lendav_rott said:
It is never wrong to prove others wrong. Just by saying "you are wrong" is not enough in my case.
OK, my apologies.
1.) is inf - inf equal to 0? No one could tell. If you assume infinities to be equal then they would have to be finite, but they're not so tough luck. Who is to say that one infinity is equal to/lesser than/bigger than the other? There is just no way you can tell.
inf - inf is undefined but not for the reasons you list.
Assuming infinities to be equal does not mean they have to be finite. I would like to know how you came to that conclusion. For example, we know that there are an equal number of real numbers and irrational numbers, and both of these sets are infinite in cardinality.
inf - inf is undefined because you run into problems defining inf + inf (or subtraction) and retaining distributive/associative laws, and other field axioms (even though an extension to infinity technically isn't a field (afaik) we would still want to keep it as "field-like" as possible and retain these simple properties.)
No operation on infinity would be left undefined just because of some "well infinity is super weird," cop-out explanation, it would be left undefined because we can't find a way to define it without coming up with undesirable consequences, and that is all.
2.) is inf times inf = inf squared or just inf? How could you add to something or multiply infinity by something if you don't know how much that "infinity" contains. Operations work with finite figures.
Infinity is not some really large, unspecified, growing value. Question: do we know how many "3s" are in the repeating decimal ##0.\overline{3}##? The answer is yes.
Operations work with
whatever the hell we want them to as long as our definition of those operations are consistent.
Infinity * Infinity is not like Infinity - Infinity. We can define it just fine and run into absolutely no problems whatsoever.
Real Projective Line
3.) is 1/infinite 0? Yes and no. In limit calculations you can say it is 0 because limits estimate the boundary, but just having 1/infinity is undefined.
1/infinity can be defined to be 0 in any extension without issue.
The only reason we say "limit as n approached infinty 1/n" in the real numbers is because infinity is not in the set of real numbers. Once we add points for infinity in the extended real line, 1/infinity means exactly the same thing as the limit. We don't even have to do anything to say that 1/infinity = 0 after extending the real line.
Furthermore, there is no reason to confine the discussion to the context of the real number line. The real number line is not the "one, true, number system." Your opinion of the real number line seems almost similar to the opinion of some ancient mathematicians' opinion of the natural numbers. I see a lot of people say that "##1/0## is meaningless because it is not defined in the real number line." No, it's not meaningless, it is meaningless
in the context of the real number line because it is not defined
in the real number line, in the same way that ##3-27## is meaningless and undefined
in the natural numbers.
The moment we start talking about operations on infinity, we are extending the real number line. It is not useful to stay in the real numbers. Your post saying that
everything involving infinity is undefined because it is not a real number is no different than me going into the Math Homework forum and telling everyone that involves complex numbers in their work that their entire assignment is undefined and makes no sense because complex numbers are not real numbers.
We can define complex numbers consistently, and so we can work with them. We can define operations on infinity consistently, so we can work with them.
4.) is inf. - any real number still inf? Undefined. Imagine if you have an expression of 3y - 2z. Try subtracting 10 elephants from 52 motorcycles - you do have the expression, but it means nothing.
Nonsense, again, you insist that the real number line is sacred. I could again apply your same logic to the complex numbers, where elephants are real and motorcycles are imaginary.
Infinity - R = Infinity
In any extension.
5,) is inf times any real number still inf? again, undefined.
Again, the real number line is somehow sacred.
Infinity * R = Infinity
Where R is not zero in the projective real line. In the extended real line, infinity is signed, and so this operation changes in the expected and natural way in the extended real line.
Don't think of infinity as some sort of number. It is just a concept - it doesn't obey any rules in maths.
Being closed minded is not how we develop new math.
Another example of the concept of infinity - how much energy do you need to reach the speed of light? An infinite amount. Wait what? How much is infinite? Exactly.
This is lacking any point.
Yes, an infinite amount of energy is required for a massive particle to reach the speed of light. What do you mean, "wait what? how much is infinite?" You seem to be expressing some sort of problem with the idea that an infinite amount of energy is required for a massive particle to reach the speed of light. That is true (afaik) according to very sound modern physics..
