.999 = 1?

[Great_Antibob]

@allotrope

Look, whether you like it or not, the term is used with monotonous regularity and in formal contexts, e.g., here.

That’s in physics. Physicists regularly use “at infinity” in a specific way that is different from decimal representations in mathematics.

Are you simply doing Google searches for the phrase “at infinity”? Because, of the three citations you’ve used so far, NONE have directly addressed decimal representations and all of them have been in different contexts (physics, projective geometry, limits) even from each other.

The closest one was the example with limits. I’ll stand by the fact that teachers use fuzzy language when they teach. Think back to some of the simplest examples.

One example is the concept of “cross-multiplication”. If we have 2/3 = x/5, several teachers teach their students to “cross-multiply” to get the solution, though formally, the concept of “cross-multiplication” doesn’t exist. It’s simply a trick.

Likewise, FOIL multiplication of binomials is a quick mnemonic device for the distributive property. The flip side is that once students have to multiply general polynomials, they often get confused because FOIL is a very context specific trick while the more general (and formal) method is not.

Even in differentiation and integration, a good calculus teacher will tell you that using ‘dx’ on its own as an individual quantity is sometimes a useful trick but that it doesn’t satisfy formalism. Tricks and shortcuts used in pedagogy have their place, but taking them too far will get you in trouble.

[Giles]

@jtart2

1 does not equal 0.99999…

They are not the same, but to make math easier to understand they are rounded and taken to be the same.

No, in the standard usage of mathematical notation they are equal, just as 1/3 = 2/6, or 1+1 = 2. There is no rounding involved in the definition.

[Indistinguishable]

@Jragon

Since we’ve been going this whole time talking about defining new systems with axioms that do whatever you want which leaves me with a question – have there been any useful systems defined where division by multiplicative zero (or whatever you want to call the transform x * y = x for all y*) exists? I mean, never mind that it makes little to no sense to divide something into zero parts (aka totally annihilate it), I’m just curious if it’s been defined in any useful system.

Sure. There are the affinely extended and projectively extended numbers. In the former, we have positive and negative infinity; in the latter, we have a single unsigned infinity. Division by zero is defined the way you probably expect. Indeed, even if you only care about “real” numbers, you reason about affinely/projectively extended numbers all the time when you manipulate limits, whether or not you choose to reify them.*

(In neither of these does division act as a proper inverse to multiplication in all cases, of course; you can’t have that nontrivially.)

[*: This is why I don’t care much for many calculus teachers’ pedantic insistence that “infinity is not a number, and you must always use the language of limits instead. You can’t say ‘1/0 is infinity’; you have to phrase it as ‘The limit as…’, blah blah blah”; it’s just an arbitrary distinction to draw. You might as well say “The square root of 2 is not a number; it’s just a limit. You can’t say ‘At the square root of 2’, you must say…”]

[Indistinguishable]

@Great_Antibob

One example is the concept of “cross-multiplication”. If we have 2/3 = x/5, several teachers teach their students to “cross-multiply” to get the solution, though formally, the concept of “cross-multiplication” doesn’t exist. It’s simply a trick.

Huh? What does it mean to say “formally, the concept of ‘cross-multiplication’ doesn’t exist. It’s simply a trick.”?

(As for whether “dx” can be used on its own meaningfully or has no formal meaning, and every other such thing, well, that all depends on the formalism. You could perfectly well make sense of it, even if that’s not the traditional way of formalizing things. So while I’m sympathetic to you in general, Great Antibob, I think you’re fighting the wrong battles with allotrope…)

[allotrope]

@Great_Antibob

That’s in physics. Physicists regularly use “at infinity” in a specific way that is different from decimal representations in mathematics.

Fortunately, I’m not too worried if I don’t convince you. If you’re really interested, I’ll let you go through a calc tutorial and see for yourself. Then we can talk.

edit: just make sure that if you are ambitious enough to go that route, you pick the most rigorous version of the course. Something like calculus and analytical geometry for the sciences should work fine.

[Indistinguishable]

@allotrope

If you’re really interested, I’ll let you go through a calc tutorial and see for yourself. Then we can talk.

C’mon, don’t be so condescending. Antibob is surely intimately familiar with calculus/analysis, in as complete rigor as you could demand… They’re just accustomed to describing uses of “infinity” in that context with slightly different language than you’d like.

[allotrope]

My apologies. then I really don’t understand what the problem with the concept of “at infinity” is.

[Great_Antibob]

@Indistinguishable

Huh? What does it mean to say “formally, the concept of ‘cross-multiplication’ doesn’t exist. It’s simply a trick.”?

