Is 1 = 0.999... ? Really?

Having studied math, the answer is obviously yes! It’s all a matter of training and mathematical inclination. If you haven’t learned basic arithmetic, you think 2 + 2 and 4 look different. After you make it through grade school, you come to recognize without a moment’s hesitation that 2 + 2 and 4 refer to the same thing. Likewise one comes to recognize that .999… and 1 as two distinct expressions for the same thing, namely the Platonic or intuitive concept of the number 1.

So all you are saying here is that “My mathematical training includes 2 + 2 = 4 but not .999… = 1.” That’s all your remark amounts to.

I’ll concede that I’m unfamiliar with the studies about what the man on the street thinks about the Peano axioms. Most people don’t think about this at all, and if asked, would think you’re weird for asking them.

So let’s leave this with you and me. I myself have a perfectly clear intuition in my mind of the endless sequence 0, 1, 2, 3, 4, … of natural numbers; and even of the “completed” set of them which we call (\mathbb N ). You do not. We could still be friends. Not everyone hears the music, as I say. Some like Picasso and some like Norman Rockwell. It’s all good.

I could ask you, though, to explore the nature of your own ultrafinitism. Do you believe there’s a largest number that has no successor? If the process 0, 1, 2, 3, … ends, where does it end?

But no, THAT IS ONE OF THE CORE CONFUSIONS of many people, pardon my shouting. I’m glad you mentioned it though.

Let’s just consider the limit of a sequence, which DOES give people a lot of conceptual trouble.

Consider the sequence of rational numbers 1/2, 1/4, 1/8, 1/16, etc. We say it has a limit of 0. This causes people who have not studied Real Analysis, which to be fair is a course only math majors take, to think that the sequence “reaches” 0 in some mysterious way.

But NO! The whole point of the formalism of limits is that we DON’T TALK ABOUT REACHING. We talk instead of getting “arbitrarily close.” You give me a small positive real number, no matter how tiny, and I’ll show you that the sequence gets closer to 0 than that. And we DEFINE that condition as being the limit of the sequence.

The entire point of the limit formalism is that we never have to think or talk about “reaching and endpoint at infinity,” which is a hopelessly muddled mess. Instead we FINESSE the whole problem by using the “arbitrarily closeness” idea. That is the brilliance of the modern approach to infinitesimals. We banish them! We don’t have to talk about them.

(I mention in passing that the hyperreals of nonstandard analysis don’t help you, because .999… = 1 is a theorem of nonstandard analysis as well).

So there is no mysterious endpoint, there is no reaching. These are mind-confusing illusions left over from your imprecise intuitions of infinity. And these intuitions are clarified and made logically rigorous in math. That’s a fact.

I apologize. I was not trying to be condescending. I am actually under the sincere impression that most adults, if pressed, will agree that there is no end to the sequence 0, 1, 2, 3, 4, 5, 6, … because you can always add one. I am under the impression that most people do feel that way, if you asked them.

I believe this must be especially true in the computer age, when many people from programmers to spreadsheet users have internalized the concept of “always add 1” or “keep adding 1.” We live in the age of algorithms and “given n, output n + 1” is a perfectly intuitive concept to many people.

Everyone who ever started to learn how to program came to understand (often with great difficulty) the concept of looping, or endless repetition. If you can do something once you can do it forever. That is one of the main grokitudes of programming!

If you genuinely don’t agree, and genuinely reject the concept of adding one, then I’m interested to learn more about what that means. And even if it’s nothing more than a convenient fiction (which is exactly what it is!) What if it’s all bullshit but Newton used it to calculate the motions of the solar system. Wouldn’t you at least grant that the mathematical formalisms are useful and therefore worthy of study?

@Magnus You’re getting a little ahead of me but I will try to catch up to all your replies to me. Just working on the other ones first! You have brought up a lot of items that I need to take time to reply to.

No worries, it’s always better for a person to take their time (:

Evidently @Ed and you both care about the topic of whether functions and numbers are essentially different. I think it’s kind of a distraction but just for sake of discussion I’ll play.

First, a function goes from any set to any other set. We always denote a function as (f : X \to Y ), meaning that (f) is a function that inputs an element of a set (X) and outputs an element of a set (Y).

Since everything in math is a set (in the standard set-theoretic formalism), a function can input anything and output anything.

For example a function can input and output functions. A familiar example is the derivative operator in one real variable. We have a function (D) that inputs a function of one real variable and outputs another. For example (D x^2 = 2x). (D sin
x = cos x), etc.

