Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

roots of equation

I do not understand the last part, how they get the iff statement. Where did 3pi/4 come from?

Could someone please explain? Thanks.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: roots of equation

Modulus-argument form is useful here.

z^4 = r(cos θ + isin θ) = r^4(cos (4θ) + isin(4θ))

e^iπ = cos π + i sin π

∴ r^4 = 1; 4θ = π (mod 2π)
r = 1; θ = π/4 (mod π/2)

So actually, in addition to the four stated, there are infinitely many other solutions to this as well.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: roots of equation

So actually, in addition to the four stated, there are infinitely many other solutions to this as well.

http://www.mathisfunforum.com/viewtopic.php?id=12635

Last edited by JaneFairfax (2009-08-28 14:06:48)

ac

Member

Registered: 2009-08-19

Posts: 8

Re: roots of equation

Can I say that a periodic function has the same value in infinite multiple values?

you can look to

as a circle, one perspective of a periodic function...

in that sense, you have infinite multiple time values with the same angle value, the solution of your equation...

If you don't reduce the angle to a circle you will count the rotations, coded in the angle value... but it is only a different representation of the periodic function... the value of the function is the same...

hope this is useful...

Last edited by ac (2009-08-28 09:11:26)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: roots of equation

Jane, you need to calm down.  I think it's clear that mathsyperson intended to convey that there are an infinite amount of ways to write these roots, especially considering the fact that e^(i*t*pi) is periodic for real variable t.

You can't get mad if someone makes a mistake.  It hurts the forum when you do, and it encourages others not to post.  But I've said this a million times before and I know you have no intention of listening this time.

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"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..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: roots of equation

Sorry, my mistake, I must have had something weird for breakfast.

I think I was probably trying to make the point that the given answer isn't the only correct one.
For example, π/4, 3π/4, 5π/4, 7π/4 would work just as well.

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: roots of equation

Hi Jane;

Why do I bother posting anything at all if no-one reads it?

You know that I read them and so does a lot of other people. Posting is not entirely for the benefit of others, by explaining we learn it ourselves. Also it is your responsibilty to help, to share what you know.

Last edited by bobbym (2009-08-29 06:15:00)

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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.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: roots of equation

Oh dear ...

Jane, could you tone down your response? A polite correction is all that is needed.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman