Wednesday, November 30, 2016

9.999... Reasons to be Skeptical about .999... equaling 1

So YouTube has a really great video entitled “9.999... reasons that .999...=1” by Vihart. It's clear, and cute, and far better than my plodding reply. (here)  But hey, I want to disagree with it in a friendly way, anyway. It's often the good stuff that is worth fighting with ...

#1) Appeal to Authority:
99.999...% of Skeptics agree you should suspend judgement on matters that are not forced upon you. When you can tell something is true, believe it. Otherwise hold you mind open and think maybe so, maybe not. I tell you “.999 = 1 is doubtful, maybe it is true, maybe not.” If “because I say so” works for you then skip the rest of my arguments. Or check out the ancient argument the “10 modes of Skepticism” see, for some very general reasons to be skeptical. Maybe you're like “.999... = 1 hmm, could be” great!
Otherwise, on to reason #2

#2) Numbers, Numerals and Expressions
So if 1 is the loneliest number, is .78+.22 just as lonely? Well, either way, they have the same numerical value. .78+.22 = 1. A number is a kind of abstract object, a kind heavily studied by mathematics. A numeral is a way of trying to express or say or write or pick out a particular number. 5, V, “Five,” “Cinco” and 11(base 4) are all different numerals expressing the same natural number. An expression is a mathematical “phrase” that uses multiple mathematical ideas to pick out a number. So “4.78+.22” or “the next natural number after 4” or “10 divided by 2” or “((2*3)-1)” are all expressions, and all of them pick out a single unique number, 5. Expressions often use operations like addition, or multiplication, or “the successor of.”

OK so yeah .78+.22 really does equal 1. It isn't just close; it is equivalent, or equi-valent, equi/equal and valent/valued, or equal in value. The value of the expression .78+.22 is the same as the value of the numeral 1. When I say or write “.78+.22=1” that is a claim or proposition or statement, it is the sort of thing that could be true or false. Philosophers say it is a potential bearer of truth-value. Its the moral equivalent of a complete sentence, rather than just being a phrase or expression, which is an incomplete part of a sentence. And the claim .78+.22=1 happens to be true. It expresses a judgment about the relation between two terms which are being compared, the expression “.78+.22” and the numeral “1” and judging them to be equal in value. The statement .73+.22=1 is also a claim. It makes sense syntactically and semantically. It is just wrong, or false. We know what both sides mean and relationship we are claiming between them, but the relationship claimed does not in fact hold. But things like “.78+lemon=1” or “.78+=1” or “.78+22fig=1” or “.78.34.6*6+.22===3” or “.78+.22={1.1, 1, .9}” these aren't even claims or statements, they are syntax errors, what we call “non-well-formed formulae.” We can't determine their meaning well enough to evaluate whether they are true or false. “.77 + ?? = 1” isn't a statement, rather it's a statement-schema. It is an incomplete statement that could be completed in various ways. One way of completing the statement will wind up making the claim true, and other ways of completing the statement will wind up making it false.

So what about “.999... = 1”? Well, 1 is numeral here surely, but what about “.999...”? what exactly is that supposed to be, another numeral? And expression? A partial expression that we need to fill in somehow before it is complete? So my second point isn't really a proof, or an argument yet, but some background and a reason to keep an open mind. It's not really very clear what exactly the “.999...” side of the equation is supposed to be.

#3) Decimal numbers
You might think that .999... is a decimal number. But it is not.
.999 is a decimal number. And so is .999999 and even .999999999999999999999999999 is a decimal number. But .999... is something else. It is a short decimal number followed by three periods.
Maybe it is an abbreviation for some decimal number? OK, but in that case, the claim comes out false. If .999... is an abbreviation for .99999, then it is exactly .000001 less than 1, and therefore not equal to 1. Similarly if we take .999... as an abbreviation for .9999999999999999999999 then still it is not quite equal to 1. Maybe .999... is an abbreviation for a number that is “too long” to write. If so then the difference between it and 1 will be “too small” to write, but there will still be one.