Or a function could input a set and output a number; for example the function that counts the number of elements of a finite set, and outputs -1 if the set’s not finite. That’s a perfectly valid function from the proper class of all sets to the natural numbers.

So functions can be completely arbitrary in terms of what they input and output. I don’t see how this sheds light.

Well numbers have properties too. Everything has properties. So that doesn’t distinguish functions from numbers.

Conceptually maybe not, but formally functions are often numbers. For example in mathematical logic, we use Gödel numbering to represent a function or a formula by a specific number.

Another example would be to use the fact that there are as many continuous functions from the reals to the reals as there are reals. So in principle there’s a mapping that inputs a continuous function and outputs a real number that can be used as a proxy for it. Instead of saying cosine we can just say #45.3. Every function has an associated number. So again, the distinction between numbers and functions is less clear to me than it is to you and @Ed.

Well … I question the relevance or point of the observation, since it’s not clear to me that it’s true, and it’s definitely clear to me that it’s a red herring in the .999… discussion. I don’t get the bit about numbers and functions. Set theory doesn’t distinguish between numbers and functions, they’re both different types of sets. So I honestly don’t know what point is being made here.

Oh but of course it is. Every decimal expression is a function (d : \mathbb N_+ \to D) where (D) is the set of decimal digits (D = {0,1,2,3,4,5,6,7,8,9 } ). That’s what a decimal expression is. You give me the number 47, I give you back the 47-th decimal digit. (Just referring to the digits to the right of the decimal point, we can patch up the idea to account for the leftward digits if needed). You give me the number 545535 and I return that digit. That is exactly what a decimal expression is, a function from the set of positive natural numbers to the set of digits.

I’m using (\mathbb N_+) which is the set 1, 2, 3, … so that the first place to the right of the decimal point is 1 and not 0 for convenience. I hope that’s clear.

You see this, right? (\pi ) is a function, (\sqrt 2 ) is a function. [After we deal with the pesky leftward digits].

What function represents (\pi - 3 )?

f(1) = 1
f(2) = 4
f(3) = 1
f(4) = 5
f(5) = 9
etc.

Hi Magnus,

What type of entity do you believe .999… to be?

Thanks Ed

The quibble over function or number reminds me of grammar.

Function is to number as verb is to noun. The definition of the word is the same, but is the definition being used differently - as a doing or as a being? Are humans “beings” or “becomings”? Well they’re still humans.

Then there’s the distinction between the definite and indefinite article, or better yet - the type/token distinction. “A human” is one specific concrete specimen, whilst “human” is abstract humanity in general. Again, the definition of human: what we’re dealing with and what specifically is meant, is the same.

The same goes for function and number - the meaning, and what we’re dealing with is the same.
What is it that is the same in this case? Quantity.

“A quantity” is what I’ve been referring to as a concrete representation. “Quantity” is abstract.
You can represent “a quantity” as a function or number, arriving at it algorithmically (the journey) or the final result of doing so (the destination). “The quantity” represented is the same. “Quantity” means the same thing either way.

So yes, this objection of whether something is function or number is superficial at best, and meaningless at worst.

@wtf, what did I say about endless repetition when trying to deal rationally with irrationality?
The irrational are like flies trying to fly through a window, trying the same thing over and over in slightly different ways, and as soon as a new angle is attempted they forget their mistake in trying the old angle, and soon repeat their attempt.
The window is never escaped, sometimes even if you literally open the window for them.

I wonder if the mental block comes from how “1” is being thought of as being arrived at from one side only? In the case of “building” the representation (0.\dot9) from (0.9) through (0.99) etc. it’s approached from below only.
If it was representationally approached from the above, would it also be “1 plus some infinitessimal” as well as “1 minus some infinitessimal” simultaneously?

The algorithmic functional “doing” to get there is superficial. How the number “looks” is superficial.

A couple of people have mentioned Cauchy sequences. (0.\dot9) is cauchy because there is no other quantity that it approaches than “1”.
Representationally, you can get arbitrarily close to “1”, but as above, whilst “the representations” can differ, “the quantity” is identical.
You can represent the quantity (0.\dot9) as different to “1” with a Dedekind cut, but again this is only representation. The “quantity” is equal.

Hi wtf,

You might be onto something here. I am thinking about Von Neumann and Godel numbering, but I need to give it some more thought.

However, with this specific example aren’t you getting into some trouble representing an uncountable object with a countable object? In this specific case I think you need sequences with limits.

Ed

Hi wtf,

I think I have screwed up my comments abount countable/uncountable. Though I am still not certain about your representation for Pi - 3.

