Exploring the Concept of "Infinite Points in a Straight Line"

[nine]

Hi
My thought might be a bit weird or maybe childish, but I can not find anything to object it or to prove that it is wrong.
As a basic definition of a straight line that it is an infinite number of points that are connected together.
Regardless the length of that straight line, it consists of infinite number of points.
I've thought about breaking that line into two halves, so I would get one half of infinity(starting from zero and going to +ve infinity and omitting the other half from -ve infinity to zero), what if I keep breaking the line for infinite times? the conclusion would be that the straight line is nothing but a point?
If it was an infinite number of points, I would always get infinity even if I break it to an infinite number of times, right?
I am really confused and would like someone to help me how to think it out.

 

[pallidin]

My 2-cents worth:

A single, non-moving point can not define a line because the concept of a vector would not exist(with respect to a "line")

You need at least 2 points.

 

[X1088LoD]

it will approach one point, but never actually be one point, it will just be an infinitesimal small line, which can be halved and halved and halved infinitly, the world could be a very small place, that's my thought at least. if you look at it as a specific size point, it may eventually get to a point where the line is so small it looks like one point, but it won't be

 

[nine]

it will approach one point, but never actually be one point, it will just be an infinitesimal small line, which can be halved and halved and halved infinitly, the world could be a very small place, that's my thought at least. if you look at it as a specific size point, it may eventually get to a point where the line is so small it looks like one point, but it won't be

From that point of view, there is no real point? because no matter how small that point is, it can be seen as a straight line from other point of view?

 

[X1088LoD]

on the contrary, i would say that there is a real point, but you cannot turn a line into a real point, and likewise you cannot turn a point into a line. Visually, yes, what you might say is the case, but eyes are deceiving. In reality, I would look at a point as a dot, zoom in a billion times, its still a dot, etc, but with a line, you zoom in enough, eventually you will see it is a line

 

[Werg22]

What does it mean to have an infinite number of points connected together? What does it mean to "see" a straight line? In analysis, a straight line is a function which rate of change is constant. The word straight is not relevant; it is just a, how would you say, visual definition.

 

[Gib Z]

From that point of view, there is no real point? because no matter how small that point is, it can be seen as a straight line from other point of view?

You're all thinking of a point as a little dot. A point is not a little dot, it is what it is called, a point. It doesn't matter how small our little dot is.

Say we want it to represent co ordinate (0,0), with a dot of radius r. It doesn't matter how tiny out dot is, it excompasses all points which satisfy $$x^2+y^2 < r^2$$. Since the reals are dense in the sense between any 2 non-equal reals in another real, it encompasses an infinite number of values.

I sure as hell hope you don't think of fundamental particles like you do points, if you do physics..

Basically, NO DOTS.

 

[Gib Z]

on the contrary, i would say that there is a real point, but you cannot turn a line into a real point, and likewise you cannot turn a point into a line. Visually, yes, what you might say is the case, but eyes are deceiving. In reality, I would look at a point as a dot, zoom in a billion times, its still a dot, etc, but with a line, you zoom in enough, eventually you will see it is a line

Did you mean, with a curve, zoom in enough and see it becomes linear? Because that is only true of differentiable functions. Try fractals.

 

[nine]

That was helpful, thanks a lot all

 

[jing]

Trying to grasp the concept infinity mathematically can be difficult at first. Well done for trying. Think about what you wrote in the OP

As a basic definition of a straight line that it is an infinite number of points that are connected together.

What do you mean by connected together? What connects the points?

I've thought about breaking that line into two halves, so I would get one half of infinity(starting from zero and going to +ve infinity and omitting the other half from -ve infinity to zero)

What is one half of infinity?

For example take the positive integers

1, 2, 3, 4, 5, 6, 7, .............

an infinite set.

These split into two halves the odds and the evens


Take the half with the evens

2, 4, 6, 8, 10, .............

now start counting them. To do this put the count above the number

1, 2, 3, 4, 5, .....
2, 4, 6, 8, 10, ...

We may have split the numbers into two halves but there is as many in each half as we started with.

You may find this site interesting

http://www.mattababy.org/~belmonte/Publications/Books/CSaW/5_infinity.html

 

[deepanshumishra]

line can't be a point

From that point of view, there is no real point? because no matter how small that point is, it can be seen as a straight line from other point of view?

How can one consider a line as a point I coulden't understood from which point of view u can consider it as a point

 

[Werg22]

It needs to be remembered that there's a difference between analysis and the physical. A point is a singularity and has thus no physical equivalent; you can't represent it with a dot. Seeing a straight line is just seeing the more or less accurate physical representation of a certain function. The word "dot" and "connected" are not relevant when it comes to analysis.

