the fundamental theorem

ray · April 21st, 2012, 08:48 PM

1) could you show me how you prove that x� = 8 has three roots?
is it (x-0) * (x+0) ....
2) Was this theorem invented after the Complex numbers, or it has a justification of its own?
3) What is the practical, concrete use of the other roots, apart from x=2 ?
thanks.

MarkFL · April 21st, 2012, 09:04 PM

1.) Write the equation as:

$x^3-2^3=0$

Use the difference of cubes formula:

$(x-2)$x^2+2x+4$=0$

So, the real root is $x=2$ and the complex conjugate roots we find with the quadratic formula:

$x=\frac{-2\pm2i\sqrt{3}}{2}=-1\pm i\sqrt{3}$

2.) I believe complex numbers were conceived of before the FTOA.

3.) To give one example, without the complex roots many theorems in classical physics governing harmonic motion would be incomplete.

ray · April 21st, 2012, 10:25 PM

Thanks, Mark.
That is how you find the roots.
My question is how you prove that a monotonously increasing function has 3 roots?

Could you give me a link re point 3? How can the root af any odd power create a problem in physics?
Thanks

ray · April 21st, 2012, 10:31 PM

Oh. I forgot,
If I got it right you cannot find the complex roots without finding the real root? is that so?
Thanks

ray · April 22nd, 2012, 12:00 AM

one more question
what is i� ? and sqrt(i)?

MarkFL · April 22nd, 2012, 12:10 AM

ray · April 22nd, 2012, 03:50 AM

Quote:

Originally Posted by MarkFL

$i^3=i^2\cdot i=-i$ $\sqrt{i}=\frac{1+i}{\sqrt{2}}$

Thanks, can you please specify if you can find the imaginary root without knowing the real one?

April 21st, 2012, 08:48 PM	#1
ray Member Joined: Nov 2011 Posts: 45 Thanks: 0	the fundamental theorem 1) could you show me how you prove that x� = 8 has three roots? is it (x-0) * (x+0) .... 2) Was this theorem invented after the Complex numbers, or it has a justification of its own? 3) What is the practical, concrete use of the other roots, apart from x=2 ? thanks.
$ray is offline$

April 21st, 2012, 09:04 PM	#2
MarkFL Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs	Re: the fundamental theorem 1.) Write the equation as: $x^3-2^3=0$ Use the difference of cubes formula: $(x-2)\(x^2+2x+4\)=0$ So, the real root is $x=2$ and the complex conjugate roots we find with the quadratic formula: $x=\frac{-2\pm2i\sqrt{3}}{2}=-1\pm i\sqrt{3}$ 2.) I believe complex numbers were conceived of before the FTOA. 3.) To give one example, without the complex roots many theorems in classical physics governing harmonic motion would be incomplete.
$MarkFL is offline$

April 21st, 2012, 10:25 PM	#3
ray Member Joined: Nov 2011 Posts: 45 Thanks: 0	Re: the fundamental theorem Thanks, Mark. That is how you find the roots. My question is how you prove that a monotonously increasing function has 3 roots? Could you give me a link re point 3? How can the root af any odd power create a problem in physics? Thanks
$ray is offline$

April 21st, 2012, 10:31 PM	#4
ray Member Joined: Nov 2011 Posts: 45 Thanks: 0	Re: the fundamental theorem Oh. I forgot, If I got it right you cannot find the complex roots without finding the real root? is that so? Thanks
$ray is offline$

April 22nd, 2012, 12:00 AM	#5
ray Member Joined: Nov 2011 Posts: 45 Thanks: 0	Re: the fundamental theorem one more question what is i� ? and sqrt(i)?
$ray is offline$

April 22nd, 2012, 12:10 AM	#6
MarkFL Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs	Re: the fundamental theorem $i^3=i^2\cdot i=-i$ $\sqrt{i}=\frac{1+i}{\sqrt{2}}$
$MarkFL is offline$

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