Ed

Perhaps some confusion can be cleared by signaling to the matter, or it’s corresponding idea relating to languages in general, mathematics being a quantifier, leading to this proposition:

'formula beginning with a quantifier is called a quantified formula. A formal quantifier requires a variable, which is said to be bound by it, and a subformula specifying a property of that variable.

Formal quantifiers have been generalized beginning with the work of Mostowski and Lindström.’

As far as generalization is concerned, it appears to pair with a tendency to integrate sets that functionally demand such, to qualify within reasonable set specifications.

So the function may be set differently,
but it tends to integrate within some mixed set consisting of both: of specified and more general characteristics.
At least this is what appears to be implied here.

I may be way off with this generalization, it seems credible.

Hello.

I take (0.999\dotso) to represent the same thing as every other decimal number which is a sum of the following form:

(\cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0 + d_{-1} \times 10^{-1} + d_{-2} \times 10^{-2} + \cdots)

Every (d_n) represents a decimal digit which is an integer from (0) to (9).

“Function” and “number” do not normally mean the same thing. You can make them mean the same thing, of course, but then, you must not equivocate.

How much money do you have? I have (f(x) = x^2) money. What does that mean? Normally, it means nothing. But of course, if you have a need to, you can make it mean something.

You can use horses to represent numbers. You can say “This kind of horse represents this kind of number”. For example, you can say that pegasuses represent number (1,000), centaurs number (100) and ponies number (10). This allows you to do arithmetic with horses. You can say “A pony multiplied by a centaur equals a pegasus” without being wrong.

You can do the opposite too. You can say “This kind of number represents this kind of horse”. You can say number (1,000) represents pegasuses, and because pegasuses can fly, you can conclude, without making a mistake, that (1,000) can fly too.

It’s all fun and games until you equivocate.

For example:

  1. All numbers are shapeless.
  2. All horses are numbers.
  3. Therefore, all horses are shapeless.

Horses qua numbers are indeed shapeless, but what is argued here is that horses qua animals are shapeless, which is not true.

In the same way, (0.999\dotso) qua limit is indeed (1) but what is being argued is that (0.999\dotso) qua sum is (1), and that is not true.

You’re not? What do you think a decimal expression like .1415926… means? It’s a map from the positive integers to the decimal digits. 1 goes to 1. 2 goes to 4. 3 goes to 1.

The expression is then mapped to a convergent infinite series by summing 1/10 + 4/100 + …

I cannot imagine you not knowing this. Please explain where you’re coming from. You have me totally confused.

Then I’m confused. If the question is, “Is .999… = 1 true in standard math?” the answer is yes without a shred of doubt. I could point you to a hundred books on calculus and real analysis. If we’re talking about standard math, how could anyone hold a different opinion?

But yes! In which case .999… = 1 in standard math and there is no question or dispute, other than to clarify for people what the notation means and why it’s true within standard math. It’s a theorem in ZF set theory. It’s a convergent geometric series in freshman calculus. It’s even true in nonstandard analysis, which some people aren’t aware of. There’s just no question about the matter.

So you really have me puzzled, Magnus. If you agree we’re talking about standard math, what is the basis of your disagreement?

Ok. So you admit that you are NOT talking about standard math, but rather about your private nonstandard use of mathematical symbols. In which case you can define .999… = 47 and I would have no objection. If that’s one of the rules in your game, I am fine with it; just as I learned to accept that the knight can hop over other pieces in standard chess.

You have already said that you are talking about standard math AND that you are talking about your own private nonstandard math. It’s not hard to misunderstand you!

Now let me talk delicately about infA. When I came to this forum several years ago, James was already a prolific poster, an ILP Legend to beat all ILP legends. I am reluctant to criticize him since he is not here to defend himself. He has far more mindshare on this forum than I do. I respect his prolific output, if not always its content.

That said, the concept if infA is confused and wrong in the extreme. The idea seems to be some sort of mishmash of the ordinal numbers, in which we do “continue counting” after all the natural numbers are exhausted; and nonstandard analysis, in which there are true infinite and infinitesimal numbers. The infA concept borrows misunderstood elements from each of these ideas and simply makes a mess.

One really valuable thing I got from this thread a few years ago is that James caused me to go deeply into nonstandard analysis, to the point where I understand its technical aspects. For that I appreciate James. But the infA concept is just bullpucky, I don’t know what else to say.

It means exactly what I’ve described to @Ed3. It’s a particular map from the positive integers to the set of decimal digits; a constant map, in fact, in which f(n) = ‘9’ for all inputs n. We then interpret this symbol as a real number as in the theory of geometric series, in which it’s proved rigorously that .999… = 1.

