

Is there infinite space between two points?




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I had a math teacher tell me in high school (35 years ago) that there is always a space between any two points. I follow that. But is it a fallacy to think that means an infinite amount of space between two points? If no matter how close two points are, you can fit another between them, where is the limit?
It doesn't seem that an infinite amount of points would mean an infinite amount of space... or does it?

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From a purely mathematical view, there is an infinite space between points, because you can reduce the decimal measurement into, essentially, infinity.
In terms of actual physical space, you are limited by the size of whatever you're trying to fit between the points.









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Yes, the notion of infinite space between two points is the basis for the mathematics known as Calculus (to some degree). The concept is you sometimes cannot get an equation to work if one number is basically infinite, but if you approach infinite by either splitting into smaller divisions or go exponentially large, you will see a real number still, that you can infer is so close to infinite, that number must be the answer if the factor was infinite. Mind you, this exists only inside the world of mathematics. Hopefully, that was a decent enough layman's explanation of it. I would leave it to others to give the technical answer.









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In the real world, there is (probably) not infinite space between two points. See Planck length  the smallest valid distance scale in our universe. How space behaves below that length is a subject of research  quantum effects may become dominant, making shorter distances invalid. Most likely we will need some new physics to describe it.









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The point to be considered is that the sum of an infinite set of numbers need not to be infinite, if the numbers tend to zero fast enough. For example for the infinitely many numbers 1, 1/2, 1/4 , ... we have 1+1/2+1/4+... =2, hence if you imagine the numbers 0, 1, 1+1/2, 1+1/2+1/4, .. as points on the real number line, then this infinitely many points will never exceed the point at 2 (in fact not even reach it)









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Originally Posted by Morpheus
The point to be considered is that the sum of an infinite set of numbers need not to be infinite, if the numbers tend to zero fast enough. For example for the infinitely many numbers 1, 1/2, 1/4 , ... we have 1+1/2+1/4+... =2, hence if you imagine the numbers 0, 1, 1+1/2, 1+1/2+1/4, .. as points on the real number line, then this infinitely many points will never exceed the point at 2 (in fact not even reach it)
Whoa.
Okay, if that number were in a decimal form, the numbers would go on without end. It might never reach the goal of "2", but there is no limit to the ever increasing number. Sort of mind blowing. That is a good illustration, thanks.

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Originally Posted by design219
Whoa.
Okay, if that number were in a decimal form, the numbers would go on without end. It might never reach the goal of "2", but there is no limit to the ever increasing number. Sort of mind blowing. That is a good illustration, thanks.
If you keep on computing this sequence forever, you'll keep getting closer and closer to π:
4  4/3 + 4/5  4/7 + 4/9  4/11 + 4/13  4/15 ...









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It depends on what you mean by »infinite«. In mathematics, there are two types of »infinite«, »countably infinite« and »uncountably infinite«. The concept of countably infinite is easier to comprehend: think of the positive integers (1, 2, 3, 4, …), these are countably infinite, because you can sit down and count them all. A set is now called »countably infinite« if you can give »house numbers« to each of its elements. The rational numbers, for instance, are countably infinite: those are numbers which are fractions q/p of two integers, p ≠ 0.
Uncountably infinite means that the set so large that you cannot number all its elements. Real numbers are uncountably infinite: it can be shown that numbers like sqrt(2) or pi cannot be written as a fraction.
So if you take any two real numbers a and b (which are not the same), then there are uncountably infinite real numbers and countably infinitely many rational numbers in between.
Although I wouldn't say there is an »infinite amount of space« in between: the notion »amount of space« is derived from a distance – and the distance is shrinking the closer the two numbers get.

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Originally Posted by Morpheus
The point to be considered is that the sum of an infinite set of numbers need not to be infinite, if the numbers tend to zero fast enough. For example for the infinitely many numbers 1, 1/2, 1/4 , ... we have 1+1/2+1/4+... =2, hence if you imagine the numbers 0, 1, 1+1/2, 1+1/2+1/4, .. as points on the real number line, then this infinitely many points will never exceed the point at 2 (in fact not even reach it)
This is how it was taught to me in grade school.
"Take an open door, and close it halfway. Then take the amount the door is open, and close it half that amount. Keep repeating this. Does the door ever close?"









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Since it is not obvious that 1+1/2+1/4+...=2 maybe an even simpler way to explain it is considering the two points 0.9 and 1: Now the infinitely many points 0.99, 0.999, 0.9999 all fit in the intervall of length 1/10.









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<pre>___0.999 = 1</pre>
That's really all you need to know.









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This is America.
We use fractions.









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Originally Posted by Morpheus
The point to be considered is that the sum of an infinite set of numbers need not to be infinite, if the numbers tend to zero fast enough. For example for the infinitely many numbers 1, 1/2, 1/4 , ... we have 1+1/2+1/4+... =2, hence if you imagine the numbers 0, 1, 1+1/2, 1+1/2+1/4, .. as points on the real number line, then this infinitely many points will never exceed the point at 2 (in fact not even reach it)
This.
If you want to see some ancient exploration of this, see Zeno's paradoxes, like his conjecture that an arrow should never reach its target because the space between it and the target is infinite as reducing points could, he held, go to an infinite number of points just like there are an infinite number of degrees between each degree on a 360 degree circle.
















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