Talk:2028: Complex Numbers

2018-08-03T21:23:01Z

Livetoad: meta-abelian vs metabelian

I assume this is strictly a coincidence, but in reference to the title-text, I'll just mention that Caucher Birkar [the mathematician whose Fields Medal was stolen minutes after he received it in Rio de Janeiro on Weds (1Aug2018)] received the award for work in algebraic geometry. [[User:Arcanechili|Arcanechili]] ([[User talk:Arcanechili|talk]]) 16:34, 3 August 2018 (UTC)

I've added a basic description of Abelian groups in the title text, and that's about as much as I know about such topics. I'm not sure what a "meta-Abelian group" is, is that an Abelian group of other groups? Also, could someone add basic descriptions of algebreic geometry and geometrical algebra? [[Special:Contributions/172.68.94.40|172.68.94.40]] 18:42, 3 August 2018 (UTC)

In the title text, since groups are a concept within mathematics, it seems odd to consider mathematics as a whole forming any sort of group within itself, which I suspect is the first part of the pun. Secondly, since groups involve the commutative property, I think the last part is a pun about the order of the words algebra and geometry, as if they're commutative themselves! [[User:Ianrbibtitlht|Ianrbibtitlht]] ([[User talk:Ianrbibtitlht|talk]]) 19:19, 3 August 2018 (UTC)
: I meant to say 'abelian' groups involve the commutative property, and the meta prefix is referring to the fact that it's about the names rather than the mathematical details - i.e. commutative in metadata only. [[User:Ianrbibtitlht|Ianrbibtitlht]] ([[User talk:Ianrbibtitlht|talk]]) 19:24, 3 August 2018 (UTC)

: I guess the joke is that informally mathematicians form a ''group'' (a number of people classed together), what would strictly be a ''set'' in mathematics. While in mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies specific conditions. --[[User:JakubNarebski|JakubNarebski]] ([[User talk:JakubNarebski|talk]]) 21:18, 3 August 2018 (UTC)

It's a false dilemma. Complex numbers ''are'' vectors (<math>\mathbb{C}</math> is a two-dimensional <math>\mathbb{R}</math>-vector space, and more generally every field is a vector space over any subfield), but that doesn't change anything about the fact that <math>i</math> is by definition a square root of -1. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 20:38, 3 August 2018 (UTC)

Fun factoid: not only is <math>\mathbb{C}</math> the unique proper field extension of finite degree over <math>\mathbb{R}</math> (since <math>\mathbb{C}</math> is algebraically closed), but the converse is true as well: <math>\mathbb{R}</math> is the only proper subfield of finite index in <math>\mathbb{C}</math>. They're like a weird married couple. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 20:53, 3 August 2018 (UTC)

Altho there are no "meta-abelian" groups there are metabelian groups. If xy=yx then the commutator [x,y]=xyx^{-1}y^{-1}=1. The group generated by the commutators -- the commutator subgroup -- is thus a measure of how far a group is from being abelian. A metabelian group is a nonabelian group whose commutator subgroup is abelian. Thus a metabelian group is one made of a stack of two abelian groups. It is "meta-abelian" in that sense. A standard example is the group of invertible upper-trianglular matrices. The commutators all have 1s on the diagonals.

explain xkcd - User contributions [en]

Talk:2028: Complex Numbers