# 0.999… revisited

The original question was: 0.999… = 1 does not make sense with respect to my conception of the number line, I do not know much about number classes but for the number line from lets say 0 to 1 there is an infinite number of points, so there is a number right next to 1 we cant write it in entirety because the decimal expansion is infinite but the difference between that 1 and that number is 1*10^(-infinity) (sorry if I am abusing notation). so that number to me should be 0.999… but it is not where am I missing the point?

Physicist: In the language of mathematics there are “dialects” (sets of axioms), and in the most standard, commonly-used dialect you can prove that 0.999… = 1.  The system that’s generally taught now is used because it’s useful (in a lot of profound ways), and in it we can prove that 0.99999… = 1.  If you want to do math where 1/infinity is a definable and non-zero value, you can, but it makes math unnecessarily complicated (for most tasks).  The way the number system is generally taught (at the math-major level, where the differences become important) is that the real numbers are defined such that (very long story short) 1/infinity = 0 and there isn’t a “next number” for any number.  That is, if you think you’ve found a number, x, that’s closer to 1 than any other number, then I can find a number half way between it and 1, (1+x)/2, that’s even closer.  That’s not a trivial statement.  In the system of integer numbers there is a next number; for 3 it’s 4, for 26 it’s 27, etc..  In the system of real numbers every number can be added, subtracted, multiplied, and divided without “leaving” the real numbers.  That leads to the fact that we can squeeze a new number between any two different numbers.  In particular, there’s no greatest number less than one.  If there were, then you couldn’t fit another number between it and one, and that would make it a big weird exception.  Point is: it’s tempting to say that 0.999… is the “first number below 1”, but that’s not a thing.

The term “real numbers” is just a name for a “sand box” of mathematical tools that have become standard because they’re useful.  However!  There are other systems where “very very very slightly less than 1” , or more precisely “less than one, but greater than every number that’s less than one”, makes mathematical sense.  These systems aren’t invalid or wrong, they’re just… not as pretty and fluid as the simple (as it reasonably can be), solid, dull as dishwater, real number system.

It is a fact, immutable and True, that every rook is safe for the next move according to the most widely accepted rules for chess.  In other games that may not be the case.  0.999…=1 in the most widely accepted rules for real numbers.

In the set of “real numbers” (as used today) a number can be defined as the limit of the decimal expansion taken one digit at a time.  For example, the number “2” is {2, 2.0, 2.00, 2.000, …}.  The “square root of 2” is {1, 1.4, 1.41, 1.414, 1.4142, …}.  The number, and everything you might ever want to do with it (as a real number), can be done with this sequence of ever-longer decimals (although, in practice, there are usually more sophisticated methods).

These sequences are “equivalent” and describe the same number if they get (arbitrarily) closer and closer to that same number forever.  Two sequences don’t need to be identical to be equivalent.  The sequences {1, 1.0, 1.00, 1.000, …} and {0, 0.9, 0.99, 0.999, …} both get closer and closer to each other and to the value “1” forever, so they’re equivalent.  In absolutely every way that counts (in terms of the real numbers), the number “0.99999…” and the number “1” or “1.0000…” are exactly the same.

It does seem very bizarre that two numbers that look different can be the same, but there it is.  This is basically the only exception; you can write things like “0.5 = 0.49999…”, but the same thing is going on.

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### 60 Responses to 0.999… revisited

1. Mark Schaal says:

A sequence is equal to its limit. For an increasing sequence its LUB is its limit. So this sequence is 1.

I have read all your posts and multiple times you have refused to use this.

2. netzweltler says:

The limit of the sequence (0.9, 0.99, 0.999, …) is 1. I agree.
Yet it doesn’t mean that 0.9 + 0.09 + 0.009 + … adds up to a point on the number line. Neither to 1 nor to any other point.

Do you agree that
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1) and not [0, 1]?

3. Alan Feldman says:

Yes, but it doesn’t matter. The definition of an infinite decimal is its LUB. Nothing you’ve said changes that, and that LUB is 1.

4. Alan Feldman says:

@ netzweltler – I think I see the problem here. The LUB of a sequence doesn’t have to be a member of it. You’re assuming it does. This seems to be the problem. Maybe.

OK, again, for the nth time:

The value of an infinite decimal is defined to be its LUB.

The value of 0.999 . . . is defined to be its LUB, which is clearly 1.

You wrote: “The limit of the sequence (0.9, 0.99, 0.999, …) is 1. I agree.
Yet it doesn’t mean that 0.9 + 0.09 + 0.009 + … adds up to a point on the number line. Neither to 1 nor to any other point.”

You need to be more precise about what you mean by “adds up to a point on the number line”.