Again I agree that if you choose to make up a new system in which .999… has some other meaning, you are perfect within your rights. After all there do happen to be many variants of chess; played on infinite boards, or with a new piece called the Archbishop, etc. If someone enjoys playing alternate versions of standard games it’s ok by me.

Yes ok. Then we are done!

I fail to follow that. Did you learn geometric series at one point? The definition of a limit? I can’t tell where you’re coming from.

Actually the limit is defined as the sum. That’s what a limit is. It’s a clever finessing of the idea of “the point at the end” or whatever. You are adding your own faulty intuition. If you would consult a book on real analysis you would find that the limit of a geometric series is defined as the limit of the sequence of partial sums; and that the limit of a sequence is defined as a number that the sequence gets arbitrarily close to. I explained all this to @Phyllo the other day. That’s the textbook definition. You’re just wrong in your impression, either because you had a bad calculus class (as most students do) or none at all. It’s not till Real Analysis, a class taken primarily by math majors, that one sees the formal definition and comes to understand that the sum IS defined as the limit of the sequence of partial sums. That cleverly avoids the confusion you’ve confused yourself with.

Ah, the evil cabal of mathematicians. I will fully agree with you that most math TEACHERS form an evil cabal. One doesn’t get all this stuff sorted out properly till one sees the formal definitions; at which point, one learns that the limit of a sequence is defined by arbitrary closeness. The belief you have is a bad intuition that mathematical training is designed to clarify. It’s sad that we don’t show this to people unless they’re math majors, and you can rest assured that when I am in charge of the public school math curriculum the teaching of the real numbers will be a lot better.

Till then, I apologize on behalf of the math community that you weren’t taught better. But limits are very rigorously defined and your idea is just wrong.

Again I hope I’m not coming on too strong, I’m criticizing your ideas and not you. I know you are sincere. Except for the part where you say you’re talking about standard math AND that you’re not. That point confused me.

As I say, due to JSS’s extreme prolificness (if I may coin a word) on this site, plus the fact that he’s not here to defend his ideas, I prefer not to argue with him due to basic fairness.

So if you could frame your points from the beginning and not ascribe them to JSS, then I won’t be in a position to try to understand the ideas of JSS, which I found faulty four years ago. Just explain your ideas to me in your own terms. Else I’m arguing with someone who can’t argue back.

Again, as I’ve said many times, if you would consult a book calculus or real analysis, you would know that .999… represents the geometric series 9/10 + 9/100 + … whose sum, as defined in real analysis, is 1. I know you have an infinite series on one hand and a number on the other, but they are indeed equal mathematical objects, and this can be rigorously proved from first principles.

I do desire to understand the nature of your disagreement. But as you keep falling back on claiming that the standard definition of a limit is a lie, without providing any more supporting details, I remain puzzled. The definition of a limit is what it is, as is the way the knight moves. Rules in a formal game. They can’t be right or wrong, they’re just formal rules that have turned out to be interesting and (in the case of math) useful in understanding the world.

That can not be rationally disputed. One need only consult hundreds if not thousands of math texts that describe the standard definition of limits.

I could see your saying that mathematicians have gotten it wrong. But I can’t see your saying that they didn’t say what they did say!

It is not necessary for a sequence or sum to “attain” its limit. That’s the whole point of limits. Attainment is NOT part of the definition. Arbitrary closeness is.

Likewise, SOME properties are preserved when we pass to the limit, and others aren’t. For example in the sequence 1/2, 1/3, 1/4, 1/5 1/6, … each term is strictly greater than 0, But the limit is 0. It is a fact taught to math majors that when we pass to the limit, (<) becomes (\leq) and (>) becomes (\geq).

The terms of 1/2, 1/3, 1/4, … get arbitrarily close and STAY arbitrarily close to 0. That is the definition of the limit. You have faulty ideas because you haven’t grokked the formal definition of a limit. That’s all I can see of your objection.

That’s just something you made up. It’s not mathematically true. Limits are based on arbitrary closeness, not attainment. It’s perfectly clear that 1/2, 1/3, … never “attains” the value 0. The limit is 0 because the terms get arbitrarily close to 0. That is the definition. And even though each element is strictly greater than 0, the limit is 0. That’s how limits work.

It’s my own psychology that I question! It’s a lot like flat eathers. I think flat earthers are harmless and usually trolling. I think “scientifically minded” people who expend energy trying to debunk flat earth theory are sillier than the flat earthers. The flat earthers are having fun and the debunkers are way too serious for their own good, like Neil deGrasse Tyson, who got into a lengthy public dispute with some rapper about flat earth theory.

