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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Important concepts in math usually show up naturally in many apparently unrelated areas.  Each area will name and define a concept that makes sense for the problems being considered.  One of the joys of math is proving that multiple, apparently unrelated, definitions are equivalent.  When definitions are equivalent you cannot pick "the one true definition" -- any of them will do.  However the principle of maximum laziness leads to the one with the easiest notion being used as a canonical definition.[[Special:Contributions/162.158.75.130|162.158.75.130]] 18:17, 7 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)
 
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)

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