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    • Thread Starter

    there is 1 question and an example, in the fp2 edexcel book, asking us to find the equation of a circle with points w on its circumference, generally looking like this:


    do the cubed roots of unity always make a circle centre 0,0 ?

    and what is the 'roots of unity' jargon ?

    • Study Helper

    Study Helper
    (Original post by Andamiriel)
    Might be useful:


    The article in Wikipaedia is a bit OTT for A-level.
    • PS Helper

    PS Helper
    An "nth root of unity" is a number which, to the nth power, equals 1. Note that you can write e^{2i\pi} = 1 = 1^k = e^{2k\pi i} for any integer value of k k, and so if \omega = e^{\frac{2k \pi i}{n}} then \omega^n = 1. Numbers of this form are the roots of unity.

    Why must they lie on the unit circle? Well if z=re^{i\theta} and z^n = r^ne^{i\theta} = 1 then what is r^n? So what is r? (Considering that r is a positive real number you're fairly limited to what it can be).

    In this case you have w^3 = -1 = e^{i\pi}, so you can proceed in a similar way, but be careful with that minus sign. Or alternatively notice that w^3 = -1 \iff (-w)^3 = 1 and go about it that way (which is probably easier).

    the cubed roots of unity are the complex numbers z such that z^3 = 1. So obviously 1 is one, but there are 2 others (there are n nth roots of unity).

    These do always lie on a circle centred around (0,0) with a radius of 1. So, all you have to do is find the z so that z^3 = -1 (this is the same as finding the cubed roots of unity called (-z)) and show that they satisfy the equation of a circle radius one on the origin.
    • Thread Starter

    right thanks for clearing that up
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Updated: April 3, 2011
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