Yet here I am doing exactly the same thing. If someone disbelieves .999… = 1 it’s harmless, for one thing, and I’ll never convince them otherwise, for another. Yet here I am. I’m nuttier than the .999… deniers. I admit it

But thanks for agreeing with me!

I have three more examples I forgot to mention earlier.

  • Number are often interpreted as operators. If we interpret the number 5 as an operator that stretches a line segment, then the number i is an operator that rotates a line segment in the plane. That is in fact the best way to view it. So it’s common fro numbers to be identified with functions.

  • We have function spaces such as the set of all continuous functions that can be added and multiplied pointwise. With these operations of plus and times, the continuous real-valued functions of a real variable become a commutative ring.

Likewise we have famous function spaces like Banach and Hilbert spaces, in which functions are points, we have inner products, and we can do linear algebra on them.

  • There is no distinction between numbers and functions in set theory. I don’t know enough about type theory to know how this is handled.

I didn’t represent an uncountable object with a countable one. There are as many continuous functions (reals to reals) as there are reals. Easy proof. That was my only point.

I’m confused about where you’re coming from. What point are you making? Morally, functions and numbers are different. But in practice we use functions as numbers and numbers as functions all the time.

No.

I disagree.

The point of contention is the standard meaning of the symbol that is (0.\dot9).

Obviously, I have to repeat it at least one more time: I am NOT talking about what I mean by (0.\dot9), I am talking about what mathematical establisment means by (0.\dot9).

That said, you might want to argue that I am wrong in my belief that mathematicians define (0.\dot9) as a sum and not as a limit.

The limit of a sum is not the sum itself.

You can call the limit of a sum by the name “sum” – and people already do that, I understand – but that doesn’t erase the difference between the concept of a sum and the concept of a limit.

Let me give you an analogy. You can call numbers horses. There’s nothing wrong with that. But you can’t say there’s no difference between numbers and horses.

The argument I’m putting forward is that (0.\dot9) represents THE RESULT OF A SUM and not THE LIMIT OF A SUM. They are two related but different things.

That said, you might want to argue that mathematicians interpret (0.\dot9) as the limit of a sum (and not as the sum itself.) By doing so, however, you would be making an exception for (0.\dot9) and similar expressions because all other decimal numbers are normally interpreted as sums.

The result of a sum is a number that is attained. The limit of a sum is a number that is approached – not necessarily attained.

Your best bet is to argue that (0.\dot9) represents a limit rather than a sum. You won’t get far by trying to deny the fact that the concept of limit and the concept of sum are two different concepts.

en.wikipedia.org/wiki/Series_(mathematics

You haven’t added anything other than reiterate that you are unfamiliar with the standard mathematical definition of the limit of a convergent series.

FWIW I’ll outline the logic.

(1) First we define the limit of a sequence a1, a2, a3, … by the “arbitrarily close” standard, which is formalized by the business about the epsilons that you may have seen. (I’ll skip writing the gory details unless requested).

(2) Then we would like to define what we mean by a notation like a1 + a2 + a3 + … The axioms for the real numbers only allow us to add up finitely many real numbers. So we have to DEFINE what an infinite sum is.

We form the sequence of partial sums: a1, a1 + a2, a 1 + a2 + a3, + … Each term is well-defined because it’s a finite sum of real numbers. Now if the resulting SEQUENCE of partial sums has a limit as given by (1), then we define the infinite sum as the limit of that sequence. It’s a definition.

That’s it. That’s really it.

Now if the question is whether that’s what mathematicians say, it is. Wikipedia agrees and so do hundreds of textbooks.

If the question is whether the mathematicians maybe got it philosophically wrong, that’s a different discussion; and one that I’m not entirely unsympathetic to. But you claim to dispute that this is how mathematicians define infinite sums, and you’re just wrong about that.

JSS started this thread. It’s his topic. That’s why I’m mentioning him. You are wondering what this thread is about. Since it’s James’s thread, it’s James who decides (or rather, who decided long time ago) what this thread is about. And what this thread is about is standard math.

I am not saying that the definition of a limit is a lie. I am not sure where you got that from. What I’m saying is that mathematicians claim that (0.\dot9) qua sum is equal to (1) and that they do so by a variety of means one of them being by pretending that an infinite sum and the limit of an infinite sum are one and the same thing. Basically, what I’m saying is that they are equivocating (not merely working with different definitions.)

I am not talking about limits, I am talking about sums. Sums aren’t based on arbitrary closeness.