• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Patterns found in complex numbers

  • Thread starter lll030lll
  • Start date
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

Science Advisor
Homework Helper
25,790
246
welcome to pf!

hi lll030lll! welcome to pf! :smile:
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 :redface:

(personally, i find degrees easier … 30°, 30° ± 120° :wink:)
 
Re: Patterns found in complex numbers URGENT!!!!

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

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top