But people who think .999... =1, don't usually think of it as an abbreviation of any specific decimal number, but as an abbreviation for an “infinite” string of 9s. It's more as a summation or series. We'll talk about series and real numbers in a minute, maybe that is one of those, but what it ISN'T is a decimal number. It has the wrong syntax for a decimal number. It is too long.

#4) the Standard Proof
The usual proof that .999...=1 goes like this
Suppose .999...=X
multiply each side by 10, and supposedly you get 9.999...=10x
Then subtract the original equation from that, so
9.999... =10x
-.999... -x
9.000... = 9x
if we imagine the 9's “all” cancel out, or we have an infinite string of 0s, that might look the same as
9=9x, if so we can divide both sides by 9 and get

OK so why do I doubt this proof? Well … try doing this on a simple calculator. What happens?
For starters, we can't even type .999... into the calculator, because … is not one of the basic symbols of algebra. It's not well-defined in basic algebra. And if you try to input an infinite string of nines your finger gets tired and the calculator overflows before you get there.

So how exactly do we know what .999... multiplied by 10 equals? I know what .999 x10 equals, but I don't know how to multiply a “…” by 10! Well it's just an abbreviation for a LOT of nines right? No if I interpret it that way things don't work as we saw earlier.
.99999 = x
9.9999 = 10x
-.99999 -x
8.99991 = 9x, and then x=.99999 not 1.

OK but what if we have an infinite number of nines? So that we never get to the last nine?
Well then I just don't know what 10 times that is. If I try to calculate it, my calculation never completes. (Indeed, if I use a device to do it, I can never even finish entering the problem into my calculation device). I get closer and closer to an answer, but I never get an answer. Indeed, if I use the standard pencil and paper decimal multiplication algorithm my kids learn in elementary school, I never finish writing the problem down. Nor do the rules of elementary algebra tell me what the answer would be. They don't have rules for dealing with infinite strings of digits at all. Forget semantics, and operations, even at the level of syntax the basic rules buckle in the face of this puppy. If we really mean .999... to be an abbreviation for an infinite string of 9s, then we are well outside of the realm of calculators and basic addition and algebra. Computers choke on infinity, and so do basic sections of math. We'll have to use something more sophisticated. As Vihart says in section 5, “elementary algebra can't deal with infinity, if you allow infinity in your elementary algebrazations you get contradictions.” All the same points apply to the subtraction step too.

This simple little proof doesn't prove what it is trying to. It assumes what is under dispute, when it assumes what happens if you multiply .999... by 10 or when you subtract .999... from 9.999..., and that means it is question-begging. We can't do either of these operations at the level of regular algebra.

More basically, what is going on is that we have two conflicting mathematical intuitions about an infinite string of nines in a decimal expansion. We think about .999, .99999, .99999999, and .99999999, and we notice 2 things: they keep getting closer to 1, and they are always just a little less than 1. So if we have an infinite number of nines, what happens then? Well one intuition says it should be infinitely close to 1, and another intuition says it should still be slightly less than 1. But without a more rigorous set of rules, we can't disambiguate whether the result is the first, the second or both.

#5) Real Numbers
So maybe instead of thinking of .999... a a string of digits in a decimal expansion, we should think of it as a real number. Real numbers aren't that much more advanced than basic algebra and even high school students routinely deal with them. There are more real numbers than can be expressed with decimal expansions of finite length. Pi and E and the square root of 2 are famous examples of real numbers that cannot be expressed precisely by any finite string of decimal digits.

The problem is that rational numbers and real numbers are both dense sets. Between any two real numbers there are an infinite number of other other real numbers. So between 3.1415 and 3.1416 there are an infinite number of real numbers, one of which is pi, and the others of which are not. Between 3.14159 and 3.14160 there are STILL an infinite number of real numbers one of which is pi, and the rest of which are not. So 3.1415... doesn't really “name” any particular real number, it picks out an infinite number of different real numbers, one of which, pi, is particularly interesting to us for various reasons.