5. netzweltler says:

@Alan Feldman

This is like saying “We _define_ 0.999… to be 1. Therefore, 0.999… equals 1.”

To me this is like comparing 1 – which is a point on the number line – to something which is not.

6. netzweltler says:

@Alan Feldman

“You need to be more precise about what you mean by “adds up to a point on the number line”.”

0.9 + 0.09 adds up to point 0.99 on the number line.
0.9 + 0.09 + 0.009 adds up to point 0.999 on the number line.

0.9 + 0.09 + 0.009 + … doesn’t add up to point 1 on the number line.
I tried to make that clear in my very first post in this thread.

Do you agree that
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1) and not [0, 1]?

7. Alan Feldman says:

Sir, just how are _you_ defining 0.999…? Obviously in a different manner. So there is no contradiction.

We are not defining 0.999… itself to be anything; we are defining what a repeating decimal is. And it is defined to be the LUB of the sequence of adding more and more digits.

What is 0.9? It is defined to 9/10.

What should we do if we require the decimal form of say, 1/3? What would you write? Most of us would write 0.333…. I suppose you would say you can’t write it down because of your points on a number line bit. The rest of us would like to write it down, so we have chosen to write a sequence whose LUB is 1/3.

_We_ have specified what we mean by a repeating decimal. You, OTOH, have talked about points being or not being on a number line. _You_ are using a _different definition_, whatever it is (you haven’t said, other than the vague idea of adding up to a point on the number line), so there is no surprise that you come up with a different conclusion.

I don’t think I can make it any clearer without spending an entire day or two on it.

8. Alan Feldman says:

“Do you agree that
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1) and not [0, 1]?”

I’ve already replied to this. Yes. But it doesn’t matter. The LUB ******DOES NOT HAVE TO BE A MEMBER OF THE SEQUENCE OF PARTIAL SUMS******. For example, the LUB of the negative real numbers is zero, which is not a negative number. And neither is 1 a member of (0,1).

Again, a infinite decimal is defined to be its LUB, which is in general _not_ a member of the set of the partial sums. All you’re saying, ABAICT (or AFAICT), is that none of the partial sums is 1. BUT THAT DOESN’T MATTER. An infinite decimal is defined to be its LUB, which in general is not a member of the set of the partial sums of the series.

By “defining” we are saying that, for example, the way to write 1/3 as a decimal is 0.333…. Multiply that by 3 and you get 0.999…, which is one.

We are defining a way to write a number in decimal form that cannot be represented by finite decimals.

I’m sorry about the “screaming”, but you keep missing this crucial point.

9. netzweltler says:

@Alan Feldman

I still think it is not clear what I am saying. To serve as an LUB, the LUB needs to be a number, i.e. a point on the number line. I cannot see how 0.999… can be assigned to a point on the number line – for the reasons given.

OTOH I fully agree, that 1 IS the LUB of the sequence (0.9, 0.99, 0.999, …). I just want to make that clear, because you are still trying to find arguments for that.

10. Mark Schaal says:

Let me try a more general approach…

Infinities are hard. They are simply impossible to deal with through brute force. It is easy to write 0.9 + 0.09 + 0.009 + … but you cannot actually add up all the terms manually. It is easy to write 1/2 + 1/4 + 1/8 + … but you cannot actually add up all of the terms manually. However there are lots of these infinite sequences, and it would be really helpful to have an agreed standard way of working with them. So how do we deal with them?

We could say: bah humbug, we do not accept infinite sequences at all and we refuse to ever deal with them. This is not useful.

We could say: every individual can make their own guesses about sequences. Then we might get:
Alice: I guess that 0.999… is equal to 1
Beth: I guess that 0.999… is equal to some number less than 1
Carrie: I guess that 0.999… is equal to 35
netzweltler: I guess that 0.999… is not equal to any number
This is also not useful.

So what happened is that the concept of “limit” was created, and for a sequence the numeric value of the sequence was defined to be the limit of the sequence. This concept of a limit of a sequence is helpful. It is pretty well behaved, you can add and subtract and multiply by a constant. It gives a well defined answer that Alice, Beth, and Carrie can all agree upon. They will even agree if they take different routes to get to the answer:
Alice: I use the definition of limit directly, do the calculation of the limit, and I see that 0.999… = 1.
Beth: I use the definition of limit to prove that for an increasing sequence the limit is the LUB, do the calculation of the LUB, and I see that 0.999… = 1
Carrie: I use the definition of limit to prove it is okay to do addition and multiplication by a constant, then I use that to do the x = 0.999…, 10x = 9 + x proof, and I see that 0.999… = 1.