What Number Comes After 101,899?

Factual Questions
1

Earlier this week, a gentleman at work asked me what number comes after 101,899. It was absolutely clear from the context that he was talking about integers. He thought maybe there was a 2 in it somewhere. I explained that the next number would actually be 101,900. This gentleman makes over $90,000 per year and has a technical position that affects the health and safety of the public. I was telling my co-worker about this today and he said that the gentleman had asked him a very similar question earlier about the number 101,799.

However, this post is about a slightly different topic. As my co-worker and I were discussing this, I said that the correct answer depends on what type of numbers we are talking about. We were both certain that the original question was about integers. Rather than getting back to work, I rambled on and asked what the next number would be if we were talking about real numbers. The difference between 101,899 and the next real number would be really, really small. No matter how small you think it is, it’s actually smaller than that. I don’t think it would be a constructible number or computable number because it could not be constructed or computed in a finite number of steps. It seems to be definable in an informal sense. I think most of us could at least figure out what the hell I am talking about. But is it definable in the formal mathematical sense? What do you all think?

2

101,899 + (the limit of 1/x as x approaches infinity)

Well, that or 101,899 + (the limit of x as x approaches zero).

3

Both of those are 101,899, not something after it at all.

4

I think the (whole) number that comes next is 101,900. And if that isn’t right then I have no idea what the hell this is all about.

Remember, not everyone on the Board has a background in math.

5

There is no such thing as the next real number. Between any two reals there’s an infinite number of reals.

There are sets other than the integers where there can be a next number. The next odd number after 101,899 is 101,901. If you’re counting in tenths, the next number is 101,899.1. The next non-prime is 101,900. The next multiple of 7 is 101,906.

Any infinite countable set of numbers can have a next number. I don’t think it’s possible to have a next number in an uncountable set. Am I right?

6

Seems pretty clear that if it’s not 101,900, then it’s someone deliberately trying to do some sort of lame semantic wordplay. The sort of wordplay of “Okay, so by common sense understanding it’s 101,900, but we’re going to nitpick the dictionary definition of what a number really is…”

7

I’m willing to agree with this.

8

The OP’s question might be addressed with something close to a Dedekind Cut. The Dedekind Cut is defined over the rationals, and is used to create the reals, but the logic of the process is close to the OP’s question.

We might reword the question to be - how can I cut the real numbers into two intervals such that one contains all real numbers up to and including 101,899, and the other, all real numbers greater than 101,899.

In the end the result is simply two intervals. We use [ or ] to denote inclusion of a bound, and ( or ) to denote a bound that is not itself part of the set.

[- \infty, 101,899] \hspace{10pt} (101,899, \infty]
This pair includes all of the reals.

The post above from @ASL_v2.0 gets you one way of defining the number required:
{\lim{\small x \to +\infty}}\hspace{10pt} 101,899 + {1 \over x}

If we constrain x to be a natural number, the definition still works and it becomes a Dedekind Cut (since {1 \over x} \in \mathbb{Q)} (This is perhaps overkill. Dedekind was concerned with constructing the Reals, so use of a real number for x would become circular.)

You can construct the intervals as
[-\infty, 101,899] \hspace{10pt} [(\forall x \in \mathbb{N} , {\lim{\small x \to +\infty}}\hspace{10pt} 101,899 + {1 \over x}), \infty]

Which really tells you little more. The intervals are closed now, we have simply shuffled the definition of the boundary to a different place.

The question of existence becomes thorny. The Reals are dense, so there are no gaps. The value clearly must exist. But it is impossible to explicitly find a value for it.

9

This is the answer. There’s no need to confuse the issue with limits or Dedekind cuts. There’s just no such thing as the next real number after a given real number. Same thing with rationals; there’s no such thing as the next rational after a given rational. It’s like asking what’s the largest integer. There’s just no such thing.

10

Personally, I’m wondering more about what was going on in the head of the co-worker who asked the original question.

11

Indeed. It looked a bit like a misremembered question about cascading rounding and mistakes made when rounding up. Was the original question expecting 102,000?

Just odd.

12

“What’s the next number after 101,899?”
“101,900.”
“No, I meant in alphabetical order.”

13

Next I suppose you’re going to tell me that 101,899.999 (repeating) is the same as 101,900.

14

This does not work. You want the real numbers to be well-ordered. You may be familiar with the axiom that every set can be well-ordered. However, the standard ordering on the reals is not a well-order, for reasons like there is no least element strictly greater than 101899.

17

Indeed. Gosh - it took me a while to remember this problem. It was a very long time ago. I suddenly feel a little old remembering those days. I feel an urge to reprise a lot of this - it was a lot of fun. And maybe make fewer mistakes. :anguished_face:

18

I hope he’s not a serial killer who can’t count.

19

That rationals are countable, but I don’t think they have an obvious next number. I suspect we would have to specify an explicit ordering. So I would say any countable set combined with an ordering rule has a next number.

20

Basically. An infinite countable set of numbers can be put in a list (i.e. matched one-to-one with the natural numbers), so for any one of those numbers, there will be a next number on the list (the one that matches with n+1). But what that “next number” is would depend on how the numbers have been placed in a list.

And since the real numbers are uncountable, that means that there is not a way of putting them all in such a list.

21

In the context of the original question it had absolutely nothing to do with rounding. That’s why the question was perplexing. All he had to do was write down the next number on a list of sequential (whole) numbers.

22

The real numbers are not countable, because there are too many of them. However, the axiom of choice implies that any set can be well-ordered, including any uncountable subset of the reals.

The story was pretty weird. Why was the gentleman in the story quizzing people about those specific numbers? Was it to see if they could add by 1 in their heads?