The same is true with .999... If we construe it as referring to those real numbers which begin with .999. It isn't the proper name of any particular real number (1 or otherwise), rather it picks out a group of real numbers. That means that .999... =1 isn't exactly true or false, if we take it to be referring to real numbers, rather it is unfinished, incomplete or perhaps it is a syntactical mismatch (comparing a incomplete expression to a complete one). It is comparing a group of real numbers with different values (with at most 1 of them equal to 1) to a single number. .999... construed this way would include .9998, and that isn't what Vihart, or anyone else meant. But because the real numbers are dense, no matter how explicit we are in decimal expansions, we will ALWAYS be picking out an infinite number of nearby real numbers. There is no such thing as the largest real number less than 1. But there is also no such thing as the smallest real number larger than .99999999999. If .999... and .99999999..., and .9999999999999999... all refer to infinite sets of real numbers that are slightly less than (or perhaps equal to) 1, then the claim .999...=1 comes out false or incomplete rather than true, so long as we take it to be about real numbers.

The traditional way to define real numbers is via a set-theoretic construction called Dedekind cuts. There will always be an infinite number of real numbers between any finite number of .9's and 1. So in the Dedekind construction, the phrase “.999...” fails to refer uniquely. It is always talk about a neighborhood of real numbers rather than a single real number. There is an alternate construction of the reals called Stevin's construction. In Stevin's construction .999...=1 BY DEFINITION. But then we can give a sensical interpretation of the rest of the reals, that is in a sense equivalent to the Dedekind construction. But notice what this means, it means that .999...=1 is OPTIONAL rather than true. If we decide to make its truth an axiom, or a naming convention, we get an completely sane and workable system. But if we decide to leave its truth undetermined we still get a sane and workable system of real numbers. There are lots of workable constructions of the reals, there is probably even one (or more) where we make .999...=1 come out false by definition.

The basic problem is that there are so many reals, that .999... just doesn't pick out a definite single real number unless we rig our naming conventions somehow, and even if we do it will still be surrounded by an infinity of unnamable near neighbors, as we'll see in a bit.

#6) Hyperreals and Surreals
Ok here is another little prooflet

Suppose there is a difference between .999... and 1
1-.999... = ?
0.00000000... now it can be tempting to think there is some kind of final 1, “beyond infinity”
say “.00000...1”
now Vihart says “Of course, if the zero repeats infinitely you never get to the 1” so “you might as well leave it off the number … There is no difference.”

Notice that if this reasoning works, then it applies equally to the prooflet in #4. You try to multiply .999... by ten and you “never get done.” You try to subtract 9.999...- .9999... and you never get done. Infinities don't get completed, they just keep going on and on.

But suppose we have the opposite intuition, that .0000...1 is a way of trying to express something infinitesimal. And we mean by this argument that there is some infinitely small difference between .999... and 1. Well then we would need a system of numbers that can deal with such things. And we have one, the surreal numbers extend the real numbers with infinite and infinitesimal numbers of various kinds.

Now in a sense, this extension doesn't settle our question either, but it can also clarify our intuitions. In the Surreal number system, there is an infinitesimal number ε, which is larger than 0, but smaller than all positive dyadic fractions. (Conway's construction of the surreals basically uses binary rather than decimals, and has to use transfinite induction to get to the decimals, but it can). But there is a sense in which the surreal number ε, is .00000...1(base2). Now we are doing mathematical poetry here, we are trying to explain rigourously defined things in one system, via loose analogies to another system. But in the same loose analogy sense, “1-ε” (which is a well-formed surreal number) is larger than .1111...(base2), but less than 1(base 2).

Essentially what this means is that we have 3 logically distinct concepts, which are strictly value-wise distinct in the surreal number system:

9/10+9/100+9/1000+9/10000 … < infinitesimally less than 1 < 1

At the level of surreal numbers these 3 number concepts are all distinct values, (and hence .999... does not equal 1, because there is s distinct value in-between them). But at the level of real numbers, these three ideas are so close together than when we try to talk about one of them, we often can't distinguish which of them we are talking about. (Without some further convention or axiom - that is our ways of talking pick out a variety of nearby real numbers in the neighborhood of of the infinite sum and 1.).

