Our resident numerologist and girl with the crunchiest numbers has inadvertently alerted us to a conundrum. If one contemplates all the natural numbers what percentage of them (to the nearest one per cent) contains a two?
4
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Feel free to reply to any comment by clicking the "Reply" button.I thought that the answer was always 42.
starwatcher-al
Level 7
July 19, 2021
2
There are scientists who spend a lifetime trying to solve questions like that.
KateOahu
Level 8
July 19, 2021
2
And then Fernapple comes along and solves the twice times table teaser in a trice.
@waitingforgodo As usual.
I’ve just looked and my 22,222 of 2 days ago…has now increased by another 22, so presently it stands at 22,244…still consistent in its constituent parts, and still perfectly divisible by 2.
Marionville
Level 10
July 19, 2021
1
A quick calculation tells me that your divisibility will come undone when you reach you next prime at 22291. I wonder why primes ( except for the first two) are always one less or one more than a number divisible by six.
Somewhere it's 2:40 sometime.
youtube.com/watch?v=y0zc7x434Aw
@waitingforgodo I think above all what this little exercise has brought home to me is that I must be pretty “pass remarkable” to make that vast number of comments in just over 3 years!
To the nearest one percent, 100%, or all of them. As the numbers get larger, more and more of them will contain a two, until a point is reached where virtually every one contains a two, certainly more than ninety nine out of a hundred, and given that you are talking about an infinite series the percent of small numbers which do not contain a two becomes insignificant. Yet the irony of that is, that even that insignificant percentage, is also itself infinite.
It is the sort of logical paradoxical mind blower which you get into when you start dealing with infinity. Because infinity is an idea really beyond the human brain, and also why the infinity based 'proofs' that religious people and creationists love, are such rubbish.
Fernapple
Level 9
July 19, 2021
3
Thanks for working that out and elucidating for me….even at this early hour it’s too hot for me to be bothered!
I think the idea of limits is useful in this context.
Based on your observation, given any positive fraction, any sufficiently large number of natural numbers will have a smaller fraction that do not contain a two. Taking that to the limit, the fraction then becomes zero, so the limit case answer to @waitingforgodo's question is exactly 100%.
An example that might be more approachable for some people is to consider the infinite sum
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....
The limit of that sum is exactly 1, but the sum of any finite number of terms is always strictly less than 1.
@anglophone No the number can never be exactly zero, since for example 378951 there you go, even if there was only that one, the count would be short of 100%, and there is a number which does not contain 2 and there must be an infinite number of them, since I can write an infinite number of numbers just using the first digit alone, 3, 33, 333, 3333, 33333....... The problem is that when you try to bring limits into the game along with several other things, you could try, then like most people who try to curb and manage the bejumme which is infinity, you are no longer talking about infinity, it is the same problem all those creationists etc. hit.
@Fernapple That's the thing about limits, it can take a while to get used to the fact that they can produce what appear to be counterintuitive results.
Edit: the Ancient Greeks had a real problem understanding (in fact they never did understand) that irrational numbers exist, which is why they found the square root of 2 to be so vexatious.
minimally 10%
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