This is probably a stupid question, so I’ll be brief. If you were to start ‘writing down’ infinity using our (‘base ten’?) numerical system, would it necessarily be a never-ending string of nines? It just seems to me that you couldn’t have a ‘7’ or ‘3’ or anything but a nine in the string of digits, as you could always move nearer to infinity with these until they become a ‘9’.
And no, I haven’t been smoking anything. I’m at work.
Well, you can get higher by using an eight. Just put it on its side.
You can’t write infinity in a decimal (or any other base) notation at all. By definition it’s larger than any finite number. I think that you’re confusing the fact that a decimal point followed by an infinite string of 9’s is equal to 1 with the idea of infinity.
There’s a flaw in your question:
Infinity is NOT “the biggest number ever” - it is a mathematical concept. In fact there is more than one “infinity”, but it is not a number that can be written down. It seems contradictory, and would seem to make sense that you have to start with the largest digit (9) but it just ain’t so.
Suppose that you did write down infinity with an infinite string of nines. Next, add 1 to that. The last 9 becomes a zero, and you carry 1; then the second last 9 becomes a zero, and you carry 1 again; and so on. You would finish up with a 1 followed by an infinite string of zeroes, if you could complete the process.
But you can’t do that, because (if we are talking about the usual meaning of infinity as the cardinal number of the set containing all the integers) when you add 1 to infinity you still have 1. So, if 999… has any meaning, it is equal to 1000… .
(That’s analogous to the infinite decimal fraction 0.999… being equal to 1, but at least in that case 0.999… has a sensible meaning).
Thanks for your answers, all. While I’ve heard of the 0.999etc as being equal to 1, before, I’ve only heard it on the SDMB, and have always skated over it in ignorant bliss. I think I may just resign myself to never really comprehending infinity in mathematical terms; when I try to wrap my layman’s head around it I feel a bit vertiginous…
On a related note I remember reading about ancient Indian mystics (gurus?) who attempted to communicate to regular (relatively uneducated) folks the concept of eternity (infinite duration, so there’s the connection to the OP).
The concept was of an enormous block of granite, say a cube 100 miles on an edge. Every 100 years a sparrow (or similar Indian small bird) lands and scrapes its bill on the block.
When the entire block is completely worn away by the little bird…that’s one day.
For any aribtrary string of nines, eg “9 999 999 999 999”
you can specify a string of threes which represents a larger value, eg “333 333 333 333 333”.
The thing about infinity that always blows my mind is that even fractions of infinity are infinite.
I’ve heard this attributed to some Mennonite group.
And this is not infinity, because it’s still a finite process which will be completed in a finite amount of time. Infinity isn’t just ‘really big’ unless you’re twelve years old. It’s fundamentally different.
Think about it this way:
If infinity could be represented as a never-ending string of the highest digit available in a number base, then in base 2 it could be expressed as 1111111…, and in base 3 it could be expressed as 2222222…
However, a string of 1s of any length in base 2 always represents an odd number, whereas in base 3, a string of 2s of any length always represents an even number, so we obtain the result that infinity must be both odd and even.
This clearly can’t be the case, therefore it makes no sense to think of infinity as a number.
It’s certainly far older than the Mennonites. I’ve already heard it attributed to Indian folklore.
It can’t be repeated often enough that infinite just means never ending. It doesn’t matter how you represent that. It doesn’t matter what base you use, either. A 9 is the largest digit in base 10, but in hex the largest digit is F. In trinary the largest digit is 2. You can convert any number to any base. The representation is irrelevant.
The numbers used don’t have to be the same either. The first hundred digits of pi is represented in base 10 by 3.141592653589793238462643383279 50288419716939937510 58209749445923078164 062862089986280348253421170679. It continues on from there forever. It would take an infinite number of digits to represent pi (true in all bases) and as far as we know there is no pattern to be found.
Yet pi can be - has to be - found by formulas. So pi is 4*(1/1 - 1/3 + 1/5 - 1/7…) That’s also infinite and exactly equal, eventually, to the 100 digits I gave above and more.
