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ⁿ√1 isn't just equal to 1! There are actually 𝑛 different complex numbers such that 𝑧ⁿ=1. Geometrically, these 'roots of unity' correspond to the 𝑛 different angles that when rotated 𝑛 times successively around the unit circle return to the same placepic.twitter.com/ifwVmxKlqi
Note: There is a bit of notational convention i didn't adhere to. The ⁿ√ symbol is usually reserved for the 'principal root' of 𝑧 so that ⁿ√ denotes a single valued function of 𝑧 .pic.twitter.com/PHW17NNoeu
This was the concept that made me drop advanced algebra my first year in college. Later on I found it to be most beautiful thing!
Nice illustration and as a Maths teacher I'll be stealing it, but don't you mean z^5 = 1 rather than the notation you use? I thought convention was that the nth root (singular) just referred to e^2ipi/5, like how sqrt1 = 1, not +/-1
A group of units, if you will ;)
Yes a nice way using complex numbers and the circle. Also: z^5-1=(z-1)(z^4+z^3+z^2+z+1)=0 The quartic eqn can also be solved algebraically and reduced to a cubic one or quadratic ones by an appropriate substitution ;-)
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