I Trouble with infinity and complex numbers

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Trouble with infinity and complex numbers, just curious.
Summary: Trouble with infinity and complex numbers, just curious.

I'm not too familiar with set theory ... but <-∞, ∞> contains just real numbers?
Does something similar to <-∞, ∞> exist in Complex numbers?
My question, is it "wrong"?
1568033300436.png
 

.Scott

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Yup, using your nomenclature, it would be (<-∞, ∞>, <-∞, ∞>).
Or a+bi where a and b are real numbers.

But one item of caution: ∞ is not a number - real or otherwise.
 
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Thanks...
And further.. when plotting functions with a imaginary axis... there can be more imaginary axis' than one I guess..
 

.Scott

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Thanks...
And further.. when plotting functions with a imaginary axis... there can be more imaginary axis' than one I guess..
If you are plotting a function with both a complex range and domain, you would need four dimensions: a+bi=f(c+di) - one each for a, b, c, and d with b and d being imaginary axis. Of course, drawing in four dimensions requires some creativity.
 
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Thanks...
And further.. when plotting functions with a imaginary axis... there can be more imaginary axis' than one I guess..
For complex numbers there is only one imaginary axis, for quarternions there are three and for octonions there are seven.
 

jbriggs444

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If you are plotting a function with both a complex range and domain, you would need four dimensions: a+bi=f(c+di) - one each for a, b, c, and d with b and d being imaginary axis. Of course, drawing in four dimensions requires some creativity.
If you are going to build a graph, you need four dimensions for the domain and another two for the range.
 

WWGD

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The thing is that there is a different way of "Approaching Infinity" as you have additional dimensions. In ##\mathbb R## , as you pointed out, you go along the +, - x-axis far right or left respectively. In , e.g., ## \mathbb R^2 ##, your set is unbounded if it is not contained in a ball of finite radius r. Similar in higher dimensions, where being contained in a ball of finite radius is equivalent to being bounded, while you "Go to infinity" by not being contained in balls of finite radius.
 
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If you are going to build a graph, you need four dimensions for the domain and another two for the range.
Is this really what you meant? For a function ##f : \mathbb C \to \mathbb C##? You have an extra two dimensions. For the domain you need only two dimensions, not four.
 
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For complex numbers it is better to add a single infinity to get the Riemann sphere.
 

WWGD

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There is a lot of material on compactifications of 1,2, etc points.
 

Delta2

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There is a way to "graph" a complex valued function similar to the way we graph a 2D vector field in just 2 dimensions. That is at every point of the domain (x,y) we plot an arrow which represents a vector that has its y-component equal to the Imaginary part of f(x+iy) and its x-component equal to the Real part of f(x+iy). I know this is not exactly a graph as we usually mean it but anyway..

Alternatively we can use 3 dimensions (where the z axis represent the real or the imaginary part of f(x+iy) )and do two graphs, one for the imaginary part and one for the real part.
 

jbriggs444

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Is this really what you meant? For a function ##f : \mathbb C \to \mathbb C##? You have an extra two dimensions. For the domain you need only two dimensions, not four.
Oops. I'd misread a function of two complex arguments.
 

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