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May 5th, 2011, 09:32 PM   #1
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Complex number

1. Determine the value of m if z= ?5+mi/4-2?5i is a pure complex number and find this real number.

2. Express (1+i)^3/(1-i)^5 in the form a+ib, where a and b are real. Hence, find the argument and modulus of this complex number.

I need help. I can't even solve one of the question. It is an exam question. Please show me step by step how you solving the question. Thank you.
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May 5th, 2011, 09:46 PM   #2
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Re: Complex number

1.) What is a pure complex number?

2.) One way to do this is through use of Euler's formula:



Now you should be able to easily compute the argument and modulus.
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May 5th, 2011, 09:55 PM   #3
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Re: Complex number

MarkFL,

I think by 'pure complex' is meant the same as 'pure imaginary', i.e. it lies on the imaginary axis.

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May 5th, 2011, 10:08 PM   #4
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Re: Complex number

I thought that was probably the case, but I wasn't sure...thanks!

In that case, it appears we should set:







I don't know what you mean by "find this real number" because z and m are imaginary.
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May 5th, 2011, 10:30 PM   #5
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Re: Complex number

MarkFL,

You are being deceived by the poor type-setting employed in the OP. I am convinced it should read



In this case, I think the answer is real.

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May 5th, 2011, 10:39 PM   #6
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Re: Complex number

Aw jeeez! This isn't the first time such a thing has happened!

I'm so entrenched in the use of parentheses for clarity, I tend to assume everyone else is too.
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May 5th, 2011, 10:44 PM   #7
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Re: Complex number

It's probably better for your health to give the benefit of the doubt as you do! I for one should be scoping out good deals on cemetery plots, given my level of faith in my fellow man.

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May 6th, 2011, 03:22 AM   #8
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May 6th, 2011, 07:44 AM   #9
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Re: Complex number

I still don't understand question 1.
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May 6th, 2011, 07:55 AM   #10
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Re: Complex number

hoyy1kolko,

If we write



then we want to "realize" (my version of "rationalize", not standard) the denominator, so multiply top and bottom by the complex conjugate of the denominator, namely 4+2?5i to get



At this point, since you want a 'pure imaginary' number, you must have the real part of the above vanish, so you set



and you are done.

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