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

In summary, the conversation covers the topic of graphing a complex number and the difficulties that come with it. The initial equation is 1/z = 1/(x+iy), and after multiplying by the complex conjugate the resulting equation is (x-iy)/(x^2+y^2). To fully graph this equation, a 4-dimensional graph with axes x,y,a,b would be needed, or two separate 3-D graphs. However, the speaker suggests that it may not be as complex as initially thought and that the graph could simply be that of x-iy since x^2 and y^2 are both whole numbers. The conversation ends with a request for clarification on what exactly is being graphed.
  • #1
james5
6
0
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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  • #2
What are you trying to graph? x versus y or what?
 
  • #3
To fully graph it, you would need a 4-dimensional graph, with axes [tex]x,y,a,b[/tex], so that [tex]x+y\imath=\frac{1}{a+b\imath}[/tex].
You could have 2 3-D graphs, [tex]z=\Re\left({\frac{1}{x+y\imath}}\right)[/tex] and [tex]z=\Im\left({\frac{1}{x+y\imath}}\right)[/tex].
 
  • #4
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...
 
  • #5
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?
 

1. What are complex numbers and why do we use them?

Complex numbers are numbers that have both a real and imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part. We use complex numbers to represent quantities that cannot be expressed solely with real numbers, such as the square root of a negative number.

2. What is the graph of 1/z and why is it important?

The graph of 1/z is a visual representation of the inverse of a complex number. It is important because it helps us understand the behavior and properties of complex numbers. The graph is a hyperbola with two branches, one in the first and third quadrants and the other in the second and fourth quadrants.

3. How do I interpret the graph of 1/z?

The graph of 1/z can be interpreted as a transformation of the complex plane. The real axis becomes the x-axis and the imaginary axis becomes the y-axis. The hyperbola intersects the axes at the points (1, 0) and (0, 1), representing the values of 1 and i respectively. The branches of the hyperbola indicate the different directions and magnitudes of the complex numbers.

4. What is the relationship between the graph of 1/z and the unit circle?

The unit circle, which represents all complex numbers with a magnitude of 1, is closely related to the graph of 1/z. The points where the hyperbola intersects the unit circle correspond to the complex numbers with a magnitude of 1, while the points inside the unit circle represent complex numbers with a magnitude less than 1. The points outside the unit circle represent complex numbers with a magnitude greater than 1.

5. How can I use the graph of 1/z in practical applications?

The graph of 1/z can be useful in many practical applications, such as in electrical engineering and signal processing. It can also be used to solve complex equations and systems of equations. Additionally, the graph can help visualize and understand complex functions and their properties.

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