It’s usually taught as a trick, and teachers often forget to teach “why” the trick works and students, even if taught, often don’t remember.

Here’s how it is usually taught:

  3/5 = x/4

—>
34 = x5
12 = 5x

The result is correct, though students cannot often explain “why” it works.

The actual reasoning involves multiplying both sides of the equation by multiplicative inverses of both denominators involved:

 (3/5)*4 = (x/4)*4
    3*4/5 = x*(4/4)
     3*4/5 = x*1
     3*4/5 = x

(34/5)5 = x5
34*(5/5)= x5
34 = x*5

Yet ask a hundred average middle school students to explain “why” it works to simply multiply the denominators “across”, and I’d be shocked if more than half could do it. Worse yet, when the same type of arithmetic gets extended to slightly more complex polynomial expressions, they get somewhat lost because they don’t understand the original reasoning for why it works at all.

Despite the fact that many students don’t really or forget understand the “why”, it’s still a useful pedagogical exercise to teach them this method.

Maybe I didn’t make the point too clearly, but students often learn shortcuts without remembering the original reasoning and definitions. And teachers often teach shortcuts without making the reasoning entirely clear*.

*That’s why I actually like it when calculus teachers pedantically insist that infinity is not a number. Sure, it’s annoying if you understand the source material, but most students don’t and aren’t likely to understand or care if you get sloppy.

[Valmont314]

@Great_Antibob

Because, of the three citations you’ve used so far, NONE have directly addressed decimal representations and all of them have been in different contexts (physics, projective geometry, limits) even from each other.

The closest one was the example with limits. I’ll stand by the fact that teachers use fuzzy language when they teach. Think back to some of the simplest examples.

One final comment before I get out the highlighter and start reviewing my 100 page (double sided) printed copy of this thread. We are not talking about decimal representations here. We are talking about trying to understand the distinction between concepts involving finite quantities, infinity, infinite precision, and infinitesimals. This thread is not about crossing ts and dotting Is. People are asserting that the sequence .999… can conceptually be thought of as the same thing as the integer 1. That is what I have a problem with.

Nowhere have I seen it written in stone that all understanding of reality occurs exclusively through mathematics. There is a symbiotic relationship between our conceptual understanding and the real world which either reenforces our understanding our shows us where its either fallacious or incomplete.

There are many occasions where the concept of what actually happens at infinity are entertained, whether they be cosmology, geometry, religion or whatever. The fact that the concept, at present, may be of little utility in applied mathematics should not make it off limits for discussion.

[Great_Antibob]

@Indistinguishable

So while I’m sympathetic to you in general, Great Antibob, I think you’re fighting the wrong battles with allotrope…)

Ok, I guess I’m being extremely pedantic, but it is one of my pet peeves, and this thread has been going far too long. I’ll stop with the pedantry (for now) but only to go onto a broader point.

One of the real problems in trying to explain many math concepts is that while something may be true in a specific context, it may be wrong in the way it is understood generally.

And we see that in some of erik150x and Valmont314’s posts. A lot of what they say makes sense - but not in the way we generally use math. In his last few posts, in particular, Valmont314 is now trying to make statements about cosmology and what not about behavior at infinity. While a lot of the discussion is interesting, it is not particularly applicable when talking about the equality of 1 and 0.999… in the everyday math sense that 99+% of us use and that has been the main topic of the thread.

The real problem is when that discussion tries to loop back around, and various posters have tried to make it applicable to the everyday sort of math we DO use. And that’s incredibly frustrating.

[Derleth]

@jtart2

@Indistinguishable

Have you read the entire thread first?
When a mathematician gives an infinite decimal as notation for a number, what they mean by it is this: the* number which is >= the **rounding downs **of the infinite decimal at each decimal place, and <= the rounding ups of the infinite decimal at each decimal place. This is the definition of what infinite decimal notation means; it’s true because we say it is, just as the three letter word “dog” refers to a particular variety of four-legged animal because we say it does.

[snip]

Just like if we call a dog’s tail a “leg” The dog now has 5 legs because we say he does, not because the tail is really a leg. Ignoring the difference doesn’t mean there’s no difference.

No. MATH IS DEFINITIONS. IN MATH, THINGS ARE WHAT THEY ARE DEFINED TO BE. That’s how math works, and if you try to make it work differently that means you are wrong.

[Trinopus]

@Valmont314

.000…1

As when another bloke tried this: please define this term. You use it cheerfully; you must be able to define it. What, exactly, does it mean?

