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

ganesh
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Registered: 2005-06-28
Posts: 21,298

Complex Numbers

CN # 1

What is the value of


where i=√(-1)?


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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
Moderator
Registered: 2005-06-28
Posts: 21,298

Re: Complex Numbers

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

Well done !!! up


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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
Moderator
Registered: 2005-06-28
Posts: 21,298

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.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

Ricky
Moderator
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: 1,905

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: 21,298

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.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,016

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
The knowledge of some things as a function of age is a delta function.

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

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,298

Re: Complex Numbers

Hi anonimnystefy,

OK. I shall post problems here too.

CN#3.  If

, then what is b equal to?


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,249

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,016

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
The knowledge of some things as a function of age is a delta function.

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

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,298

Re: Complex Numbers

Hi bobbym and anonimnystefy,

CN #4. The complex number

lies in which quadrant?


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,249

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.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,016

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
The knowledge of some things as a function of age is a delta function.

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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: 21,298

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!


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,298

Re: Complex Numbers

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

.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,249

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,016

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
The knowledge of some things as a function of age is a delta function.

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

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,298

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

?


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 105,249

Re: Complex Numbers

Hi ganesh;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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#24 2016-06-02 05:44:41

zetafunc
Member
Registered: 2014-05-21
Posts: 1,351
Website

Re: Complex Numbers

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