 

[nine]

well, I need two known points to draw a straight line, this is clear, but when this straight line is drawn, there in an infinite number of points connecting the two points I have connected together.
for example
f(x) = 2x + 5
f(0) = 5
f(1) = 7
now, connect these two points on the graph paper, you get a straight line, right?
how many points lay on that line? the answer is infinity
now, resubstitute in the equation
f(0) = 5
f(0.5) = 6
now the line is broken into half, but still having infinite number of points
if and I say if I substitute this way
f(0) = 5
f(0+s) = 5
where s a very small fraction that tends to be zero.
so, what I've got here?
two points, but no straight line?

 

[JasonRox]

What does it mean to have an infinite number of points connected together? What does it mean to "see" a straight line? In analysis, a straight line is a function which rate of change is constant. The word straight is not relevant; it is just a, how would you say, visual definition.

How does Analysis come into play here?

What he means that all the points connected is that if you cross the line with another line, it will intersect at some point. Of course he didn't say that, but it's kind of implied. It's all semantics. He's not a mathematician.

For example, if the line were filled with just rational points, it is not "connected" in the OP's definition. Why?

Anyways, let's move on with the OP's question.

The answer is no it never becomes a point.

Say for example the length of the line is 64. The distance between one end point and the other. (You'll understand in a second why I picked this nice number.

Divide that line in half and we get 32.

Again, and we get 16.

Again, 8.

Again, 4.

Again, 2.

Again, 1.

Again, 1/2.

Again, 1/4.

Again, 1/8.

Again, 1/16.

Again 1/32.

See what's going on here? The line ALWAYS has length. It does converge to 0 though, but it is never 0. So, the line always has endpoints, call them a and b. They certainly are different points because b = a + e, where e is the length of the line. But how many points are there between a and b? There are infinitely many of course!

If this line contained only the rationals, then it still has infinitely many points and that's not even fully "connected" in the OP's definition!

 

[JasonRox]

The word "dot" and "connected" are not relevant when it comes to analysis.

Really?

The real line is connected, and Analysis does in fact take note of this. And connectedness is a big property and not something irrelevant. Also, "dots" are also relevant because "dots" are limit points to sequences.

 

[JasonRox]

What does it mean to have an infinite number of points connected together? What does it mean to "see" a straight line? In analysis, a straight line is a function which rate of change is constant. The word straight is not relevant; it is just a, how would you say, visual definition.

Straight or linear, it's the same thing. Like I said, it's all semantics.

 

[nine]

Say for example the length of the line is 64. The distance between one end point and the other. (You'll understand in a second why I picked this nice number.

Divide that line in half and we get 32.

Again, and we get 16.

Again, 8.

Again, 4.

Again, 2.

Again, 1.

Again, 1/2.

Again, 1/4.

Again, 1/8.

Again, 1/16.

Again 1/32.

See what's going on here? The line ALWAYS has length. It does converge to 0 though, but it is never 0. So, the line always has endpoints, call them a and b. They certainly are different points because b = a + e, where e is the length of the line. But how many points are there between a and b? There are infinitely many of course!

If this line contained only the rationals, then it still has infinitely many points and that's not even fully "connected" in the OP's definition!

So, if I take the limit of x when x tends to zero, what do I get?

 

[JasonRox]

So, if I take the limit of x when x tends to zero, what do I get?

Um... 0.

 

[Werg22]

Really?

The real line is connected, and Analysis does in fact take note of this. And connectedness is a big property and not something irrelevant. Also, "dots" are also relevant because "dots" are limit points to sequences.

But the way he used connected pertains to the visual. Certainly if connectivity is analogous to continuity, analysis does take this into account. Because of continuity you see the visual representation as being "connected". However, if you see a "connected" line, it doesn't necessarily mean that it represents a continuous function; it could very well be discreet and defined on very close x's. And by dots, I meant big dots made with pencils. Not singular points. What I'm trying to point out is that the visual interpretation of things in the physical world are not necessarily translated to analysis. Simply saying "connected" and "dots" is not true to the definitions of analysis. But this debate is kind of useless; it's more philosophical than mathematical really.

 

[billiards]

My 2 cents, and i don't know how qualified I am to say this (probably not very!) but a line is not an infinite number of dots. To me, a dot is a point, it has 0 dimensions, no length shape or size, nothing - in that sense it doesn't really physically exist, except as a mathematical tool to define a position, I suppose it fits into that other mathematical realm which is order. If you have infinite many dots, you just have infinite points in space, because each of these points have no length then they cannot make up the entire line, because the line must have a length > 0, and the sum of the lengths of all the points would be 0.

 

[JasonRox]

But the way he used connected pertains to the visual. Certainly if connectivity is analogous to continuity, analysis does take this into account. Because of continuity you see the visual representation as being "connected". However, if you see a "connected" line, it doesn't necessarily mean that it represents a continuous function; it could very well be discreet and defined on very close x's. And by dots, I meant big dots made with pencils. Not singular points. What I'm trying to point out is that the visual interpretation of things in the physical world are not necessarily translated to analysis. Simply saying "connected" and "dots" is not true to the definitions of analysis. But this debate is kind of useless; it's more philosophical than mathematical really.

