Trouble with infinity and complex numbers

In summary, the trouble with infinity and complex numbers is that there is no "wrong" answer, but using the nomenclature given, the two concepts are different. Additionally, when plotting functions with a imaginary axis, there can be more imaginary axis' than one.
  • #1
Troxx
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TL;DR Summary
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
 
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  • #2
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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  • #3
Thanks...
And further.. when plotting functions with a imaginary axis... there can be more imaginary axis' than one I guess..
 
  • #4
Troxx said:
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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  • #5
Troxx said:
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.
https://en.m.wikipedia.org/wiki/Octonion
 
  • #6
.Scott said:
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.
 
  • #7
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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  • #8
jbriggs444 said:
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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  • #10
There is a lot of material on compactifications of 1,2, etc points.
 
  • #11
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.
 
  • #12
Mark44 said:
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.
 
  • #13
jbriggs444 said:
Oops. I'd misread a function of two complex arguments.
I thought that might be it.
 

1. What is infinity and why is it considered a problem in mathematics?

Infinity is a concept that represents something that is boundless or endless. In mathematics, it is used to describe a quantity that has no limit or endpoint. The problem with infinity arises when it is used in calculations or equations, as it can lead to undefined or contradictory results.

2. How are complex numbers different from real numbers?

Complex numbers are numbers that have both a real and imaginary component, represented as a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Real numbers, on the other hand, are numbers that can be represented on a number line and do not have an imaginary component. Complex numbers are used to solve equations that cannot be solved with real numbers alone.

3. What is the difference between a finite and an infinite set?

A finite set is a set that has a limited number of elements, while an infinite set has an unlimited number of elements. For example, the set of all even numbers is an infinite set, while the set of all letters in the alphabet is a finite set. In mathematics, the concept of infinity is used to describe the size of an infinite set.

4. Can infinity be used as a number in calculations?

No, infinity cannot be used as a number in calculations. It is a concept that represents something that is unbounded or endless. Using infinity as a number can lead to undefined or contradictory results. In some cases, it can be used as a limit in calculus, but it is not considered a number in the traditional sense.

5. How do mathematicians deal with the concept of infinity in their work?

Mathematicians have developed various theories and techniques to deal with the concept of infinity in their work. One approach is to use limits and approximations to avoid using infinity as a number. Another approach is to use set theory and cardinality to compare the sizes of infinite sets. Additionally, complex numbers and other mathematical tools have been developed to handle calculations involving infinity.

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