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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

CN # 1

What is the value of

where i=√(-1)?

Character is who you are when no one is looking.

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

"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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

Well Ricky, that was a cryptic answer.

You got it right!

*Well done !!!*

Character is who you are when no one is looking.

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

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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

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

You can express this so:

<==+1

IPBLE: Increasing Performance By Lowering Expectations.

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

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

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

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

hi ganesh

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,528

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi ganesh;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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