Talk:Algebraic number

The are some disagreement between these two statements in wiki pages:

algebraic numbers: "A real algebraic number of degree 2 is a quadratic irrational."

quadratic irrational: "When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers."

I suggest, to skip confusion, modify this page to "An algebraic number of degree 2 is a quadratic irrational." — Preceding unsigned comment added by 88.3.70.40 (talk) 09:29, 1 January 2017 (UTC)

Recently someone put pi as an example of algebraic number, either because he/she did not understand the reference or a grief. Since it causes direct contradiction, I removed it for now. --Raxu360 (talk) 00:30, 30 June 2019 (UTC)

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently, by clearing denominators, with integer coefficients).

As with so many mathematical definitions in Wikipedia, this definition can only be understood by those who already know what the term means, and likely not all of them. (I have seen mathematical definitions in Wikipedia where I already knew what the term meant, but still couldn't understand the definition.)

The sole purpose of an encyclopedia is to convey information to those who do not already have it. This definition does not perform that function.

Koro Neil (talk) 21:06, 8 October 2021 (UTC)

:I would include that it is not the same as number representable in radicals. Valery Zapolodov (talk) 05:56, 25 December 2022 (UTC)

: Do you have a proposed alternative? My suggestion would be to just get rid of all of the parentheticals as they make the sentence very awkward. Something like: “An algebraic number is a complex number that is the solution to a single-variable polynomial equation with rational coefficients. [... examples and further explanation unpacking some of those terms ...]” It looks like the definition has changed somewhat since this comment though. Did those changes solve your issue? –jacobolus (t) 00:43, 26 December 2022 (UTC)

Says "The set of algebraic numbers is countable (enumerable)," with two old print refs. Can we link to an online proof ? - Rod57 (talk) 22:40, 19 September 2022 (UTC)

:What online proof? Math. Stack. is not allowed, cause it is not a reliable source. Valery Zapolodov (talk) 05:54, 25 December 2022 (UTC)

The article says "The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental." Yet a stronger statement would be to replace "complex" with "real". As worded, it implicitly suggests that the statement may not be true of the reals. So, is there a reason it says "complex" here, and not "real" ? 140.177.255.59 (talk) 13:54, 21 September 2024 (UTC)

:{{fixed}}. Thank you for pointing this out. — Anita5192 (talk) 15:03, 21 September 2024 (UTC)

There are a number of what appear to be server side error messages embedded in the Properties section under "Degree of simple extensions of the rationals as a criterion to algebraicity".

The previous edit didn't touch that section and it was fine in the previous version, so I've no idea what's causing this or how to fix it.

Benkelly314 (talk) 12:28, 15 May 2025 (UTC)

:Is this still appearing for you? I don't see any. This is a temporary server error, and if you are editing an article the way to prevent readers from seeing it is to click "preview" on your edit a couple times until all such messages disappear; there's some caching of the rendered math, so if there aren't any messages in the preview there won't be in the final page either. –jacobolus (t) 15:19, 15 May 2025 (UTC)