Multiple replies to this thread follow (as I came in late, initially).
This reply is enormous, but it covers many different topics, so it might be worth skimming if you find this interesting. Just skip over the stuff you hate.
12th Archivist wrote:
I can even take it a step in the opposite direction, too. Zero to infinity in degree one could be a single value of a larger set, this particular set having an infinite number of these single values.
It's important to realize here that "
infinity"
is not a number. Infinity is only a concept that means "
this goes on forever without bound"
. (The different "
sizes of infinity"
are actual numbers as defined through set theory, but the non-numeric concept of "
infinity"
still arises there all the time, and it is important to understand the difference.)
I think what you're trying to get at is not representable in the reals, because there is no "
smallest"
real value, no "
least"
, no "
greatest"
, etc. These are, in fact, some of the main properties which define the reals. You might want to take a look at the surreal numbers, which attack this issue in a formal way:
http://en.wikipedia.org/wiki/Surreal_numberskell wrote:
No two infinities are the same infinities, we had quite a lot of fun reducing expression with infinities on polytechnics to make them from: inf / inf ^2 to 0.5 (just a non-working example).
To be specific, you're not using infinities. You're working with limits that approach infinity, which is what analytical calculus handles. This is not the same thing.
True infinity cancellation, called "
normalization"
, does occur in some branches of physics. It is still (for a minority of mathematicians and scientists) mildly controversial in its application.
Someone Else wrote:
12357 <-> 0.75321
234900 <-> 0.009432
...
Take the real number n.a(1)a(2)a(3)...a(m)... (n is any number of digits, a(m) is one digit for each a(m)). It corresponds to the counting number P(n+1)^(...a(m)...a(3)a(2)a(1)), where P(n) is the nth prime.
The reason these strategies fail is that they can only count real numbers which terminate. Real numbers that continue infinitely would not produce a natural number (no natural number has an infinite number of digits).
da rogu3 wrote:
As for lowest possible number - for whole numbers, it's zero; for integers and real numbers, it's -infinity.
No. -∞ is not a number. There is no lowest possible number: that's what infinity means. More accurately, the integers and reals are
unbounded below. The
infimum of each, meaning the greatest number that is less than or equal to the integers or reals, does not exist. At least, it is not a real number; you could say it is "
-∞"
without too much confusion, which basically means it does not exist in the domain of ℝ.
In some forms of numerical analysis, there is occasionally a number line used, called the
extended number line, that includes ∞ and -∞ as distinct points. But these are still not numbers in the usual sense. These two objects can be treated as true numbers in the construction of a
wheel (a group where everything can be divided, even zero), but this is rarely done, and additional "
numbers"
like
0/0 must also be created to satisfy the necessary properties.
Most of this math is new, owing to the fact that it has been less than 150 years since people realized what mathematics actually is: the study of the results of arbitrary choices of logical rules. Prior to this everyone was trying to fit math directly to reality, and failing to discover some extremely interesting ideas (many with direct applications to science).
For more on wheels, see
http://en.wikipedia.org/wiki/Wheel_theoryThe spitemaster wrote:
What I find interesting is infinite sets of numbers. For example, prime numbers. If positive integers is countably infinite what does that mean for prime numbers which are a degree of magnitude less that those numbers?
As da rogu3 indicated, these are the same "
size"
. As it turns out, they're even the same "
length"
(ordinal). If we strip out all numerical information, the only difference between "
naturals"
and "
primes"
is the labels we choose to use for the numbers, which are arbitrary. Both sets are indistinguishable in terms of the way they are ordered.
This is even true for something that is truly "
smaller"
in a sense. Consider the set of all squares of naturals versus the set of all naturals (we'll eliminate 0 for reasons I will explain later). By the time we reach 1, 100% all numbers are squares. At 4, 50% of numbers are squares. At 9, 33% of numbers are squares. At 100, 10% of numbers are squares. At 10,000, 1% of numbers are squares. This proportion shrinks without bound, so we would say that the squares in their usual order of appearance are "
0% dense"
on the naturals.
This was in fact the very argument Newton made for why attempting to reason about infinity was absurd (and Newton, like everyone in his time, would not have treated 0 as a natural number). Of course, Newton didn't realize the difference between the "
number"
of something and the way it is
ordered or paired, but he didn't need to. The contributions needed during his time were the ones he produced, and were far more important than pure theoretical math.
As an aside, in Newton's day they were forced to use infinitesimals for calculus (they had not yet discovered the epsilon-delta definition for limits), but mathematicians wisely avoided coming to conclusions about them as numbers. Today we have a formal understanding of infinitesimals, and ironically we no longer need them for basic mathematics. Math is filled with these sorts of ironies: applications which used to be controversial, after 100 years of work we learned to formalize them, and now we discover we never really needed them for what they were originally used: infinitesimals, quaternions, the Axiom of Choice, proofs requiring uncountably infinite sets, and the list goes on.
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