What does continuity have to do with connectedness? Who said a "connected" line must be continuous? This makes no sense at all because there is no domain, range, or even function to say anything about continuity.

However, if you see a "connected" line, it doesn't necessarily mean that it represents a continuous function; it could very well be discreet and defined on very close x's.

Can you give me an explicit example? Remember, it must look as though it's "connected" in your definition, but actually theoritically be discontinuous.

And stop bringing Analysis into this. The question has nothing to do with it. It's basically an ancient Greek mathematics question, and that is exactly how I interpreted it. So, it's more like Geometry.

 

[Werg22]

Ok, I give up. There's no point taking the thread nor the question there...

 

[JasonRox]

because each of these points have no length then they cannot make up the entire line, because the line must have a length > 0, and the combined length of all the points would be 0.

This is where it gets tricky. I'm not 100% sure how to answer it, but logic fails here though.

Because what you're saying here is that you're counting all the zeroes and it adds to zero. The truth is that you can't even count all the zeroes! There are uncountably many points in a line, so you can't count them all even though they're all zero.

 

[JasonRox]

well, I need two known points to draw a straight line, this is clear, but when this straight line is drawn, there in an infinite number of points connecting the two points I have connected together.
for example
f(x) = 2x + 5
f(0) = 5
f(1) = 7
now, connect these two points on the graph paper, you get a straight line, right?
how many points lay on that line? the answer is infinity
now, resubstitute in the equation
f(0) = 5
f(0.5) = 6
now the line is broken into half, but still having infinite number of points
if and I say if I substitute this way
f(0) = 5
f(0+s) = 5
where s a very small fraction that tends to be zero.
so, what I've got here?
two points, but no straight line?

No! That is not two points. What are the points?

You said as s tends to 0, the function tends to 5, but then you have f(0+(0))=5.

Which is basically one point back again and that is only 0. Your logic fails because you found the limit of the function as s goes to 0 and you forgot to take the limit of s as s goes to 0, which is 0 itself.

 

[JasonRox]

Ok, I give up. There's no point taking the thread nor the question there...

Personally, I think it's a great question.

 

[Werg22]

No! That is not two points. What are the points?

You said as s tends to 0, the function tends to 5, but then you have f(0+(0))=5.

Which is basically one point back again and that is only 0. Your logic fails because you found the limit of the function as s goes to 0 and you forget to take the limit of s as s goes to 0, which is 0 itself.

But here I think theory and representation overlap incorrectly... Distance is defined as the square root of the sum of the squares of the respective translations needed for one point to become the other. Adding the length of an infinite number of points does not mean anything to me.

 

[JasonRox]

Adding the length of an infinite number of points does not mean anything to me.

Points have no length, so it is 0 length each point.

So, this doesn't make sense to you:

0+0+0+0+...?

The answer is clearly 0.

 

[Werg22]

Hummm. What I meant is that it doesn't mean anything to add points as if you were adding infinitesimals. Distance is to be taken as the "he square root of the sum of the squares of the respective translations needed for one point to become the other". All other definitions of distance are subsequent to that first definition. I don't believe "adding points" takes away or adds anything to that definition.

 

[JasonRox]

Hummm. What I meant is that it doesn't mean anything to add points as if you were adding infinitesimals. Distance is to be taken as the "he square root of the sum of the squares of the respective translations needed for one point to become the other". All other definitions of distance are subsequent to that first definition. I don't believe "adding points" takes away or adds anything to that definition.

http://en.wikipedia.org/wiki/Metric_space

Check the definition of metric. Basically, it's a distance function.

So, what happens when you measure the distance from x to x?

Well, d(x,x)=0.

 

[Werg22]

http://en.wikipedia.org/wiki/Metric_space

Check the definition of metric. Basically, it's a distance function.

So, what happens when you measure the distance from x to x?

Well, d(x,x)=0.

d(x,x)=0, what's the point of bringing this up? No one contested that.

 

[JasonRox]

d(x,x)=0, what's the point of bringing this up? No one contested that.

You're saying it doesn't make sense to measure a point. That is a one right there.

 

[jing]

f(0) = 5
f(0+s) = 5
where s a very small fraction

No f(0+s) = 2s+5 and if s =/= 0 then f(0+s) =/= 5 no matter how small s is.

As s tends to 0 then f(0+s) tends to 5 but is only =5 when s=0

Look at JasonRox's example for $$1/2^n$$ n an integer

as n gets larger $$1/2^n$$ tends to 0 but no matter how large n is $$1/2^n$$ will still be none zero

 

[HallsofIvy]

Points have no length, so it is 0 length each point.

So, this doesn't make sense to you:

0+0+0+0+...?

The answer is clearly 0.

The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.

 

[Werg22]

You're saying it doesn't make sense to measure a point. That is a one right there.

Let me clarify: it doesn't make sense to get the measure of a point for the purpose of using it as a part of a distance.

 

[JasonRox]

The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.

That's what I talked about. I was just talking about that case.