Talk:Irrational number

all square roots of natural numbers, other than of perfect squares, are irrational

Can we provide a source for this? I want to read the proof Immanuelle (talk) 23:41, 4 April 2022 (UTC)

Read the article and follow the link: "The square roots of all natural numbers that are not perfect squares are irrational and a proof may be found in quadratic irrationals". Specifically, Quadratic irrational number#Square root of non-square is irrational. Meters (talk) 23:57, 4 April 2022 (UTC)

2^e

It was stated by Alexandru Froda in his Sur l'irrationalité du nombre 2^e. Just recently, Amiram Eldar claimed that the number is irrational. More at https://oeis.org/A262993. Question is, did he really prove that this constant cannot be expressed as a/b with a and b being positive integers? Kwékwlos (talk) 21:49, 19 June 2023 (UTC)

It does say in "http://www.bdim.eu/item?fmt=pdf&id=RLINA_1963_8_35_6_472_0" (in French) that the irrationality of 2^e is assumed. But then he proceeds to demonstrate that the assumption that 2^(e-1) is rational leads to the same number being irrational, contradicting the hypothesis. He finishes by saying that since 2^(e-1) is irrational, 2* 2^(e-1) = 2^e is also irrational. Dhrm77 (talk) 16:43, 20 June 2023 (UTC)

So, as far as I understand, we should change the section about open questions. Is this a reliable source with a proof that $2^{e}$ is irrational? D.M. from Ukraine (talk) 12:30, 22 June 2023 (UTC)

If so, this would imply that any number of the form rational^transcendental, assuming that the transcendental is not a logarithm of a rational with the base being the aforementioned rational, is irrational. Kwékwlos (talk) 23:35, 23 June 2023 (UTC)

I don't think so. I think that proof (if correct) is only valid for that number. The same way you can form an integer from 2 rational numbers, or a rational number from 2 irrational numbers, I believe there might be cases where rational^transcendental is not irrational. But besides 1^e or 1^pi, I can't find a non-trivial example. Dhrm77 (talk) 00:57, 26 June 2023 (UTC)

But the main question is whether the proof about just 2^e is situated in a source reliable enough for us to change the Wikipedia article. D.M. from Ukraine (talk) 13:33, 29 June 2023 (UTC)

Irrational numbers

If it is 5, you can write as 5/1(five upon one or five divided by 1). It is because every number multiplied by 1 is number itself. If π or √2 are irrational, can't we write them as ratio of π to 1 or √2 to 1? If it is wrong and ratio needs co-prime numbers, then what is ratio of 2 or 5? 2402:A00:401:B896:4923:CB28:C2C3:C3A7 (talk) 17:01, 4 July 2024 (UTC)

We can write π as π/1, but for a number to be rational it is necessary to have possibility to write it as a ratio of two INTEGERS (WHOLE numbers). The number π is not an integer. It is not important for the numbers in the ratio to be co-prime, important is to be integers. The number π is irrational, 2/5 is rational. Do you ask about the notion "irrational number" or about possibility to divide numbers? :) D.M. from Ukraine (talk) 14:31, 11 July 2024 (UTC)

infinity as irrational

i like the idea that since..

a*b = c therefore b = c/a, is our axiom..

if infinity is in fact a valid number, and we treat it as such,

1/0 = inf, 1/inf = 0,

0 * inf = 1.. (identity) so infinitely scaling "nothing" is equal to ONE.

and infinity can not be expressed as p/q quotient of integers or natural numbers, therefore it is irrational.

we respect our axioms of algebra, we deem infinity an irrational number, and we allow 0*inf = 1.

stating that infinity is irrational makes sense on all levels. and i think this is elegant and respects our logical foundations. 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 02:07, 16 January 2025 (UTC)

Infinity is not a valid number in the usual sense, so none of this applies.—Anita5192 (talk) 03:30, 16 January 2025 (UTC)

okay but who said that infinity is not a valid number? i mean how many natural or rational numbers exists? it must be a quantity, therefore we can respect our axioms and treat infinity as an irraional number.

its a puzzle piece that fits perfectly, i mean all irrational numbers have infinitely continuing decimals and thus has brought infinity itself to the attention of mathematicians. its only elegant that infinity itself may be a kind of irrational. 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 13:29, 16 January 2025 (UTC)

i mean i understand it's not usual, but it's a very nice way to tie infinity formally into our mathematics without logical breaks. 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 13:30, 16 January 2025 (UTC)

Apart from what Anita5192 said above, I would like to point out that rational neednt be a ratio of natural numbers, but a bit more generally a ratio of integer numbers. This allows us to have negative rational numbers, too. And if so, and you explicitly said yourself $\infty =1/0$, thus including zero among valid denominators (opposite to what standard arithmetic assumes)... It clearly means you defined infinity as a quotient of two integer numbers, hence it certainly must be rational. --CiaPan (talk) 13:43, 16 January 2025 (UTC)

hm, yeah perhaps the logic kind of breaks, it would have been nice though to accept infinity as a valid number so u can get algebraeic results from it.. such as 0*infinity=1.

and i figured that maybe zero isn't really quite a natural number, or its a weird integer and so it doesnt really apply to say it can be expressed as a quotient of natural integers..

but it would also have weird implications on the infinite monkey theorem 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 13:49, 16 January 2025 (UTC)

isn't it possible that we made poor assumptions thousands of years ago though? maybe to think outside of the box can yield interesting results 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 13:53, 16 January 2025 (UTC)

i just felt like maybe stamping "undetermined" on things is a workaround that doesnt really tie everything up perfectly.. if you used limits to consider these results it would have been trivial, but everything breaks exactly when hitting the bounds of the number line or when using zero as a divisor.. 2A02:A46E:D6AB:0:E03E:572F:C70E:EA59 (talk) 14:14, 16 January 2025 (UTC)

Pi

Is there a reason pi isn't on here? Lemur3215 (talk) 13:30, 1 July 2026 (UTC)

Second paragraph of the lead. - DVdm (talk) 13:35, 1 July 2026 (UTC)