[Jragon]

@Derleth

No. MATH IS DEFINITIONS. IN MATH, THINGS ARE WHAT THEY ARE DEFINED TO BE. That’s how math works, and if you try to make it work differently that means you are wrong.

In addition, linguistics is arbitrary – albeit somewhat more resilient against random redefinition. That said, saying that legs are now called tails doesn’t make them the same thing – it just means that tail has two definitions (tail has two different intentions that point to different two extension sets, if you’re into that sort of thing).

[Jragon]

@Trinopus

As when another bloke tried this: please define this term. You use it cheerfully; you must be able to define it. What, exactly, does it mean?

Yes. The problem I have with it in these discussions is that people start off by saying that “.999…” never terminated, but it’s a “process” so it only gets arbitrarily close to 1… forever. But I say that then in “.000…1”, since the “1” comes after the “…”, the zeroes never end, they only get arbitrarily close to ending… forever, so that number is actually zero.

[Frylock]

@Valmont314

.000…1

And if your response is that .000…1 is not a number please provide references to support your claim. Remember, you didn’t say it had to be a real number, you simply said number. Once you are done with providing those references you can provide further references to support the claim that .000…1 is not a valid concept that can be seen as the difference between the concepts .999… and 1.

You can do that. And you’re right it wouldn’t be a real number. And you’re right that you can do it anyway.

You’re wrong to think this is controversial (though you’re right to think most non-mathematicians are unaware that you can do this).

You’re wrong to think that the dispute between you and others has anything to do with what’s “really true” about numbers. You guys are just talking about different kinds of numbers.

You’re wrong to think (if you think this) that you’ve said everything that needs to be said in order to show rigorously that what you’re doing will not lead to some kind of internal contradiction.

[Frylock]

@jtart2

At the end of the day the real question is…what difference does it make (pun intended) :smack:
Joe

Agreed. But since you’re the one making a distinction others say isn’t there, it’s your task to tell us what difference it makes.

[CookingWithGas]

@Jragon

Yes. The problem I have with it in these discussions is that people start off by saying that “.999…” never terminated, but it’s a “process” so it only gets arbitrarily close to 1… forever. But I say that then in “.000…1”, since the “1” comes after the “…”, the zeroes never end, they only get arbitrarily close to ending… forever, so that number is actually zero.

It’s true that .000… is zero, but if the dots mean “an infinite number of zeroes” then it is meaningless to add “followed by a 1” since by definition you can never reach the 1. If you could, then the number of zeroes would be finite.

[Valmont314]

@Trinopus

As when another bloke tried this: please define this term. You use it cheerfully; you must be able to define it. What, exactly, does it mean?

I will try my best to do that, but first, stop and ask yourself this question:

Go back to When you first studied Calculus and recall the first time you were asked to evaluate a limit.

Say Lim 1/x as x -> ∞

What were you asked to do?

Did the teacher tell you to make it a homework assignment, go home, grab your abacus or bag of shells and use them to somehow solve the problem and demonstrate to yourself that the answer is correct and then check your work afterwords by performing the inverse operation.

You can certainly do all of this if the teacher asked you for an answer to 2 + 2 or any finite arithmetic operation.

When we are asked to evaluate a limit, and this applies every single time we do anything that involves the concept of infinity, we are asked to use our intuition and do our best to simply see and judge what the right answer is and write it down and deem it to be correct because we can all agree that it just “looks right”.

There is no inverse operation to that, its a thought experiment. The solution can never be demonstrated directly in the finite world. The reason we believe it to be true is because as others in this thread have pointed out the results are demonstrable but in a more indirect way. The symmetry between the math and reality is there but it is more indirect. We can fire a projectile and use Calculus to calculate the trajectory and see that it matches up very well based on our measurements, but measuring the flight of a projectile and saying it corresponds exactly to our conceptual understanding is not quite as convincing as looking at 3 pears and 2 apples and seeing that we have 5 total pieces of fruit.

We verify our concepetual understanding by sampling reality. But there is no way to directly perform this sampling when the concept of infinity is involved, so we are relegated to what is essentially performing a thought experiment which not only we can all agree upon and which seems to make intuitive sense, but also seems to correspond with reality when it is applied in more indirect ways.

The main point is that the rigor that we perceive to be there and that we invoke when substantiating our arguments is directly proportional to how well that which is conceptual matches up with reality. One can certainly devise rigorous conceptual systems that have no bearing on reality. That does not make them wrong it simply makes them useless. Nothing conceptual is ever proven via our sampling of reality, it is simply reenforced until we stumble across a real world experiment that invalidates our theory which is the point of divergence. So again, we prove nothing conceptual, we only fail to disprove. And the more abstract a concept is, the more indirect the method of verification is.