Now the surreal numbers are cool, but also tricky, and fairly newly discovered (1974, first well explored in 1976 with plenty of discoveries about them since). In essence a lot of older mathematicians don't distinguish between “.999...”, “infinitesimally less than 1” and “1” because the surreals are pretty recent, and the distinction doesn't make a lot of sense in the real numbers, although Vihart clearly gets this distinction.

#7) Neighborhoods

Vihart argues “If .999... and 1 were distinct real numbers there would have to be a real number higher than .999... and lower than one.” And that is true. There would have to be, because real numbers are dense. Further we don't have an easily nameable distinct real number in-between the two ideas in the way that there IS such a distinct surreal number. But the problem is that .999... isn't a distinct real number at all. It is an incomplete description of a rough neighborhood of many different real numbers, many of which are distinct from 1 (which is a real number). Or if you prefer it is an infinite summation or series that converges to a neighborhood including many distinct real numbers.

In calculus we play this game of “neighborhoods.” Say how accurate you want me to be, and I'll give you a whole slew of numbers that are at least that close to the target. 1+ or - .001, no problem! 1 + or minus .0000001? no problem, lots of real numbers are that close to 1.

So when we try to figure out what the sum of the series .9+.09+.009+.0009+.00009 … = ?? we can't actually give a real number. If we try to do the addition, we never complete the task. But what we CAN do is say what it is CLOSE to.
[my textprocessor is choking on the standard notion here but]
The Sum from x=1 to infinity of 9/(10^x) “converges to” 1. What does “converges to” mean? Well roughly that if you pick any non-zero margin of error δ, I can find an even smaller positive real margin of error ε, and prove that the answer is at least that close to the target. Now “converging to” 1 isn't and can't be the same as just plain equaling 1 (if it were it would ruin it's usefulness in calculus where the whole point is it allows us to almost but not quite divide by zero). “Converging to” 1 means that we get as close as we like to 1, but can never quite say if we equal 1 or not.

So we can't say that .9+.09+.009+.0009 … =1 at the level of real numbers. Maybe it does, maybe it doesn't. We can't prove it either way. We CAN say that it gets close to 1. That it gets as close as we like to 1. That it is in a very small neighborhood that includes 1. But because the real numbers are dense, there are always also other numbers which are different from one, which are ALSO in the very small neighborhood around 1. And any of those MIGHT be the value of .9+.09+.009+.0009 … We can't give these other real numbers finite, unique names, because there are just too many real numbers to give them all names like that. Indeed, any real number we CAN give a finite definite name to other than 1, is guaranteed not to be the proper sum. But there are all these unnameable real numbers around 1, that might be the proper sum. Heck, we can't prove that .9+.09+.009... doesn't equal 1 at the level of real numbers either, it always might. But no matter how precise we get, 1 is always one of an infinite number of (mostly unnameable) real numbers that MIGHT be the proper summation of the series. And of course at the level of surreal numbers we can prove the distinctness of the 3 concepts (sum of the series of nines < number infinitesimally less than 1 < 1), so it is often hard to tell which of the concepts we were groping for when we could only use real number language and limits and neighborhoods and such to talk about them.

#8) Ok, now I'm just rebutting instead of explaining my own views ...
Here is another purported prooflet “Take .333...= 1/3, multiply both sides by 3, you get .999...=1”
As noted this one is very simple, and relies only on the idea that .333...=1/3 and that .333...*3=.999...
But we've seen we have reasons to doubt both of these claims.
.333... “converges to” 1/3, that means it is as close as you like to 1/3, but that doesn't mean that it is equal to 1/3. .333... gets so close to 1/3 that we can't give names to any of the other real numbers near enough to also be contenders, but there always will be other contenders besides 1/3.
.333... *3 is an operation that never terminates, it never finishes computing. It is not always well-defined within the reals (unless we make it equal to 1/3 by definition at which point our argument is question-begging again). Again, if we want, we can estimate the value of .333...*3, and we can prove that it will converge to .999... or that it will converge to 1, but we can't prove what it equals.

