the dart throw, it seems to me the probability of getting an irrational is 0.

That depends on the number line you're using. You can say that irrationals don't exist and you won't lose anything. But if your number line includes the reals, then the rationals form 0% of it.

> Invoking the number of possible outcomes has bad feeling implications.

That isn't how this is measured. You don't want to compare a count to an area. For probability, you need to compare like with like. A number line is one-dimensional, so we consider one-dimensional areas, or "lengths".

The interval from 0 to 50 has length 50. How much of that length is occupied by rationals, and how much by irrationals?

Each value is a point with no length. So, to measure the rationals, we assign to each rational point an interval that contains it. We will estimate the total length occupied by the rational numbers within the interval as being no greater than the total length of the intervals we put around each one.

Since there are only countably many rationals, we can use an infinite series with a finite sum to restrict our total-length-of-intervals to a finite amount. (Rational number one gets an interval 3 units wide. Rational number two gets one 0.3 units wide. Number three gets one 0.03 units wide. What do all these intervals add up to? Four thirds.) We can scale those intervals however we like. We will scale them down. If our first set of intervals had total length 20, we can multiply them all by 1/400 and now they'll have total length 1/20. The limit of this process is a total length of zero, which is our upper bound on how much of the length of our interval is occupied by rational numbers.

Since zero is also a lower bound on any length, we know that the total length of the interval occupied by rational numbers is exactly equal to 0. It is then easy to calculate the probability that a randomly chosen value from this interval will be rational: it is 0 (the amount of length occupied by rationals) over 50 (the total amount of length).

> Like as a stupid example say we only include the irrational numbers between 9 and 10 and pick a random point between 1 and 10. If the random method uniformly sampled a point along the line by distance we would suggest a 90% chance of getting a rational number <= 9 and a 10% chance of getting an irrational number above 9.

This seems to be just you being confused over the concept of a uniform distribution.