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# infinity? watch

1. I'm vaguely amused by the fact that the real number line is a closed set, yet its boundary is empty. It's somewhat mind-boggling.
2. There is no actual end to the number line. However, there does seem to be some coonfusion (slighlty off topic) as to why a circle's circumference is not exact.
A rational number can be expressed in the form (p/q). So, taking a rational number (such as 2) and splitting it half. And then one of those pieces in half again. And so on... you can create an infinite series that will CONVERGE to 2.

The difference with pi is that it is IRRATIONAL. A series can be used to converge to pi, but you cannot terminate the series, or add up infintely many terms. So, the circle's circumference can be 'trapped' between values, yet will not have an exact value.
3. It has an exact value, defined by the limit of the series. It just so happens that value is not rational.
4. (Original post by DeanK2)
There is no actual end to the number line.
Loosely speaking, lies at both ends of the number line following one-point compactification.

A rational number can be expressed in the form (p/q). So, taking a rational number (such as 2) and splitting it half. And then one of those pieces in half again. And so on... you can create an infinite series that will CONVERGE to 2.
I don't know what you think this is proving; it's not exactly hard to produce an infinite series that will CONVERGE (sic) to pi, y'know.
5. It isn't hard to produce one at all. When using pi in calculations, you will ALWAYS obtain (unless dividing pi by pi etc) an irrational number hence circumference can only be trapped between two values on the number line, but can never be thought of as a point.
6. But the definition of convergence means that the limit is trapped between two values that can be arbitrarily close, therefore, it can be considered a point on the real number line.
7. But pi is a transcendental number and therefore is not required to be given a point on the number line even though the sequence converges.
8. I don't know what number line you have in mind, but I can assure you that is a point on the real number line.
9. But if it did have a point wouldn't that imply it was rational?
10. No. You don't actually know what the real number line is, do you?
11. (Original post by DeanK2)
But if it did have a point wouldn't that imply it was rational?
lol? the real numbers include the rational and irrational ones
12. Just because they are in the set of real numbers does not mean an exact value (and hence point) can be given to pi.
13. (Original post by DeanK2)
Just because they are in the set of real numbers does not mean an exact value (and hence point) can be given to pi.
Er, it does, actually. As you'd know if you actually knew what a real number is.
14. By claiming that it cannot be defined on the real line, you are either assuming that it isn't a real number, or it is more than one number. Both are absurd.
Just because it is irrational, doesn't mean it's not on the line, in fact, there are infinitely many more irrational numbers on the number line than rational ones.
15. I understand the set of real numbers contains both rational and irrational numbers, but am confused as the how the number line can have a point for pi - even though it does make sense that there should be a point. I guess I'm confused as to how a number cana be irrational yet represented by a point.
16. (Original post by Zhen Lin)
It has an exact value, defined by the limit of the series. It just so happens that value is not rational.
yes the value defined, but, but not rational....
17. (Original post by DeanK2)
I understand the set of real numbers contains both rational and irrational numbers, but am confused as the how the number line can have a point for pi - even though it does make sense that there should be a point. I guess I'm confused as to how a number cana be irrational yet represented by a point.
it is defined, and as such it has a definite position on the line. it just happens to be irrational. it is a little hard to conceive, as, the more decimal places of pi we accept, the value of pi 'jiggles' around a little, but mathematically it has a definite position, and so can be represented as a point.
18. I think he is assuming that because the decimal representation of the numbers goes on forever, it cannot have an exact value. This is of course doesn't really make sense if for instance we consider a more obvious example. If we had three pieces of wood that are one metre long each and we put them together, we'll have a 3m piece of wood. If we define that big piece as being called 10 DeanK2s, then each smaller piece of wood are 3.3333333r DeanK2s long. Even though the decimalization goes on forever, the length of the wood isn't suddenly trapped between two values.
19. Just because you can't get a pencil, take out your ruler and place pi exactly on the line doesn't mean it's not there, just that you'll be a little inaccurate.
20. I suppose you can think of it as marking a point on a line with a pencil and paper...but your pencil is too thick, so magnify into the line and mark again, still not accurate enough - repeat forever

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