3 cube roots, 4 fourth roots, and N nth-roots of -1

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Discussion Overview

The discussion revolves around finding expressions for the cube roots, fourth roots, and general nth roots of -1, with a focus on the use of complex numbers and graphical representations. The context includes challenges faced by a student in a computational class that does not assume prior knowledge of advanced mathematics.

Discussion Character

  • Homework-related
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the professor's instructions regarding the use of graphs and vectors to understand the roots of -1.
  • Another participant suggests that understanding imaginary numbers and complex arithmetic is essential for solving the problem.
  • A different participant mentions that representing all nth roots of -1 is challenging without using imaginary numbers.
  • One participant proposes forming a polynomial equation to find the roots through analytic factoring, noting that this is manageable for small n but becomes complex for larger n.
  • Several participants share links to external resources, including articles on roots of unity and Argand diagrams, although one participant finds these resources unhelpful.

Areas of Agreement / Disagreement

There is no consensus on how to approach the problem, with participants expressing varying levels of understanding and suggesting different methods. Some emphasize the necessity of complex numbers, while others focus on polynomial equations.

Contextual Notes

Participants highlight the lack of assumed mathematical knowledge in the course, which may contribute to the difficulties in understanding the concepts discussed.

MWidhalm18
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My "Computational World" class is supposedly an intro to computer science without programming, but the questions are all over the place. I'm completely stuck on this question:

Show expressions for:
(a). 3 cube roots of -1
(b). 4 fourth roots of -1
(c). In the general case, N nth-roots of -1

My professor said that we could use graphs with vectors. He showed us an example of the 3 cube roots of 1, but I have no idea what any of that meant. I unfortunately have no work to show for my attempt because I have no idea what he means. Please help
 
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The Electrician said:
Everything you need to know is explained here:

http://en.wikipedia.org/wiki/Root_of_unity

Sorry, but that actually doesn't help at all. I have no idea what 98% of that article is saying
 
Do you know what imaginary numbers are? Have you studied complex arithmetic?
 
The Electrician said:
Do you know what imaginary numbers are? Have you studied complex arithmetic?

The class is a 1000 level course and the professor assumes absolutely no mathematical or physics knowledge beforehand. I have taken algebra, trig, and calc but it was at least 2 years ago and I don't want to make the problem harder than I need to. It's very basic
 
Boy, this is going to be a tough one. I don't know how you can represent all the nth roots of -1 without the use of imaginary (or complex) numbers.

To find these roots uses the square root of -1, otherwise you just can't do it.

Here are some links to show the use of graphs and vectors to represent imaginary numbers. Maybe they will help.

http://en.wikipedia.org/wiki/Argand_diagram

http://www.daviddarling.info/encyclopedia/A/Argand_diagram.html

http://demonstrations.wolfram.com/ArgandDiagram/
 
form the corresponding polynomial equation and find all the roots through analytic factoring. for n=1...4 this isn't too difficult, but for larger n, i wouldn't do this. at the very least you need to know what complex number is and why they come up in polynomial equations.
 

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