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June 26th, 2012, 04:20 AM  #1 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Complex complex numbers
Hi, complex numbers again, 1.) Where 1 and 2 are subscripts, Another way of proving (z1/z2)* = z1*/z2* is to start with (z1/z2)* x z2*. Show how this can be done. By letting z1 be a+bi and z2 be c+di, and then expanding LHS and factorizing again, I got RHS therefore proving (z1/z2)* = z1*/z2* My problem comes with the second part, when I multiply (z1/z2)* by z2*, z1*/z2* x z2* Canceling common factors, Isn't it just z1*?! How do I prove that (z1/z2)* x z2*= z1*/z2* ? 2.) Prove that for all complex numbers z and w, zw* z*w is purely imaginary or zero. By letting z be a+bi and w be c+di and expanding, I got the imaginary part, which was 2(bcad)i. How do I prove that its zero..Is it because it lies on the imaginary yaxis, which includes 0? Greatly appreciated, Thank you so much! 
June 26th, 2012, 09:43 AM  #2 
Senior Member Joined: Aug 2008 Posts: 113 Thanks: 0  Re: Complex complex numbers
Hello Tutu, For your own convenience, please try to solve both the problems using polar representation of the complex numbers. (1) Try to show that (z1/z2)* x z2* = z1* (2) If you get the result 2(bcad)i, where a, b, c, and d are reals, your result must be either purely imaginary or zero. Let me know if you could follow the argument. Best regards, 
June 26th, 2012, 10:36 AM  #3 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Complex complex numbers
Hi thanks a lot! For question 1.) I got z1*, but to me the question seems to be asking for me to prove z1*/z2* from (z1/z2)* x z2*. That was how I understood the question, which I think is wrong because (z1/z2)* x z2* gets me z1* and not z1*/z2* . For question 2.) I understand that with 2(bcad)i as a result, my result is purely imaginary because of the i, but I dont see how it equals to 0.. Thank you so much! 
June 26th, 2012, 11:00 AM  #4 
Senior Member Joined: Aug 2008 Posts: 113 Thanks: 0  Re: Complex complex numbers
Let me try to clarify: 1. If you can prove that (z1/z2)* x z2* = z1*, then it is equivalent to prove that (z1/z2)* = z1*/z2* provided z2* is not equal to 0. Hence if you could prove that (z1/z2)* x z2* = z1*, you are done! 2. The result can't be purely imaginary AND 0 at the same time. That is why you have been asked to prove that the result is either purely imaginary OR zero. Have you understood the argument now? Best regards, 
June 26th, 2012, 11:08 AM  #5 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Complex complex numbers
Ahh yes. Can I ask, is 0 real or imaginary?

June 26th, 2012, 11:23 AM  #6  
Senior Member Joined: Aug 2008 Posts: 113 Thanks: 0  Re: Complex complex numbers Quote:
However, in the context of your problem, if ad = bc, then you get a zero and otherwise you get a pure imaginary number. Best regards,  
June 26th, 2012, 12:07 PM  #7 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Complex complex numbers
Thanks! So 2(bcad)i is my result, it is a purely imaginary number. Can I say that since ad does not equal to bc, it is a purely imaginary number? Actually, which part tells that ad does not equal to bc? Sorry and thank you! 
June 26th, 2012, 12:56 PM  #8  
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Complex complex numbers Quote:
If bcad is nonzero then your result is purely imaginary. If bcad is zero, then your result is zero. So, your result is purely imaginary, or it's zero.  
June 26th, 2012, 01:07 PM  #9 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Complex complex numbers
I get it, thanks! 
June 26th, 2012, 01:15 PM  #10 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Complex complex numbers
Conventionally, do I have to write all that as a conclusion to the question, just o prove that it's zero OR imaginary? a,b,c,d could be any real numbers. So bcad may or may not be zero. If bcad is nonzero then your result is purely imaginary. If bcad is zero, then your result is zero. So, your result is purely imaginary, or it's zero.[/quote] 

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