Editing Talk:2028: Complex Numbers
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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) | 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) | ||
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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) | 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. | 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. | ||
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