Complex numbers: don't understand graph of 1/z

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Discussion Overview

The discussion revolves around understanding how to graph the function 1/z, where z is a complex number represented as x + iy. Participants explore the implications of this representation and the challenges of visualizing it in a graph.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about how to graph 1/z after manipulating it into the form (x - iy)/(x^2 + y^2).
  • Another participant questions what exactly is being graphed, whether it is x versus y or something else.
  • A different participant suggests that a 4-dimensional graph is necessary to fully represent the function, proposing the use of two 3-D graphs for the real and imaginary parts.
  • One participant claims that the graph is not complex and describes it as a diagonal line, suggesting it resembles the graph of x - iy, while also mentioning that x^2 and y^2 are "basically a whole number."
  • A later reply challenges the clarity of the previous statement, asking for clarification on what is meant by "basically a whole number" and reiterating the need for a clearer explanation of the graphing process.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on how to graph the function or what the graph represents, with multiple competing views and some confusion remaining in the discussion.

Contextual Notes

There are unresolved assumptions regarding the definitions of the terms used and the nature of the graph being discussed, as well as the mathematical steps involved in the graphing process.

james5
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1/z is 1/(x+iy)

however, i then multiply by the complex conjugate and get:

(x-iy)/(x^2+y^2)

now, how do i graph this?

thanks.
 
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What are you trying to graph? x versus y or what?
 
To fully graph it, you would need a 4-dimensional graph, with axes x,y,a,b, so that x+y\imath=\frac{1}{a+b\imath}.
You could have 2 3-D graphs, z=\Re\left({\frac{1}{x+y\imath}}\right) and z=\Im\left({\frac{1}{x+y\imath}}\right).
 
well, i think it's not that complex since the graph i made that is correct is just one that goes down diagonally... so, it's basically the graph of x-iy since x^2 and y^2 are basically a whole number...
 
james5 said:
well, i think it's not that complex since the graph i made that is correct is just one that goes down diagonally... so, it's basically the graph of x-iy since x^2 and y^2 are basically a whole number...

Perhaps it would be better if you explain what you are talking about! What do you mean by "basically a whole number"? And, as you were asked before, what exactly are you graphing?
 

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