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What is de Moivre's theorem?

de Moivre worked out a brilliant and beautiful way to solve complex equations.

If you for example have z3= 1 and you want to find all real and complex z that satisfy this equation. i is just a complex number written in rectangular form; z = 1 + i * 0 = cos(µ) + i sin(µ) . If you remember your specific angles for sine and cosine you want an angle that gets cosine(µ) = 0 and sine(µ)=1 so µ = 0+ 2nπ. Now, the + 2nπ is especially important for reasons you should see soon.

Rewrite i in euler form 1 = ei * (0 + 2nπ) = z3.  Now, if we take the cube root of both sides, that will be the same as taking it to the power of 1/3.

Remember your power rules: (ab)= ab * c. This will give you z = ei(0+n2π)/3. This is where the + 2nπ gets really important. If not, the answer would just be one. Put in the different values for n (n=0, n=1, n=2, etc) gives you your different angles. Plug them on your Argand diagram, and you get three different solutions: 

z= 1 ; z= ei 2π/3 and z2 = ei 4π/3

Hope it helped. If not, well take it in the session

Answered by Frederik Dahl M. Maths tutor

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Answered by Frederik Dahl M.
Maths tutor

172 Views

See similar Maths IB tutors
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