.999 = 1?

[Great_Antibob]

@7777777

Let’s accept what James Tanton writes that
…999999999 = -1

Why should we accept this? I certainly don’t accept this is true under a standard formulation of the real numbers.

Once again, you can use a nonstandard real number system, but if you choose to do so, it is incumbent on you to make sure it is consistent. There’s no moral failing to using this alternate system where …9999 is an actual thing, but it doesn’t really affect the other question, either.

And once again, if you use some nonstandard real number system, nothing you state about it makes any real difference to our every day, humdrum, standard real number system.

[Andy_L]

@7777777

I think that
…9999999999.99999999999…
…0000000000.00000000000…
are not the same because every digit is different

But you accept that …99999999999999 is the same as -1, even though every digit is different.

[7777777]

@Andy_L

But you accept that …99999999999999 is the same as -1, even though every digit is different.

You can think 2 and 1 are the same even though every digit is different
if it feels right. In this way you can try proving that 2=1.

[Andy_L]

@7777777

You can think 2 and 1 are the same even though every digit is different
if it feels right. In this way you can try proving that 2=1.

You’re the one that says that …9999= -1, even though all the digits are different but .999… can’t equal 1 because all the digits are different. Until you settle that issue, I’m not interested in your proof that 2 equals 1.

[7777777]

@Andy_L

You’re the one that says that …9999= -1, even though all the digits are different but .999… can’t equal 1 because all the digits are different. Until you settle that issue, I’m not interested in your proof that 2 equals 1.

0.99999999…can’t equal to 1 because every digit is different just like 2 cannot
equal to 1 because every digit is different.

…9999999 can be equal to -1 because James Tanton writes so even though
Andy L thinks all digits are different. That seems to be your issue if you see
it as a problem. I don’t know if I am interested in settling that issue.

[Telemark]

@7777777

0.99999999…can’t equal to 1 because every digit is different just like 2 cannot
equal to 1 because every digit is different.

Does 6/3 = 8/4? Every digit is different.

…9999999 can be equal to -1 because James Tanton writes so even though
Andy L thinks all digits are different. That seems to be your issue if you see
it as a problem. I don’t know if I am interested in settling that issue.

Since you’re relying on this point to bolster your proof and it seems logically inconsistent with your end result you might want to become a bit more interested.

[Exapno_Mapcase]

@7777777

On this document Mr.Tanton is writing about how …999999=-1 and I found how to use it in my proof.

Please tell me that you understand the following things about that .pdf.

1. It is a basic language instructional document for pre-calculus students.

2. It assumes all the way through that 0.999999… = 1.

3. More than that, it gives some very basic and unshakable proofs that 0.999999… = 1.

4. One of those proofs (p3) is a variation of the way to think about the problem I gave here in post #1092.

5. Albert is intended to be an idiot, one who eventually comes around to knowledge.

6. Albert is used as a device to demonstrate some typical arguments that students make but are completely wrong.

7. It agrees that humans are not capable of counting infinities but that they can be expressed and manipulated by careful math.

8. It shows that notation is important because only by agreeing on the definition of terms before one starts can one do anything in math.

9. It ends with a calculus professor walking in because the manipulation of infinities is what calculus is all about.

10) The whole thing is an attack on people just like you who try to argue against the notion that 0.999999… = 1.

[7777777]

@Telemark

Does 6/3 = 8/4? Every digit is different.

You see, every digit being different does not always mean that the numbers are different.
So …9999999=-1 can be the same even though every digit is different.
Brilliant, you proved it! The issue is settled.

@Telemark

Since you’re relying on this point to bolster your proof and it seems logically inconsistent with your end result you might want to become a bit more interested.

Yes, I already became. Every digit can be different, and the numbers can be still
the same. It is all contextual.

[Great_Antibob]

@7777777

…9999999 can be equal to -1 because James Tanton writes so

I can see that you don’t fully understand what I’ve been posting nor what Indistinguishable posted before, so I’m going to try one more time.

First, let’s start simple. We know that 6+8 = 14, right?

Well, that’s not really right. We made some assumptions here. The big one is that we’re assuming we’re working in base ten.

