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#1 2006-03-05 16:43:05

ganesh
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Registered: 2005-06-28
Posts: 15,130

Complex Numbers

CN # 1

What is the value of


where i=√(-1)?


Character is who you are when no one is looking.

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#2 2006-03-05 16:51:07

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Complex Numbers


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-03-05 17:18:53

ganesh
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Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Well Ricky, that was a cryptic answer.
You got it right!

Well done !!! up


Character is who you are when no one is looking.

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#4 2006-03-05 17:21:41

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Complex Numbers

I thought we were doing complex numbers.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2006-03-05 17:40:07

ganesh
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Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Yes, we are, Ricky.
You gave Euler's identity as the solution, I was referring that.

CN # 2

Find the fifth roots of unity and their sum.


Character is who you are when no one is looking.

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#6 2006-03-05 17:41:08

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Complex Numbers

Heh, yea, it was a bad pun.  I gave a complex solution...


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2006-03-05 19:04:24

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Complex Numbers

You can express this so:


<==+1


IPBLE:  Increasing Performance By Lowering Expectations.

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#8 2011-07-25 17:35:23

namealreadychosen
Member
Registered: 2011-07-23
Posts: 16

Re: Complex Numbers

a^2+b^2=1
b/a=tan(n*360/5)

1, 0.31+0.95i, 0.81+0.59i, -0.31-0.95i, -0.81-0.59i
sum=1

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#9 2011-07-25 17:57:01

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Hi namealreadychosen,

In polar form, 1 = cos(2πk) + i sin(2πk) for any integer k.
==> The 5 fifth roots of unity are given by
cos(2πk/5) + i sin(2πk/5) for k = 0,1,2,3,4.
----------------------
If you want the answer not in trigonometric form, we need to be more crafty.

Since x^5 - 1 = 0 is the equation for the fifth roots of unity:
(x - 1)(x^4 + x^3 + x^2 + x + 1) = 0.

The first factor yields x = 1.

As for the second factor, rewrite it as
x^2 + x + 1 + 1/x + 1/x^2 = 0 (divide both sides by x^2)
==> (x^2 + 1/x^2) + (x + 1/x) + 1 = 0
==> [(x + 1/x)^2 - 2] + (x + 1/x) + 1 = 0.

Letting z = x + 1/x yields
z^2 + z - 1 = 0.

Now, we have a quadratic in z!
==> z = (-1 ± √5) / 2.

Now, we solve for x.
Since z = x + 1/x = (-1 ± √5) / 2,

2x^2 - x[-1 ± √5] + 2 = 0.

The plus sign yields
x = [(1 - √5) ± sqrt((6 - 2√5) - 16)] / 4
= [(1 - √5) ± i * sqrt(10 + 2√5)] / 4.

The negative sign yields
x = [(1 + √5) ± sqrt((6 + 2√5) - 16)] / 4
= [(1 + √5) ± i * sqrt(10 - 2√5)] / 4.

In summary, the five fifth roots of unity (in radical form) are
x = 1, [(1 - √5) ± i * sqrt(10 + 2√5)] / 4, [(1 + √5) ± i * sqrt(10 - 2√5)] / 4.


Character is who you are when no one is looking.

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#10 2011-07-26 02:49:52

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: Complex Numbers

hi ganesh

why don't you continue posting problems here?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#11 2011-07-26 16:10:05

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Hi anonimnystefy,

OK. I shall post problems here too.

CN#3.  If

, then what is b equal to?


Character is who you are when no one is looking.

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#12 2011-07-26 21:00:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,482

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#13 2011-07-26 21:14:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: Complex Numbers

hi ganesh

thanks


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#14 2011-07-26 23:45:34

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Hi bobbym and anonimnystefy,

CN #4. The complex number

lies in which quadrant?


Character is who you are when no one is looking.

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#15 2011-07-26 23:52:36

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,482

Re: Complex Numbers

Hi ganesh;

Yes, you are right, I had that and forgot about the root of 3! A really stupid mistake!


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#16 2011-07-27 00:09:15

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: Complex Numbers

hi ganesh


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#17 2011-07-27 00:15:54

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Complex Numbers

Hi ganesh,



"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#18 2011-07-27 00:30:53

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Hi bobbym, anonimnystefy, and gAr,

The solutions CN #3 and CN  #4 are correct. Brilliant!

I shall post more problems here soon!


Character is who you are when no one is looking.

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#19 2011-07-28 00:01:40

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

CN #5. Find the real and imaginary parts of the complex number

.


Character is who you are when no one is looking.

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#20 2011-07-28 00:42:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,482

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#21 2011-07-28 02:26:41

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: Complex Numbers

hi ganesh


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#22 2011-07-29 00:37:49

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,130

Re: Complex Numbers

Hi bobbym and anonimnystefy,

The solution CN#5 is correct. Neat job!

CN #6. If

is a cube root of unity, then what is the value of

?


Character is who you are when no one is looking.

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#23 2011-07-29 03:01:36

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,482

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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