"0.999... === 1" is at the very least internally consistent

Introduction

I sort of think of math as two separate things:

1. a set of objective truths, completely independent of human kind (not everyone accepts that this has any validity, but you don’t have to for the sake of this conversation I think)

2. the system of symbol manipulation humans use to try our best to understand and model #1.

Now, if we go into this ‘infinite decimals’ question with any hope of understanding how it relates to #1, I don’t think we’ll get very far. We can find examples in the real world that we all call “1” or “3” or whatever, but I don’t think we can find an object that everyone on this forum would agree to describe as “0.99999…”, so proving “0.999… === 1” using a real world example isn’t possible in the way that proving 1 + 2 = 3 is.

So let’s ignore the question of #1 for now, let’s ignore if this question of infinite decimals is a part of the “set of objective truths”, and exclusively concern ourselves with this question:

If we accepted that 0.9999… === 1, would that at least be internally consistent with the rest of our system of symbol manipulation we call math?

I posit that it would.

Note that the point of this isn’t to prove that anybody should accept the claim “0.999… === 1”, but rather that you should accept that treating infinite decimals in the way standard mathematics treats it leaves our mathematical systems of symbol manipulation in tact and consistent with each other.

My actual point

The whole debate last time started with the wikipedia proof that 0.999… === 1. en.wikipedia.org/wiki/0.999….

So, treating infinite decimals in the way they’re treating it here is at least internally consistent with the above-shown basic algebra pieces of Mathemtical symbol manipulation. I have another example that’s also internally consistent.

If we are going to treat infinite decimals as an acceptable representation of certain values, which is what this is all about, then we would also be accepting that 1/3 === 0.333…

And it would intuitively follow that 0.333… + 0.333… + 0.333… = 0.999…

And mirroring the above expression, we have 1/3 + 1/3 + 1/3 = 1.

So, putting all the premises together,

(a) if we accept the premise that infinite decimals are possible and an acceptable representation of certain values

(b) and we accept then that 1/3 can be acceptably represented as 0.3333…

(c) and we accept the Wikipedia proof that 0.999… = 1

Then what we’re left with is… an entirely in-tact sense of mathematical symbol manipulation, where these ways of representing values are completely consistent and compatible with the rest of the rules of symbol manipulation we call math. We don’t actually lose anything by accepting that 1/3 is perfectly represented by 0.333…, and we also don’t lose anything by accepting that 0.999… = 1. Those statements, whether you agree with them at some deeper philosophical level, at the very least fit in our systems without breaking anything.

PostScript

I actually came up with the idea for this post at lunch today. It occurred to me that all of human mathematics is a set of rules that we agree on on how we’re allowed to manipulate mathematical symbols - we agree on those rules because they seem to mirror the behavior of other things that happen in reality, and the idea is to keep this mirroring in a way that we can use our symbol manipulation rules as tools.

For example, we have a rule that says “X * 1 = X” – if you multiply anything by 1, you get the same result back. That’s a manipulation that you’re generally allowed to do across mathematical disciplines. We have rules like, if X = Y then 2X = 2Y, and X + 2 = Y + 2 – if you accept the equality of two things, then you accept that the two things remain equal when you do the same operations on them. These are all within the set of allowable symbol manipulations. These are of course only a couple of the basic parts of allowable manipulations, just for illustrative purposes.

And then I thought, it’s a bit odd that a big chunk of this forum doesn’t accept that 1/3 can be represented symbolically as 0.333… I mean, I get it to some degree. I get that it’s not exactly nice to accept an infinite decimal representing a finite value.

But that’s just it. It’s not NICE. But if you put that to the side for a second, and just let yourself accept it, regardless of how nice or unnice it is… then what?

So we have people on this forum who do accept that 1/3 is a totally valid fraction to have (Motor being the primary disagreer there), but these people don’t accept that 1/3 can be represented as 0.3333…, and I think the major thought that kept jumping out at me is, if they’re not accepting it, it seems to me that they’re not accepting it as a matter of taste. As in, they just prefer not to treat the symbolic meaning of 0.3333… as being equal to 1/3. And that’s what this post is all about - to show that the opposite allowance, the opposite preference, leaves every mathematical system of symbol manipulation in tact with no negative consequence.

