anonymous
  • anonymous
If z being a complex number lies on the unit circle and the sum of z with its reciprocal is 1, then find z^235
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
wt bt this??
amistre64
  • amistre64
z^235? thats alot
anonymous
  • anonymous
theres no answer given bt frm what i know it comes out to be a simple answer

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amistre64
  • amistre64
z + 1/z = 1?
anonymous
  • anonymous
yeah
amistre64
  • amistre64
z^2 +1 ------- = 1; z^2 +1 = z; z^2 -2 +1 = 0 Z
amistre64
  • amistre64
(z-1)^2; z = 1
amistre64
  • amistre64
z^235 = 1 then ?
anonymous
  • anonymous
but z isnt 1, its a complex number...
amistre64
  • amistre64
typoed it again
anonymous
  • anonymous
z is the 3rd root of -1. because z^2-Z+1=0; z^3=1 z^235=z=-1+isqrt(3) /2
amistre64
  • amistre64
z^2+1 = 1 z^2 = 0 z=0 ?
anonymous
  • anonymous
Z^3=-1 (z+1)(z^2-z+1)=0 Z^2-z+1=0;
amistre64
  • amistre64
so that means the the complex part is either up or down
anonymous
  • anonymous
z+1/z=1 z^2-z+1=0.
anonymous
  • anonymous
i think dipankars right, did u apply the demoivres theorem??
anonymous
  • anonymous
no need to apply it....
amistre64
  • amistre64
dip is usually right ;)
anonymous
  • anonymous
i have given the steps....
anonymous
  • anonymous
z is the complex roots of -1.
anonymous
  • anonymous
so but then howdu get z^235??
anonymous
  • anonymous
u must realize amistre m doing wht u often do;)
amistre64
  • amistre64
guess wildly lol
anonymous
  • anonymous
z^235=z^234*z=(z^3*78)^z=1*z=z
anonymous
  • anonymous
no not that...like wht u were doin with the tangent plane question...
anonymous
  • anonymous
ok dipankar....
amistre64
  • amistre64
ahhh... :)
anonymous
  • anonymous
with all the three ;)
amistre64
  • amistre64
its good to be smart :)
anonymous
  • anonymous
him i think u have understood the process...
anonymous
  • anonymous
yeah right...
amistre64
  • amistre64
so, when I find z=0, the complex parts have to be pure imaginaries then right? like "i"
anonymous
  • anonymous
ill tell u what
anonymous
  • anonymous
z + 1/z =1 let z=cosθ + isinθ 1/z = cosθ - isinθ z + 1/z = 2cosθ = 1 cosθ = 1/2 hence z = 1/2 + √3/2
amistre64
  • amistre64
z^235 = i^235...235/4 = 58 and 3 remaining i, -1, -1 z^235 = -i?
anonymous
  • anonymous
√3/2 x i sorry
anonymous
  • anonymous
no man look above z = 1/2 + i√3/2
amistre64
  • amistre64
lol...stab in the dark then :) i see what you did
anonymous
  • anonymous
now du wanna know the other one as well?
amistre64
  • amistre64
i do, but at this late hour it aint gonna stick to well ;)
anonymous
  • anonymous
hahaha...wts d time there??

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