Solving De Moivre's Theorem: Help Needed

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Discussion Overview

The discussion revolves around the application of De Moivre's Theorem to simplify a complex number raised to a power, specifically focusing on the expression \([- (1/2) + i \sqrt{3}/2]^3\). Participants explore the steps involved in separating the expression into real and imaginary parts and the potential discrepancies in their results.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion over consistently arriving at the answer "-1" when using De Moivre's Theorem, while another method yields "1".
  • Some participants clarify that the expression is already separated into real and imaginary parts, identifying the real part as \(-1/2\) and the imaginary part as \(\sqrt{3}/2\).
  • Another participant suggests that the goal may be to express the complex number in the form \(a e^{i \theta}\), although this is met with some disagreement.
  • One participant correctly identifies the expression as \([- (1/2) + i \sqrt{3}/2]^3\] and confirms that it means the square root of 3.
  • Another participant applies De Moivre's Theorem, calculating the right-hand side and concluding it should equal "1".
  • There is a correction regarding the imaginary unit, with a participant noting that "i" is the standard term rather than "iota".
  • One participant points out a potential mistake in the interpretation of the complex number, suggesting that a mix-up may have occurred in the signs of the components.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct interpretation or simplification of the expression. Multiple viewpoints and potential errors are discussed, indicating ongoing uncertainty about the correct application of De Moivre's Theorem in this context.

Contextual Notes

There are unresolved issues regarding the interpretation of the original expression and the steps taken to simplify it. Participants express different understandings of the problem, leading to conflicting results.

sarah786
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I have practiced solving de moivre's theorem... but i don't know why i keep getting the wrong answer in a particular question although i have checked a hundred times and havn't found a mistake...
this is the question i am talking about:
Simplify the following ( separate into real and imaginary parts ):
[-(1/2) + iota*(sq.root3)/2]

I have tried a hundred times using de moivre's theorem and i get "-1" each time... but when i do the same question using the formula (a+b)^3, i get "1" ... moreover, i have got a solution lest and there it says the answer should be "1" ... please if you can do the steps for me, i'll be very thankful and it'll be a big help... i will then be able to check where i am wrong... thank you...
 
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Isn't it already separated?

Real part = -1/2
Imaginary part = root(3)/2
 
no, i have to remove that power "3" somehow... i have to write a complex number to the power 1...
 
i have to "simplify" it and write the "simplified form" as a complex number...
 
From the looks of it, they want you to go the other direction and write it in a form like a e^{i \theta}, is that what the question is asking for?
 
olivermsun said:
From the looks of it, they want you to go the other direction and write it in a form like a e^{i \theta}, is that what the question is asking for?

no... i think i have very clearly stated my question...
 
Please write the expression in a different way. I don't see any power of three.

If (sq.root3) means \sqrt{3} then it's already answered. If it means something else, then it's not obvious.
 
oopsss... sorryy... yeahh thanks for telling ... the question is:
[-(1/2) + iota*(sq.root3)/2]^3
 
and yea... it means square root of 3
 
  • #10
de Moivre's theorum:
[cos(x) + i sin(x)]n = cos(nx) + i sin(nx)

n=3
cos(x) = -1/2
sin(x) = sqroot(3)/2
x=120deg

the RHS is
cos(3*120deg) + i sin(3*120deg) = 1

So 1 looks OK.
 
  • #11
By the way, in the standard English, at any rate, the imaginary unit is "i" not "iota".
1/2+ i\sqrt{3}/2 in polar form is cos(\pi/3)+ i sin(\pi/3).
Its third power is given by cos(3(\pi/3))+ i sin(3(\pi/3))= cos(\pi)+ i sin(\pi)= -1

As Unrest says, perhaps you made the same mistake I did, and used 1/2+ i\sqrt{3}{2} instead of -1/2+ i\sqrt{3}/2!
 
Last edited by a moderator:
  • #12
HallsofIvy said:
cos(\pi)+ i sin(\pi)= 1

Now, as to why you are getting "-1"..

Maybe sarah did the same as you! ;)
 
  • #13
Unrest said:
Maybe sarah did the same as you! ;)

Yes.. i did this same mistake :smile:
 

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