What if we’re actually working in hexadecimal? Then 6+8 = E. It’s pretty similar to base ten in many ways but different in other ways.

So, different assumptions = different results.

The point I’ve been trying to make (possibly poorly) is that your assumptions determine your system.

Under our every day standard real number system of arithmetic, there is no …9999. That number doesn’t exist. Just using it means you’re using some alternate number system that looks a lot like our regular number line in many ways but is quite different in other ways.

James Tanton, whatever else he is trying to describe, isn’t really describing the normal real number line. And that’s ok! Using alternate real number lines is not fundamentally wrong. It’s not a moral failing.

But while it’s ok to use a different number system, things you figure out in that system don’t necessarily transfer over to our regular standard real number system.

It’s ok to say that E-8=6 in hexadecimal. But it becomes a problem when you want to say that E-8 = 6 in base ten, because the numeral E doesn’t really exist in base ten (though the number fourteen exists in both).

And that’s what’s happened here. Whatever else is true, there is no …9999 in the regular standard real numbers. So bringing it up isn’t really relevant to whether 1 = 0.999… in the standard real numbers.

@7777777

Yes, I already became. Every digit can be different, and the numbers can be still
the same. It is all contextual.

That’s well and good, but you need to provide that context. What number system are you using (because it’s not the regular standard real number system) and how are you defining equality under this system? Basically, what are the arithmetic rules of this system, particularly for these constructs like …9999 that don’t have conventional definitions we could be expected to implicitly know?

[Great_Antibob]

I don’t recall if it got posted earlier in the thread, but the Ask Dr Math FAQ has a pretty good entry on this topic, including references to some textbooks with as much mathematical rigor as you could want (I’m partial to building up the reals a la Rudin’s “Principles of Mathematical Analysis” by Dedekind cuts).

[Indistinguishable]

I’ll switch to my other mode, posting things that are interesting to me, rather than aimed at thread-resurrectors:

We can easily enough make sense of decimal expansions which continue infinitely to the left rather than the right, manipulated according to all the usual rules of decimal arithmetic: these are the 10-adics [for what it’s worth, the p-adics were mentioned previously by Lumpy in this post], and in this context, we will have that …999 = -1.

Working with bidirectionally infinite decimal series is a little trickier: addition and subtraction remain straightforward, but multiplication is only straightforward for cases where one factor terminates to the left and another terminates to the right (this prevents infinitely many digit-by-digit multiplications from landing at the same digit location, as would happen with, say, …111 times 0.111…). Still, those multiplications cover a lot of ground anyway.

Note that our interpretation of bidirectoinally infinite decimals can be made to accord with various different interpretations of rightwardly infinite decimals. For example, if we wanted to preserve the intuition that 0.999… is slightly different from 1.000…, we could choose to interpret these in terms of hyper-10-adic-rational (extending the hyperrational interpretation previously noted).

On the other hand, if we wanted to preserve the cleanliness of the reasoning leading to 0.999… = 1, we could take ourselves to be working with an appropriate amalgam of the 10-adic rationals with the reals. A bidirectionally infinite series could be taken to be a pair (L, R) where L is a 10-adic rational corresponding to the digits from some point leftward and R is the real corresponding to the remaining rightward ray of digits. If we standardize on always splitting at the decimal point, this means L will be a 10-adic integer and R will be a real in [0, 0.999…].

Addition and subtraction in this presentation will be done component-wise, of course. As noted above, multiplication will be in general tricky, but will be well-defined so long as one factor terminates to the left and the other to the right. For coherence, we will need (L1, R1) to be considered equal to (L2, R2) even in some cases where the individual components are not equal; of course, the question reduces to which differences should be considered zero.

Specifically, we can get away with considering a bidirectional decimal to be zero just in case it is completely cyclic (e.g., …813813.813813…, or the aforementioned …999.999…). This is in accordance with the observation that these values are invariant under multiplication by a suitable power of 10. The nice thing is that this is the only failure of uniqueness of decimal representation we need to accommodate.

