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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

CN # 1

What is the value of

where i=√(-1)?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

Well Ricky, that was a cryptic answer.

You got it right!

*Well done !!!*

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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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: 1,905

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

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.

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,049

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

Hi anonimnystefy,

OK. I shall post problems here too.

CN#3. If

, then what is b equal to?Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,049

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

Hi bobbym and anonimnystefy,

CN #4. The complex number

lies in which quadrant?Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,049

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

Hi bobbym, anonimnystefy, and gAr,

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

I shall post more problems here soon!

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

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

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,049

hi ganesh

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,753

Hi;

The solution CN#6 is correct. Excellent, bobbym and zetafunc!

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

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