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This involves De Moivre's Theorem, heads up. So, given a number that you're supposed to find all of the complex roots for, in this case, the cube roots of 1000, i'm failing to understand where an instructional i'm on suddenly gets this:
"The number 1000, or 1000 + 0i, is a complex number written in rectangular form. The polar notation for a complex number a + b*i is r(cos[theta] + i*sin[theta]), where r is the absolute value of the complex number and theta is the angle in standard position whose terminal side passes through the point (a.b).
(cont'd below)
 one year ago
 one year ago
This involves De Moivre's Theorem, heads up. So, given a number that you're supposed to find all of the complex roots for, in this case, the cube roots of 1000, i'm failing to understand where an instructional i'm on suddenly gets this: "The number 1000, or 1000 + 0i, is a complex number written in rectangular form. The polar notation for a complex number a + b*i is r(cos[theta] + i*sin[theta]), where r is the absolute value of the complex number and theta is the angle in standard position whose terminal side passes through the point (a.b). (cont'd below)
 one year ago
 one year ago

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SchrodingerBest ResponseYou've already chosen the best response.0
"...The complex number 1000, or 1000 + 0i, is shown below in polar form. 1000(cos [0 degrees] + i sin [0 degrees])." How did they get to theta being zero, all of a sudden? I just don't understand how they got the angle, at all. I'm assuming it has to do with 0i?
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
first off to find the cube roots of 1000 is identical to finding the cube roots of 1 and then multiplying the result by 10
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
I'm not concerned with that at the moment, mainly just the angle part.
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
the angle is 0 because in the complex plane the real numbers live on the \(x\) axis
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
dw:1352818446497:dw
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
I don't know what that means, I just don't understand the significance of that statement at all. xP
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
(I'm selfteaching, so i'm not totally aware of everything that's being talked about.)
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
ok you want the angle right? and the way you make an angle for unit circle trig is you start here dw:1352818720308:dw
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
meaning that in the complex plane, all positive real numbers make an angle of 0 and all negative real numbers have an angle of \(\pi\)
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
it is also of course true that \(\cos(0)=1\) and \(\sin(0)=0\) so for any positive number say \(1000\) you can write \[1000=1000\left(\cos(0)+i\sin(0)\right)\] in a way it is a rather silly thing to write, except that you are about to use it to find cube roots
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
I'll have to look up whatever the "complex plane" is, but that can't be true, in the way that i'm typically thinking of it. And positive real numbers of what? How do numbers by themselves simply form an angle? I can imagine a number being represented as a vector just forming a line, but numbers themselves don't generate an angle, unless that's basically what you mean when you say all positive real numbers produce an angle of zero.
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
also notice that the representation is not unique, since it is also true that \[\cos(2\pi)=1\] and \[\sin(2\pi)=0\] meaning \[1000=1000\left(\cos(2\pi)+i\sin(2\pi)\right)\]
 one year ago

satellite73Best ResponseYou've already chosen the best response.2
the real line lives inside the complex plane
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
I don't quite get it yet, but that's just me. Thanks for your help so far.
 one year ago

phiBest ResponseYou've already chosen the best response.0
Here is a little background http://www.khanacademy.org/math/algebra/complexnumbers/v/complexnumberspart1
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
Thanks. Sorry, I would reply promptly but for some reason, Openstudy causes my computer to start randomly tripping balls as hard as possible.
 one year ago

phiBest ResponseYou've already chosen the best response.0
This little app shows the complex plane http://www.khanacademy.org/math/algebra/complexnumbers/e/the_complex_plane notice that you can use polar coordinates (r (distance) and theta (angle from the xaxis) to define a point on the plane, rather than x+ iy (rectangular form) the distance r (use pythagoras) is sqrt( x^2+ y^2) the angle is the arc tangent of (y/x)
 one year ago

SchrodingerBest ResponseYou've already chosen the best response.0
Okay, this helps a lot. I thought that was potentially how coordinates would be plotted in the imaginary plane, but I didn't want to assume. Thanks!
 one year ago
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