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## Main Question or Discussion Point

I think a lot a users have vague concepts about the roots of unity.

I try to post a link to WolframAlpha, which calculates all the second roots

of unity

http://www.wolframalpha.com/input/?i=sqrt(1)

There you can see the input [itex]\sqrt{1}[/itex] and the plot of all roots in the complex

plane.

The roots are lying on the unit circle [itex]\ e^{i\alpha}[/itex] = cos[itex]\alpha[/itex]+i[itex]\dot{}[/itex]sin[itex]\alpha[/itex]

There are two real roots as you can see on the plot:

[itex]\sqrt{1}[/itex] = +1 (principal root)

[itex]\sqrt{1}[/itex] = -1

Wikipedia says that there exists a mathematical fallacy of the following kind:

1= [itex]\sqrt{1}[/itex] = [itex]\sqrt{(-1)\dot{}(-1)}[/itex] = [itex]\sqrt{-1}[/itex][itex]\dot{}[/itex][itex]\sqrt{-1}[/itex] = i[itex]\dot{}[/itex]i = -1

the fallacy is that the rule [itex]\sqrt{x\dot{}y}[/itex] = [itex]\sqrt{x}[/itex][itex]\dot{}[/itex][itex]\sqrt{y}[/itex] is not valid here according to Wikipedia:

http://en.wikipedia.org/wiki/Mathematical_fallacy#Positive_and_negative_roots

WolframAlpha implies no error. Which one should we trust? My guess is

Wikipedia is just wrong and WolframAlpha is correct.

I try to post a link to WolframAlpha, which calculates all the second roots

of unity

http://www.wolframalpha.com/input/?i=sqrt(1)

There you can see the input [itex]\sqrt{1}[/itex] and the plot of all roots in the complex

plane.

The roots are lying on the unit circle [itex]\ e^{i\alpha}[/itex] = cos[itex]\alpha[/itex]+i[itex]\dot{}[/itex]sin[itex]\alpha[/itex]

There are two real roots as you can see on the plot:

[itex]\sqrt{1}[/itex] = +1 (principal root)

[itex]\sqrt{1}[/itex] = -1

Wikipedia says that there exists a mathematical fallacy of the following kind:

1= [itex]\sqrt{1}[/itex] = [itex]\sqrt{(-1)\dot{}(-1)}[/itex] = [itex]\sqrt{-1}[/itex][itex]\dot{}[/itex][itex]\sqrt{-1}[/itex] = i[itex]\dot{}[/itex]i = -1

the fallacy is that the rule [itex]\sqrt{x\dot{}y}[/itex] = [itex]\sqrt{x}[/itex][itex]\dot{}[/itex][itex]\sqrt{y}[/itex] is not valid here according to Wikipedia:

http://en.wikipedia.org/wiki/Mathematical_fallacy#Positive_and_negative_roots

WolframAlpha implies no error. Which one should we trust? My guess is

Wikipedia is just wrong and WolframAlpha is correct.