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

ganesh
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Registered: 2005-06-28
Posts: 14,415

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: 14,415

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: 14,415

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
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Registered: 2005-06-28
Posts: 14,415

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: 15,522

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
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Registered: 2005-06-28
Posts: 14,415

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: 86,436

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

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: 14,415

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: 86,436

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

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: 14,415

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
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Registered: 2005-06-28
Posts: 14,415

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: 86,436

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

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: 14,415

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: 86,436

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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