The point isn’t to change anybody’s preference, but rather I suppose to show that that’s all it is: a preference. And the opposite preference still works.

Felix wrote:

“We can find examples in the real world that we all call “1” or “3” or whatever,”

Ecmandu replies:

I don’t mean to sound like a post modernist … because I’m not.

All numbers are irrational numbers.

Everything is an approximation.

In a physical world, approximations work so well that we can send people to space with them.

Another thing about numbers is that if you hold infinite bases in your head… every number is a whole number.

But when you look deeper, even whole numbers are irrational.

So your argument amounts to -

• If you don’t know that I am lying - then I am not lying
• It does hurt for the US government to borrow another 7 trillion dollars - so accept it
• Wikipedia has the proof (even though invalid)

x = 0.999…
10x = 9.999…
—is invalid (and incorrect)

On a different board James had a list of about 6 proofs for why 1 is NOT equal to 0.999… (I forgot what board it was). One of his revelations was that any number that ends with “…” isn’t actually a number in the sense that it doesn’t represent a finite quantity.

There is finite – “1.0”
And there is in-finite (not-finite) – “0.999…”

Literally by definition they are not the same – one finite – the other in-finite.

The fact that you do not care does not dictate truth.

…
x = 0.999…
10x = 9.999…
—is invalid (and incorrect)
…

My point wasn’t to prove that it’s correct per se, just consistently usable with the rest of mathematical symbol manipulation.

Regardless, if you have a way to prove its not correct, I’m open.

That Wiki proof claims:

That “proof” is garbage.

x = 0.999…

That doesn’t change because you add and subtract from that. “x” will always equal 0.999…, because that is how it is defined, that x = 0.999…

“10x = 9.999…” is simply moving a decimal point to the right one place. OK, great. Whatever. It means 10 x’s equal 9.999…
That is: x x x x x x x x x x = 9.999…

Then it claims
10x = 9 + 0.999…
That is false!
10x = 9 x’s + 0.999…
9 x’s = 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999…

Then it claims
10x = 9 + x
That is also false!
10x = 9 x’s + 1 x

Then it claims
9x = 9
That is also false
9x = 9 x’s, not 9.0
9 x’s = 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999…
10x’s = 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999… + 0.999…

Then it claims
x=1
That is false!
As stated at the very beginning, x = 0.999…

Actually it is not consistent with the rest of maths manipulation either - although right now I can’t remember the numbers that display the inconsistency - so moving on –

It is easy for me to prove that 10x is NOT equal to 9.999…
The problem is for you to prove that it is (since you proposed it).

I imagine that your effort to prove would be something along the lines of -
“to multiply by 10 - simply shift the decimal point” – ?

If so - that isn’t valid. You will have to show what happens to every decimal digit - good luck with that last one.

But I will save you the trouble of your positive proof by provided my negative proof -

If a number is equal to one value - it cannot also be equal to different value.
It is by deduction that a conclusion can be drawn concerning the destiny of an infinite series.
The number of relevant decimal places must remain the same for true equality –
x = 9.0: 10x = 90
x = 0.9: 10x = 9.0
x = 0.99: 10x = 9.90
x = 0.999: 10x = 9.990
x = 0.9999: 10x = 9.9990
— ad infinitum to a destiny of 9.999…9990.

Note that the representation “0.999…” has no zeroed decimal location at all - it does not end with “…9990”. In fact it does not end at all – 9’s forever and ever - never concluding - certainly not with a 0.

But contrary to that destiny - the above sequence reveals that 10x of anything must have a “0” at the end after the operation even though typically left out for fractional decimals.

That difference may seem irrelevant - and in ordinary operations it would be - but in trying to be rigorously precise - 10 times an infinite decimal is an unknowable value - except that there must always be a new 0 just before “infinity”.

9.999… is not the same as
9.999…9990.

You will have to show what happens to every decimal digit - good luck with that last one.

They all move one place to the left relative to the decimal.

