Infinite set of points on a line.

In summary: When we talk about limits, we are talking about a point where the function or number becomes undefined. For example, if we were to take a limit as n-->∞ on the line segment from 0 to 1, the line segment would become undefined at n=∞.
  • #1
redrum419_7
58
0
If a line segment of size L, is made up of an infinite amount of points. Then divided into half, and then half again, divided into half to infinity, would this mean that the resulting line segment is a point made up of an infinite amount of points?

Or would it be ∞/∞?
 
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  • #2
redrum419_7 said:
Then divided into half, and then half again, divided into half to infinity

What does this mean?? What does "to infinity" mean??
 
  • #3
What does it mean to do a thing to infinity? Until you are clear on that your question makes no sense.

There is an accepted way to define what you are trying to express which is the intersection of all the line segments.

Let us consider the case where we start with the line segment from 0 to 1 and always choose the left half when we divide. So,
Let L0 be the line segment from 0 to 1.
Let L1 be the line segment from 0 to 1/2.
Let L2 be the line segment from 0 to 1/4.
...
Let Ln be the line segment from 0 to 1/2^n
...


Now let L be the set of points obtained by doing this infinitely many times whatever that means. Clearly whenever we divide we can only remove points, never add. Thus if a point is in L, then it must also be in
L1,L2,L3,...
since L is contained in all of these.

So what points are contained in
L1,L2,...?
Is -1 in all these? Is 0? Is 1/2? Is 1? Is 2?
Once you answer this question I think you should have an idea of how to answer your question in most simple cases.

BTW infinity/infinity doesn't really make any sense in this context, or in most mathematical context for that matter.
 
  • #4
Line is made up an infinitely amount of points right? Then wouldn't half of that line, still being a line, be made up of an infinitely amount of points?
 
  • #5
micromass said:
What does this mean?? What does "to infinity" mean??

Taking a limit as n-->∞
 
  • #6
redrum419_7 said:
Line is made up an infinitely amount of points right? Then wouldn't half of that line, still being a line, be made up of an infinitely amount of points?

Yes.

redrum419_7 said:
Taking a limit as n-->∞

And what does that mean?? Limits is usually defined for functions or numbers. You're working with sets. So what do limits mean?
 

What is an infinite set of points on a line?

An infinite set of points on a line refers to a collection of points on a straight line that extends infinitely in both directions. This means that there is no end to the number of points that can be placed on the line.

How is an infinite set of points on a line different from a finite set of points on a line?

A finite set of points on a line has a specific and limited number of points, whereas an infinite set has an unlimited number of points. Additionally, the spacing between points in an infinite set is equal, while in a finite set the spacing may vary.

What is the importance of an infinite set of points on a line in mathematics?

Infinite sets of points on a line play a crucial role in mathematical concepts such as geometry, calculus, and number theory. They allow for the exploration of concepts such as continuity, limits, and infinity, which are essential in many areas of mathematics.

Can an infinite set of points on a line be visualized?

While it is impossible to physically visualize an infinite set of points on a line, we can use mathematical techniques such as graphs and equations to represent and understand these concepts.

How is an infinite set of points on a line related to the concept of infinity?

An infinite set of points on a line is closely related to the concept of infinity because it represents an uncountable number of points that extend indefinitely. This infinite concept is crucial in many areas of mathematics and helps us understand the vastness of the universe.

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