Infinity, and its related concepts of infinite precision and infinitesimal are amongst the most abstract of mathematical concepts.

So in answer to your question, the best I can do is ask you what you see, the same way your first Calculus teacher asked you to see what you see when evaluating the limit above.

Some questions (ranked in the order of clarity with which I myself can see the answer to - your view may differ from mine)

1. Can you see that 1 is the boundary or limit of the .999.. sequence much in the same way the X axis is the boundary of the function above?
2. Can you see that as 9s are added to the sequence .999… the sequence gets closer and closer to 1 but never appears to get there?
3. Can you see that as more and more 9s are added the finite precision number .999 EXACTLY approaches (and becomes at infinity) the infinite precision sequence .999…?
4. Can you see that as more and more zeros are inserted into the finite precision number .001 it EXACTLY approaches (and becomes at infinity) the infinite precision representation of the infinitesimal as originally denoted by Newton as 1/∞ (which I have represented in decimal notation as .000…1)?

Now back to logic (and reason when necessary):

A. If you agree with 2 above does it make sense to deduce that there must be some interval which can be represented (somehow) as a magnitude between the .999… sequence and 1 ?
B. If it makes sense that there is an interval what is the most likely candidate for that interval?
C. It can be demonstrated that the candidate, for every other case leading up to infinity is the quantity identified in 4 above.
D. Does it seem reasonable that the candidate in C. above would be the best choice for the answer to A. ?

Now, a little rigor:

The question of what .000…1 is can be explored via long division (yes you can demonstrate this for yourself using nothing more than a piece of paper and a pencil)

The long division of 1/3 yields, after 3 rounds, two numbers, the quotient and remainder .333 and .001/3

Back to intuition:

5. Can you see that after an infinite number of steps the finite precision number .333 approaches (and EXACTLY becomes at infinity) the infinite precision sequence .333…
6. Can you see that after an infinite number of steps 3 times .001/3 approaches (and EXACTLY becomes at infinity) the infinite precision sequence .000…1
   The last bit of Logic:

E. If the correct answer to 1/3 is, for every known case demonstrable, the quotient plus the remainder, would it be reasonable to deduce that 1/3 = .333… + .000…1 ?
That’s the best I can do, this is not simply a math or even a physics question…its deeper than that and those who insist on confining it to the realm of mathematics are never going to allow themselves to see anything other than what they have been taught.

I must say, I find it amusing when there is a collective agreement that something is undefined, and yet the same people who are part of this agreement vehemently oppose any efforts to seek an understanding on the grounds that it does not immediately gratify their desire for a definition.

Erik150x was right on target when he pointed out that most of the emphasis for this discussion should be placed on attempting to understand what 1/∞ represents and how it relates to other mathematical concepts.

OK, I was on my way out the door before I was asked this question so I will go in peace and come back to continue this discussion when I have read the entire thread.

[Exapno_Mapcase]

Bull. This is not a word problem; it is a math problem. All math problems are worked out in a world of rigorous definitions, axioms, and proofs. Once you provide the rigor, the answers received can be agreed to by every person in every time and every place who understands the system. The rigorous approach to infinity was worked out 150 years ago. Some examples of the rigorous math has been given in this thread. Because they understand this rigorous system, in this example, every mathematician says that 0.9999~ is identical to 1.0. Period. Not equal at infinity. Equal.

Anybody outside the system who doesn’t understand and wants to substitute intuition instead can scream at the walls for an infinite amount of time without changing anything but our opinion of him.

[Valmont314]

@Exapno_Mapcase

Bull. This is not a word problem; it is a math problem. All math problems are worked out in a world of rigorous definitions, axioms, and proofs. Once you provide the rigor, the answers received can be agreed to by every person in every time and every place who understands the system. The rigorous approach to infinity was worked out 150 years ago. Some examples of the rigorous math has been given in this thread. Because they understand this rigorous system, in this example, every mathematician says that 0.9999~ is identical to 1.0. Period. Not equal at infinity. Equal.

Anybody outside the system who doesn’t understand and wants to substitute intuition instead can scream at the walls for an infinite amount of time without changing anything but our opinion of him.

Excuse me, but did you even bother to read a single word I said?

The burden of proof is not on me.

The burden of disproof is on you.

I assert that .9999… + 1/∞ = 1

If you have all the answers and everything is crystal clear in your head you should be able to cite numerous examples of how that assertion leads to contradictions.

So I suggest you spend more time enumerating those contradictions as opposed to making hyperbolic proclamations. Thanks.