#9) Infinitely protected, but infinitely unclear ...
So you might argue that .9+.09+.009+.0009 … =1 is going to be wrong because at every step of the way the sum is less than 1. And that is true, but it is also getting closer to one. The tension here is between our competing number intuitions, that the sum is always less than one and that it is also always getting closer to one (which in the surreal case turn out both to be correct). But Vihart takes a different tack, she argues “but infinity has got your back,” “your sum is always less than one so far, but you also always haven't gotten an infinity of terms added yet.” And that is right, but it doesn't mean that if you could add an infinite number of terms you would get to 1, it just means that we always can't tell yet. Infinity protects, but it also mystifies. Can't tell yet. Still can't tell... At the level of real numbers neither side can prove their point, that the sum is 1 or that it isn't. We are in the classic skeptical position of suspending judgment.

#10) But .999...=1 WORKS!

Well, yeah, it does, but it doesn't work as WELL as some of the alternatives.
The Stevin construction of the real numbers works. If you wanna just define .999... as equal to 1, rather than trying to argue for it, nothing particularly bad happens. It can be done consistently. But the Dedekind construction of the reals works too. If you want to stay neutral on whether .999... equals 1 at the real number level, nothing bad happens either.

It is like the axiom of choice. Some mathematicians are strongly convinced of its axiomatic truth, others are suspicious. Or even better, there is something called the generalized continuum hypothesis (which is actually about the nature of infinity). It works. But the negation of it works too. So does simply staying neutral on it. In math we occasionally have options ...

It isn't very common that mathematicians fight, but it can happen, and part of what is cool about .999...=1 is that it is one of the easier cases to understand where mathematicians have legitimate cause to disagree.

If you said 5/3 is unsolvable over the whole numbers, you'd be right. That answer works. But the answer 5/3=1 2/3 is just plain a better answer most of the time. Harder, not always the best approach (if there really is a reason to be restricting yourself to whole numbers), but generally it is the better answer. Negative numbers, imaginary numbers, irrational numbers, quaterion numbers. Each of these seemed wacky when first developed, and they are harder, but they are also often BETTER answers than other also correct answers.

The surreal numbers are like that too. They work. But they also give better answers than we could before to a lot of questions about transfinite values, infinitesimals, games, and so on. And in the surreal numbers .999...=1 is just plain wrong.

“The square root of negative one has no answer (over the reals)” is correct but missing the point. To my mind “there is no provable difference between .999... and 1 (over the real numbers)” is equally correct but equally missing the point. There IS a difference over the surreal numbers, and that difference captures the conflicting intuitions that we had been trying to express all along, about .999... always being close to 1 but not quite equal to it.

Vihart say. “The rules of elementary algebra and real numbers can't tell the difference between .999... and 1” and that's true, (well truish). But that doesn't mean that there ISN'T a difference, just that the difference is too subtle to be a big deal for those systems. (Depending on how you set up the rules they might be able to tell the difference at the syntactical level of one number being a formula of finite length, and the other being a non-finite symbol string.)

But Vihart and I largely agree on the moral of the story. Math is beautiful and cool, and there is more than one way to understand it and think about it. It is so common for there to be a single right answer in math, 2+2=4, positive square root of 121 is 11, etc., that we can forget that it isn't always so. That some questions in math can be sources of legitimate disagreement by smart people that understand what is going on. The real numbers are beautiful and useful, and you can do all kinds of neat things with them. But so are the split-octonions. Or the surreal numbers. Things like “.999...” that aren't exactly expressions or phrases, that don't exactly fit into the rules, but are still so close to the rules as to be suggestive, to hint at the spirit animating the rules, to egg us on to coming up with even deeper rules ... these are fun! And bickering about them in a productive way might even help us to better understand the concepts nearby them in interesting ways, and that pleasure of sudden new understanding is what math is all about.

By Dr. B. P. R. Morton. Reach me at if you'd like to reply. (I can't even claim to be a counter-example to your 99.999...% of mathematicians believe .999...=1, because I'm not really a mathematician. I used to be a mathematical logician, which is basically a philosopher trying to dress up as a mathematician and hang around with mathematicians.  When I wrote this in 2013, I was a housewife.  Now I'm a high school math teacher.)

Also see my friend Dr. Axel Barcelo's arguments on the deductivity of this here.