Two questions on complex numbers. If both 2 and (- I + i (3^.5)), respectively, are cube roots of 8, does that mean the latter expression = 2?
ok I will ask a different question:
\(x^2=4\\ x=+2 \;\; or \;\; x=-2\\ \text{so take away the positive bit which is just a convention}\\ \sqrt4=+2 \quad or \;\;-2\\ \text{Does that mean that +2=-2 ?} \)
or looking ati it slightly differently
\((-3)^2=9\\ (+3)^2=9\\ \therefore\;\;-3=+3\)
Second question, in the book the Mathematical Universe by W. Dunham, on page 292 he says 'It is easy to see that i ^ 2 = (-1^.5)^2 = -1'. Why is this 'easy' to see? Is it because the i squared is defined as -1?
He means it is easy to see because i^2=-1 but he is wrong because he has not used brackets properly.
By not using brackets properly he has not followed proper convention.
He has overlooked this because in the context of what he was talking, his meaning was clear, he thought so anyway :)
\((-1^{0.5})^2=(-1*1^{0.5})^2=(-1*\sqrt1)^2=(-1*1)^2=(-1)^2=1\)
but
\(((-1)^{0.5})^2=(\sqrt{-1})^2=(i)^2=-1 \)
I have checked and this calculator does return 1. It is correct.
I like questions like this because it means you are really trying to meanings sorted properly :)
