I would like to ask a fundamental question with regards to the imaginary number and it is something many beginners are told are wrong, but I would like to seek a rigorous proof of why it is wrong. It is a question I faced when my student asked me this. Take for example, (−1)1/6. We can compute this in 2 different ways:
(−1)1/6=[(−1)2]1/12=1
or
(−1)1/6=[(−1)1/3]1/2=(−1)1/2=i
I understand both the methods above are wrong, and the typical response is that you cannot square −1. Is there a rigorous proof as to why this method is flawed? Perhaps using abstract algebra or Galoise Theory?
Thank you!
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