Patterns found in complex numbers

[lll030lll]

Patterns found in complex numbers URGENT!

• use de moivre's theorem to obtain solutions to z^n = i for n=3, 4, 5
• generalise and prove your results for z^n = 1+bi, where |a+bi|=1
• what happens when |a+bi|≠1?

Relevant equations[/b]
r = √a^2 + b^2
z^n = r^n cis (nθ)


This is what i have done:
z^3=i
z^3=i cis(0)
z^3=cis(π/2+2kπ),k=0,1,2
z=cis(π/6+2kπ/3),k=0,1,2
z=cis(π/6),cis(π/6+2π/3),cis(π/6+4π/3)
z=√3/2+0.5i,-√3/2+0.5i,- √3/2-0.5i

but the 3rd solution is incorrect, should be -i. what have i done wrong?

 

[tiny-tim]

welcome to pf!

hi lll030lll! welcome to pf!

z=cis(π/6+2kπ/3),k=0,1,2
z=cis(π/6),cis(π/6+2π/3),cis(π/6+4π/3)
z=√3/2+0.5i,-√3/2+0.5i,- √3/2-0.5i

but the 3rd solution is incorrect, should be -i. what have i done wrong?

dunno, but π/6+4π/3 = 9π/6

(personally, i find degrees easier … 30°, 30° ± 120° )

 

[lll030lll]

anyway, got it right in using a+bi=re^iθ