# A Complex Number Question

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I need help with this question, already discussed with my friend but both of us had no clue

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Here is the solution:

Tell me if there is anything unclear in the solution. Cheers!

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Okay so, w = 1/(1-z)

= (1-z)/(1-2z+z^2) [multiplying the numerator and denominator by (1-z)]

Now, z^2 = |z|^2 = 1 (I think so, I can justify it in my head)

So, w = (1-z)/(1-2z + 1)

w = (1-z)/(2-2z)

= 1/2 (1-z)/(1-z)

= 1/2

P.S. I Just saw that Vioh posted an answer up there and it's pretty good.

Also, lmao looking back, I don't know why i multiplied both the numerator and denominator by 1-z. I meant to do the conjugate, but clearly, the mistake led me to the solution.

Edited by Ossih
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Here is the solution:

Tell me if there is anything unclear in the solution. Cheers!

Thank you this helped so much

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Okay so, w = 1/(1-z)

= (1-z)/(1-2z+z^2) [multiplying the numerator and denominator by (1-z)]

Now, z^2 = |z|^2 = 1 (I think so, I can justify it in my head)

So, w = (1-z)/(1-2z + 1)

w = (1-z)/(2-2z)

= 1/2 (1-z)/(1-z)

= 1/2

P.S. I Just saw that Vioh posted an answer up there and it's pretty good.

Also, lmao looking back, I don't know why i multiplied both the numerator and denominator by 1-z. I meant to do the conjugate, but clearly, the mistake led me to the solution.

I'm not sure whether z^2=|z|^2, but I know zÂ·z*=|z|^2. Thanks you anyways!

ps. let z = a+bi, z^2=(a+bi)^2=a^2 + 2abi - b^2, whereas |a+bi|^2 = a^2 + b^2, so i don't think z^2 = |z|^2

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Okay so, w = 1/(1-z)

= (1-z)/(1-2z+z^2) [multiplying the numerator and denominator by (1-z)]

Now, z^2 = |z|^2 = 1 (I think so, I can justify it in my head)

So, w = (1-z)/(1-2z + 1)

w = (1-z)/(2-2z)

= 1/2 (1-z)/(1-z)

= 1/2

P.S. I Just saw that Vioh posted an answer up there and it's pretty good.

Also, lmao looking back, I don't know why i multiplied both the numerator and denominator by 1-z. I meant to do the conjugate, but clearly, the mistake led me to the solution.

I'm not sure whether z^2=|z|^2, but I know zÂ·z*=|z|^2. Thanks you anyways!

ps. let z = a+bi, z^2=(a+bi)^2=a^2 + 2abi - b^2, whereas |a+bi|^2 = a^2 + b^2, so i don't think z^2 = |z|^2

Hmm, well caught. In that case, I guess I got lucky . I don't know why, it made sense in my head.

Thank you then for correcting me

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