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 http://en.wikipedia.org/wiki/Sextus_Empiricus, 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
1=x=.999....
Voila!
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... = ?
1.00000000...
-.99999999...
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 bprmorton@gmail.com 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.
