3 ms·

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually someth

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc. And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanation, everyone more or less gets it.

p1necone 5mo ago
INFINITY PLUS 1
- freetime2 5mo ago
  Uncountable infinity
samplatt 5mo ago
And then the next kid says "infinity plus two", which is a perfectly acceptable progression, and the cycle starts again.
- rmunn 5mo ago
  When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two". Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).
  - ndriscoll 5mo ago
    This is one of my life goals is to prepare my kids to troll their math teachers with the dual numbers and the claim that .999... is obviously 1-ε. Goal is to convince the teacher .999...≠1. Bonus points if they instead convince the teacher to doubt that complex numbers exist.
    - srean 5mo ago
      That would be both fun and correct. It really comes down to what semantics we attach to "=" when one of the sides is an infinite series. The "equals to" sign that we have used prior to that mental exercise was for finite terms only, we had not had to deal with infinitely many terms before that leap in thought. So now we have to extend the notion in a way that is backward compatible. A convenient one is it is equal to its limit if it exists.
      - freehorse 5mo ago
        > semantics we attach to "=" when one of the sides is an infinite series I would say that the semantics are about what an infinite series itself is, not about the equal sign. Once we have the common analytic notion of convergence of an infinite series, then the equality makes sense. The issue is that an infinite series is not an actual sum, but, formally, it is a sequence (of the partial sums). As you say, we represent the limit of the sequence of the partial sums with the same notation and only in the case that we have absolute convergence, but that's basically because we use the same notation for two different things (the sequence of the partial sums, and the limit of that). If we know we refer to the limit, I don't think there is any semantic complication with the equal sign.
  - deleted 5mo ago
    [deleted]
archargelod 5mo ago
The obvious way to win in this game, that probably many kids discover is to define your number as "whatever number the other kid says, plus 1".
nkrisc 5mo ago
The game is to name a larger number. Is infinity a number? Some would say, “no”. Eventually, one of the kids will name a largest number, because no one else will name another and the game ends with a largest number. It is possible that aliens exist, so is that proof aliens exist? Is is possible to create ever larger numbers, but is that proof that infinity exists other than as a fanciful idea in our minds?
tomjakubowski 5mo ago
24 is the highest number. Where you gonna go from there?

p1necone 5mo ago

INFINITY PLUS 1

freetime2 5mo ago
Uncountable infinity

samplatt 5mo ago

And then the next kid says "infinity plus two", which is a perfectly acceptable progression, and the cycle starts again.

rmunn 5mo ago
When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two". Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).
- ndriscoll 5mo ago
  This is one of my life goals is to prepare my kids to troll their math teachers with the dual numbers and the claim that .999... is obviously 1-ε. Goal is to convince the teacher .999...≠1. Bonus points if they instead convince the teacher to doubt that complex numbers exist.
  - srean 5mo ago
    That would be both fun and correct. It really comes down to what semantics we attach to "=" when one of the sides is an infinite series. The "equals to" sign that we have used prior to that mental exercise was for finite terms only, we had not had to deal with infinitely many terms before that leap in thought. So now we have to extend the notion in a way that is backward compatible. A convenient one is it is equal to its limit if it exists.
    - freehorse 5mo ago
      > semantics we attach to "=" when one of the sides is an infinite series I would say that the semantics are about what an infinite series itself is, not about the equal sign. Once we have the common analytic notion of convergence of an infinite series, then the equality makes sense. The issue is that an infinite series is not an actual sum, but, formally, it is a sequence (of the partial sums). As you say, we represent the limit of the sequence of the partial sums with the same notation and only in the case that we have absolute convergence, but that's basically because we use the same notation for two different things (the sequence of the partial sums, and the limit of that). If we know we refer to the limit, I don't think there is any semantic complication with the equal sign.
- deleted 5mo ago
  [deleted]

archargelod 5mo ago

The obvious way to win in this game, that probably many kids discover is to define your number as "whatever number the other kid says, plus 1".

nkrisc 5mo ago

The game is to name a larger number. Is infinity a number? Some would say, “no”. Eventually, one of the kids will name a largest number, because no one else will name another and the game ends with a largest number. It is possible that aliens exist, so is that proof aliens exist? Is is possible to create ever larger numbers, but is that proof that infinity exists other than as a fanciful idea in our minds?

tomjakubowski 5mo ago

24 is the highest number. Where you gonna go from there?