Editing Imaginary unit (section)

==={{math|''i''}} vs. {{math|−''i''}}===
{{anchor|i and -i}}
Being a [[quadratic polynomial]] with no [[multiple root]], the defining equation {{math|1=''x''{{isup|2}} = −1}} has {{em|two}} distinct solutions, which are equally valid and which happen to be [[additive inverse|additive]] and [[multiplicative inverse]]s of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled {{math|+''i''}} (or simply {{mvar|i}}) and the other is labelled {{math|−''i''}}, though it is inherently ambiguous which is which.

The only differences between {{math|+''i''}} and {{math|−''i''}} arise from this labelling. For example, by convention {{math|+''i''}} is said to have an [[Argument (complex analysis)|argument]] of <math>+\tfrac\pi2</math> and {{math|−''i''}} is said to have an argument of <math>-\tfrac\pi2,</math> related to the convention of labelling orientations in the [[Cartesian plane]] relative to the positive {{mvar|x}}-axis with positive angles turning [[anticlockwise]] in the direction of the positive {{mvar|y}}-axis. Also, despite the signs written with them, neither {{math|+''i''}} nor {{math|−''i''}} is inherently positive or negative in the sense that real numbers are.<ref>{{cite book |first1=Apostolos K. |last1=Doxiadēs |first2=Barry |last2=Mazur |year=2012 |title=Circles Disturbed: The interplay of mathematics and narrative |page= [https://books.google.com/books?id=X9Uoug4lNWkC&pg=PA225 225] |edition=illustrated |publisher=Princeton University Press |isbn=978-0-691-14904-2 |via=Google Books |url=https://books.google.com/books?id=X9Uoug4lNWkC}}</ref>

A more formal expression of this indistinguishability of {{math|+''i''}} and {{math|−''i''}} is that, although the complex [[field (algebra)|field]] is [[unique (mathematics)|unique]] (as an extension of the real numbers) [[up to]] [[isomorphism]], it is {{em|not}} unique up to a {{em|unique}} isomorphism. That is, there are two [[automorphism|field automorphisms]] of the complex numbers <math>\C</math> that keep each real number fixed, namely the identity and [[complex conjugation]]. For more on this general phenomenon, see [[Galois group]].