Sketch in an Argand Diagram

I have worked the values out to be:

I just do not know how to draw this.

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- January 18th 2009, 08:54 PMronaldo_07[SOLVED] Sketch z^6 = 1 on an Argand diagram
Sketch in an Argand Diagram

I have worked the values out to be:

I just do not know how to draw this. - January 18th 2009, 09:17 PMJhevon
i suppose you mean ??

in that case, your values are wrong.

when you find the correct ones (can you?), see if you can continue after looking at this - January 18th 2009, 09:22 PMronaldo_07
- January 18th 2009, 09:36 PMJhevon
first draw (faintly) a circle of radius 1. you are going to draw the points that correspond to each complex number on that circle. the x and y coordinates correspond to the real and imaginary parts of the complex number respectively. the angles you find (called "arguments") tell you how to locate the complex number based on its angle from the positive x-axis.

example, one of the answers is . this would be your in the diagram you see on the site i gave you. begin by measuring an angle of anti-clockwise from the positive x-axis, indicating in with a line going from the origin as you see in the diagram. then, wherever it touches the circle, that's the complex number. plot a point there, and you're done - January 18th 2009, 09:38 PMJhevon
- January 18th 2009, 09:40 PMronaldo_07

=

because I need I divide by 6

was I wrong to do this? - January 18th 2009, 10:05 PMJhevon
- January 18th 2009, 10:16 PMronaldo_07
**This is the full question**

How many complex numbers are there such that Calculate them all and sketch them in the Argand diagram.

I think there are 6 complex numbers. is this correct?

For this question do I sketch with valuse or the x values?

Sorry to cause confusion - January 18th 2009, 10:35 PMJhevon
yes. an (non-constant) nth degree polynomial over the complex numbers has n roots (Fundamental Theorem of Algebra)

Quote:

For this question do I sketch with valuse or the x values?

Sorry to cause confusion

i will continue using the form you have shown me, but i believe another form would be easier to work with, that is, Euler's equation: . working with would have made the process easier to see.

anyway, you have

you want this to equal 1. you should note two things. first, note one solution, the most basic and obvious one is the one you should think of (here, that would be x = 0). the second thing you should note, is that if an angle is a solution, then that angle plus any multiple of is also a solution. thus, the solutions are of the form, , or simply , where is an integer.

so you have:

but now, to solve for , simply divide the angles by 6. we have

now, let take on the values 0, 1, 2, 3, 4, and 5. (it will start to repeat after that. in general, for something like , you would proceed in this manner and let take on the values )

got it?

now finish up. i'm going to bed (Yawn) - January 18th 2009, 10:46 PMronaldo_07