Understanding Infinity

[Positor]

However, it's not clear to me that 1/3 would count as a number (an arithmetic value) since it might have to be expressed as an infinite succession of 3's following the decimal point.

In base 3, or any base divisible by 3, it could be expressed finitely. E.g. in base 3 it would be 0.1; in base 12 it would be 0.4.

[owleye]

However, it's not clear to me that 1/3 would count as a number (an arithmetic value) since it might have to be expressed as an infinite succession of 3's following the decimal point.

In base 3, or any base divisible by 3, it could be expressed finitely. E.g. in base 3 it would be 0.1; in base 12 it would be 0.4.

Yes, I'm aware of this, though I admit not thinking about more than decimal or binary arithmetic at the time. The "it's not clear to me" referred to the position being taken by Raziel. The argument I ultimately would rely on for the topic is to transform infinity into something manageable, as is done by adopting an appropriate mapping. Raziel's x-y plane example probably made it seem like the problem is intractable without differentiating infinity from other numbers in some important way. As Lincoln is fond of pointing out, infinity should be considered a challenge, one that requires us to put on our thinking caps (though he may have a slightly different meaning in mind).

James

[Raziel]

This is where mathematics steps over the boundary between theory and application. 1/3 is a rule to find a value. The rule cannot be finished because the operation (calculation) is recurring. If the rule is impossible to finish then there can't possibly be a value. Binary is theoretical.

[jchook]

I like that you realized every point is the center of infinity. That's super fun to think about.

Reminds me, my hometown claims to be "center of the universe" haha.

Re: math, I am still trying to get over the idea of different density infinities. Example: the set of all real numbers has members that occur more frequently along a number line than the set of all integers, yet they are both of infinite size. http://en.wikipedia.org/wiki/Natural_density

One good exercise is to imagine Hilbert's Hotel:
Imagine there is a hotel with an infinite number of rooms, yet when you try to check in, they are all occupied. No worries, we'll just move the person in room 1 to room 2, and the person in room 2 to room 3, and so forth.

Brilliant.

[moranity]

a friend came up with this analogy to visualise infinity:
imagine standing on the circumference of an infinite circle, the line would look straight, and no matter how far you walk along the circumference, the path you have traced will never measurably deviate from a straight line.