The simple series 1/2 + 1/3 + 1/4 + 1/5 + … is infinitely large. It takes longer to get to really big numbers than that huge rock does to disappear but it will eventually surpass any number that you can think of in any notation.
And you can think of infinity in other ways as well. The set of all possible curves is infinite. You could number them but you don’t need to.
The big leap is getting rid of that notion that infinity is the largest number, or a number of any kind. That holds you back from any possible thinking on the subject. Once you do it becomes easier to see how the even numbers can be infinite and the odd numbers can be infinite and the combination of the two is infinite in exactly the same way. Why? Because you can put 1,3,5,7,9… and 2,4,6,8,10… in a one-to-one correspondence with 1,2,3,4,5,6,7,8,9,10… They don’t need to “catch up” because the series never ends. It works with squares, 1,4,9,16,25 and cubes 1,8,27,64… and fourth powers and everything else. If infinity were a number, this wouldn’t work.
Infinity will blow your mind, but it can be played with. Cast out the nines and start from there.*
*Yes, a math joke.
Can you? I assume by “number them” you mean put in 1 to 1 correspondence with real numbers, not just integers, but even then my first thought would be that this was a bigger infinity than the reals.
With apologies to George Orwell: All infinities are equal, but some are more equal than others.
That is to say, it is correct to say that there are infinitely many integers. And there are even the “same number” of infinitely many rational numbers, even though there are infinitely many rational numbers simply between 1 and 2. This “countable” infinity is distinct from the transcendental irrational numbers, of which there are “more” infinitely many than the algebraic numbers.
And there may be a third, or more, level on top of those, but this is where I get off the infinity bus.
It might depend on your definition of “curve”, but if it includes all continuous real functions then it’s going to have a cardinality at least as great as that of the real numbers.
Yes, I see that the way I wrote that sentence implies a countable infinity. I didn’t mean that, and I do know the difference between countable and uncountable infinities. I meant to show that there are ways of thinking about infinities that don’t involve strings of nines or any digits at all.
The concept you’re close to stumbling onto is called two’s complement.
Here’s how I would explain it, using base ten. Let’s take as a given that …999999 is how we represent “infinity” (the cardinality of the set of natural numbers, if it matters, but I don’t think it does). One less than that is …999998, two less is …999997, etc. This can be extended to any value less than infinity.
But how can we match these numbers up with the natural numbers, 1, 2, 3, etc? We can’t, because there’s no “crossover” point between the two sets. However, we could match them up with the negative numbers:
…999999 ~ -1
…999998 ~ -2
…999997 ~ -3
And so on. This is ten’s complement.
Why would we want to do something strange like this? Well, there’s actually a physical phenomenon that lends itself to this kind of numbering: negative temperature. And of course there’s chip-design reasons to use a finite-digit version of it on digital computers.
If by “ancient Indian gurus” you mean Terry Pratchett. But it’s a lousy analogy anyway. Granite has a density of 2.75 times water, and if we approximate it as being made entirely of silicon (atomic mass 28), a block of granite a hundred miles on a side contains 2.5*10[sup]44[/sup] atoms. If we make the assumption that the bird only removes a single atom with each beak-scraping (far too low a figure, but let’s stretch this analogy as far as it can go), then it’ll take 2.5*10[sup]46[/sup] years to completely wear away the block. That’s not only a finite time, it’s not actually all that impressive a finite time: The largest black holes in the Universe, for instance, will take over 10[sup]97[/sup] years to disappear entirely. So if, every time the granite block were worn away, you replaced it, and scraped a single atom off of a second granite block, the second granite block would be entirely gone before that black hole was even a tiny fraction of the way through its lifetime.
And that’s a mere pittance compared to some of the finite numbers used in mathematics. I was able to express that number fairly easily in scientific notation. There’s a number called Skewes’ Number, though, that’s so long that, if I were to attempt to write it in scientific notation like that, it would take me longer than the lifetime of that black hole to finish writing it. And then there’s something called Graham’s number, that takes a full page of explaining an entirely new sort of notation just to be able to write it at all. And those numbers are both finite, too.
I think that one way to conceptualize infinity is this:
Picture yourself in line at the DMV…
Yes, but I think it actually has a greater cardinality than the reals.