[complex numbers] smallest positive integer? ? (1 Viewer)

[PG5]

In pg. 39 of Cambridge, example 20 b), could someone explain and show me how the book got n = 12?
and also... what's that k as an integral all about?

question is
b) If z = (1 + i)/(root3 - i), find the smallest positive integer n such that z^n is real, and evaluate z^n for this integer n

thnx again

 

[McLake]

(1 + i)/(root3 - i)
=(1+i)(root3+i)/(3 + 1)
= (root3-1 +i(root3+1))/4

Make this into polar form

so |z| = root((4 - 2root3)/16 + (4 + 2root3)/16)
= root(8/16)
= 2root2/4
= root2/2
= 1/root2

and arg(z) = arctan((root3+1)/4) / ((root3-1)/4)
= arctan(1 + 2/(root3 - 1))
= 5pi/12

so z = 1/root2 cis 5pi/12

so for z^n

@ = 5kpi/12n

for real we want @ = 0 or pi


hmm ....

 

[SoFTuaRiaL]

michael, u still got ur cambridge book with u
man, u must luv 4u !!

 

[McLake]

Originally posted by Da Monstar
michael, u still got ur cambridge book with u
man, u must luv 4u !!

No, I'm just working off the question PG5 wrote up ...

 

[ND]

Originally posted by PG5
In pg. 39 of Cambridge, example 20 b), could someone explain and show me how the book got n = 12?
and also... what's that k as an integral all about?

question is
b) If z = (1 + i)/(root3 - i), find the smallest positive integer n such that z^n is real, and evaluate z^n for this integer n

thnx again

its supposed to be k integer, not k integral.

"if z^n is real, then arg(z^n) = k*pi"

that is because a CN is only real if it has an arg that is a multiple of pi (or 0).

arg(z^n) = n*argz (from de moivre's theorem).
"therefore n*(5pi/12) = k*pi where k = 0, 1, 2, 3 etc."
".'. n = 12k/5 where k = 0, 1, 2, 3 etc."

so the smallest possible integer that n can be is 12, and that is when k = 5 (but you don't need to say that).

 

[PG5]

hmm... i gets.... thnx!