So what do you propose concerning that last 9?
If you are going to propose that every 9 shifts to the left - you cannot add any additional 9’s to make up the hole being left - so what happens at the infinitesimal destiny? How would still be an infinity of only 9’s?

• There is no last 9 therefore – how can it be shifted without adding another?
• The last 9 shifts to the left – leaving a zero?

There is no last 9. There’s infinite 9s.

Obsrvr.

James was not the smartest person in the world.

I am.

Here’s a quote from you Obsrvr :

“ isn’t actually a number in the sense that it doesn’t represent a finite quantity.”

Ecmandu replies:

You can’t even do base conversion without infinite equalities. Superposition.

Shit. How do I explain this to a human?

Every number equals every number.

The reason repeating decimals come up is because existence is infinite.

How do I explain this to a human?

Umm… the patterns for what triggers a repeating decimal are totally arbitrary.

I’ve given you the proof for what creates a rational repeating decimal in every base.

But you have to understand superposition.

All numbers are all numbers.

There was an infinity of 9’s
Now there is an infinity of 10*9’s

How do you show it when every 9 becomes a 90?

You are required to multiply (or shift) every 9 - what do you do about 10 times the most infinitesimal?

Obsrvr. You’re funny.

We call that:

0.909090…

Continue.

I don’t know what any of that means

I know exactly what it means. It’s a good argument actually. It’s confused… but it’s good.

He’s making the double infinity argument for 1-base.

He’s trying to win it through disproof through contradiction.

It’s a primitive argument but it’s good.

He’s basically saying that…

0.999…000…

Is impossible.

What it means is that every 9 in the string gets shifted to the left - leaving a 0 to its right - which gets filled with the next 9 which left a 0 - which gets filled with a 9 - which left a 0 – ad infinitum.

And that means that the “last decimal place” - no matter where you stop - must be a 0.
And if you do not stop (truly infinite) the tail must be 9 and also must be 0.

Transcendental numbers are interesting (Pi, sqrt2) because they must be either - both odd and even - or more sensibly neither odd nor even.

In the case of multiplying an infinite decimal that would cause a shift - the total number can be neither the original nor the remainder that would stem from multiplying each digit.

So is it fair to say that you (or “we”) do not really know what to do about the tail end of a 10 times infinite decimal? We can’t just add another number to make up for the shift - nor can we leave any space implying an end. We shifted - we had to shift FROM somewhere - leaving a space behind.

Flannel.

I know you’re a moderator so you see my messages still.

Obsrvr can’t explain himself.

I articulated something in a few sentences that he’ll never approach his whole life.

Don’t worry about it

I don’t think that’s fair to say obsrvr. There is no tail end as far as I’m concerned. I don’t feel any pressing need to do anything about it - I’m fine just shifting the decimal.

So what you are saying is that it is convenient (“nice”) for you to just assume there is no need to look further - no need to prove your position?

Isn’t that what you said about the opposition?

I have offered you a proof (several now). You have not offered any proof.

Can you point out what is wrong with this proof –

It is fair to say to him.

He’s using a timeless disproof.

I’m not mad at him. He’s just wrong.

He’s using the double infinity disproof as a disproof through contradiction.

I corrected him and stated that 0.909090…

Solves the problem.

I additionally tried to explain that superposition makes every whole number an infinite decimal and vice versa.

Flannel. I do this because I’m bored.

I don’t actually feel like I need to prove that you can multiply .999… by 10 to get 9.999…, for a couple reasons:

1. The point of this thread isn’t that it’s “correct” per say, but just consistent within the symbolic manipulations of standard math - I don’t need to prove it because it’s truthfulness is largely outside the scope of what I’m actually talking about.

2. It in fact IS the position of the vast majority of standard mathematicians - this is the standard position among experts

3. You explicitly said it’s easy for you to prove otherwise, which means you’ve explicitly accepted the burden of proof there.

So far it seems like your proof relies on me agreeing with you that multiplication by 10 must be done in one particular way and not in other ways. But I don’t accept that and I dont agree with that. That feels to me like it’s an arbitrary rule you made, similar to the arbitrary rules Motor makes in his conversation with you in the other thread. “No you can’t do it that way, you have to do it my way.” I don’t see why I have to do it in your particular way.