Note that in this system, negating a value amounts to simply subtracting all its digits from 9, as 0 = …999.999… . In particular (just as in the ordinary p-adics), there is no need for any explicit signs. Everything is just done in terms of digits, in a uniform way.

[watchwolf49]

@Itself

Those are just the axioms for a commutative ring with unity. (They hold for R, but don’t characterize it.) There is no first-order characterization of R; a common set of axioms is that it’s the unique ordered, complete field.

Those are the axioms for a vector space, not a ring.

[Indistinguishable]

@Indistinguishable

but multiplication is only straightforward for cases where one factor terminates to the left and another terminates to the right

Er, rather, where it is NOT the case that one factor extends infinitely to the left and the other extends infinitely to the right.

[Indistinguishable]

@watchwolf49

Those are the axioms for a vector space, not a ring.

True, but Itself’s main point stands, in that those axioms do not come close to comprising a complete system, in the sense of settling all questions in the relevant language “Yes” or “No”.

(Furthermore, in fully axiomatizing a vector space, one will need to axiomatize the field (or, let us say more generally, the ring) structure of its scalars. And once one has done this for the reals as scalars, there’s no need to separately worry about their vector space structure in formalizing real arithmetic; they’ve already been given all the relevant structure qua ring)

[Itself]

@Indistinguishable

I think most posters taking 0.999… to be infinitesimally smaller than 1 have never been taught about the “real numbers” in any formal detail, and have also never really given much consideration to the idea that number systems could be formalized in various different ways. So they assume whatever preformal intuition they have for what numbers amount to (developed not necessarily unreasonably from their schoolchild experience with decimal calculations) describes the One True System of numbers; that there could be multiple systems of numbers, so that a numerical question has not one fixed answer, but only an answer relative to a particular system, never crosses their mind (again, this is just what we would expect, given the way they are taught in the first place).

Sounds right to me. I seriously doubt they know anything about mathematical infinitesimals or nonstandard analysis, or even what a real number is. Instead, their idea of a real number is an infinite decimal expansion. The numbers 0.999… and 1.000… have different digits, so they’re different numbers. Their difference would be 0.000…1 (whatever that means), which is also just a string of decimal digits.

I realize people have some sort of intuition on that subject, but this is math, not guessing. If you don’t know what the definition (pick any of the half-dozen or so equivalent ones you like) of a real number is, or what it means for a space to be complete, or what the archimedean property is, or etc., then your opinion on the subject is worthless. This isn’t philosophy or guessing, and it’s not even physics, where you can at least start to understand the matter through intuition and analogy. Words in math have specific, precise meanings. If you don’t know what those meanings are, you don’t get to play.

[Itself]

@watchwolf49

Those are the axioms for a vector space, not a ring.

Oh, a and b are scalars in some fixed field, not elements of the space itself. Sure, my mistake.

[Telemark]

@7777777

Every digit can be different, and the numbers can be still the same. It is all contextual.

Then provide some context, because as it stands you’ve made no reasonable case. As it stands, you’ve only demonstrated a tenuous grasp of how to make a logically consistent proof. Since it’s all contextual, and you’ve provided no context (other than sometimes it’s true, sometimes it’s not) it’s a pretty weak foundation for your proof.

So far, you’ve got nothing. Just a reference to a doc that you don’t seem to understand; and that actually says the opposite of what you are trying to prove.

[Andy_L]

@7777777

0.99999999…can’t equal to 1 because every digit is different just like 2 cannot
equal to 1 because every digit is different.

…9999999 can be equal to -1 because James Tanton writes so even though
Andy L thinks all digits are different. That seems to be your issue if you see
it as a problem. I don’t know if I am interested in settling that issue.

It’s your claim that …9999999 = -1. If you don’t believe it’s a valid assertion, then your “proof” falls apart.

[The_Hamster_King]

I’m still waiting for **7777777 **to tell us what the decimal expansion of 1/3 is.

1/3 = ?

[Indistinguishable]

I actually read 7777777’s last post as a sincere change of mind (“Yes, I already became” and so on…), such that there’s no need to continue arguing with them. I could easily be wrong, though.