I agree that a careful thinker confronted with this puzzle for the first time should eventually conclude that the crux is what exactly the expression “0.999...” actually means. At this point, if you don’t know enough math to give a rigorous definition, I think a reasonable response is “I thought I knew what it meant to have an infinite number of 9s after the decimal point, but maybe I don’t, and absent me actually learning the requisite math to make sense of that expression I’m just going to be agnostic about its value.”
Here’s an argument in favor of doing that. Consider the following proof, nearly identical to the one you present. Let’s consider the number x = …999; in other words, now we have infinitely many 9s to the left of the decimal point. What is this number? Well,
10x = …9990
x − 10x = 9
-9x = 9
x = −1.
There are a couple of reasonable responses you could have to this argument. Two of them require knowing some math: one is enough math to explain why the expression …999 describes the limit of a sequence of numbers that has no limit, and one is knowing even more math than that, so you can explain in what sense it does have a limit (the details here resemble the details of 1 + 2 + 3 + … but are technically easier). I think in the absence of the requisite math knowledge, seeing this argument side by side with the original one makes a pretty strong case for “stay agnostic about whether this notation is meaningful.”
And on the third hand, I can’t resist saying one more thing about infinite sequences of decimals to the left. Consider the following sequence of computations:
5^2 = 25
25^2 = 625
625^2 = 390625
0625^2 = 390625
90625^2 = 8212890625
890625^2 = 793212890625
It sure looks like there is an infinite decimal going to the left, x, with the property that x^2 = x, and which ends …890625. Do you agree? Can you find, say, 6 more of its digits, assuming it exists? What’s up with that? Is there another x with this property? (Please don’t spoil the answer if you know what’s going on here without some kind of spoiler warning or e.g. rot13.)
Actually, gur zbfg vzcbegnag ahzore-gurbergvp cebcregl gung qevirf guvf curabzraba vf gur snpg gung gra unf zber guna bar cevzr snpgbe. Vg pna’g unccra va n cevzr be cevzr cbjre onfr (naq gurfr ner onfvpnyyl gur fnzr guvat sbe gurfr checbfrf naljnl). Gur ehyr qrfpevovat juvpu vavgvny qvtvgf znxr guvatf jbex vf zber pbzcyvpngrq; sbe fgnegref, gel fvk va onfr gra, gura gel onfr svsgrra.
If you want to look up more, gur xrljbeq vf c-nqvp ahzoref. Urer jr’er jbexvat va n ahzore flfgrz pnyyrq gur gra-nqvp ahzoref. Gur gra-nqvp ahzoref sbez n pbzzhgngvir evat, juvpu onfvpnyyl zrnaf lbh pna nqq naq zhygvcyl gurz naq gur hfhny ynjf bs nytroen lbh’er snzvyvne jvgu jvyy nccyl. (Guvf vf fbzrguvat lbh pna irevsl sbe lbhefrys: gung vg nyjnlf znxrf frafr gb nqq naq zhygvcyl vasvavgr qrpvznyf gb gur yrsg. Lbh whfg xrrc pneelvat shegure naq shegure gb gur yrsg.) Ohg hayvxr gur erny ahzoref, gurl qba’g sbez n svryq, juvpu zrnaf lbh pna’g nyjnlf qvivqr ol n abamreb gra-nqvp ahzore.
Let’s consider the number x = …999; in other words, now we have infinitely many 9s to the left of the decimal point.
My gut response (I can’t reasonably claim to know math above basic algebra) is:
Infinite sequences of numbers to the right of the decimal point are in some circumstances an artifact of the base. In base 3, 1⁄3 is 0.1 and 1⁄10 is 0.00220022..., but 1⁄10 “isn’t” an infinitely repeating decimal and 1⁄3 “is”—in base 10, which is what we’re used to. So, heuristically, we should expect that some infinitely repeating representations of numbers are equal to some representations that aren’t infinitely repeating.
If 0.999… and 1 are different numbers, there’s nothing between 0.999… and 1, which doesn’t jive with my intuitive understanding of what numbers are.
The integers don’t run on a computer processor. Positive integers can’t wrap around to negative integers. Adding a positive integer to a positive integer will always give a positive integer.
0.999… is 0.9 + 0.09 + 0.009 etc, whereas …999.0 is 9 + 90 + 900 etc. They must both be positive integers.
There is no finite number larger than …999.0. A finite number must have a finite number of digits, so you can compute …999.0 to that many digits and one more. So there’s nothing ‘between’ …999.0 and infinity.
Infinity is not the same thing as negative one.
All I have to do to accept that 0.999… is the same thing 1 is accept that some numbers can be represented in multiple ways. If I don’t accept this, I have to reject the premise that two numbers with nothing ‘between’ them are equal—that is, if 0.999… != 1, it’s not the case that for any x and y where x != y, x is either greater than or less than y.
But if I accept that …999.0 is equal to −1, I have to accept that adding together some positive numbers can give a negative number, and if I reject it, I just have to say that multiplying an infinite number by ten doesn’t make sense. (This feels like it’s wrong but I don’t know why.)
I think you mean “they must both be positive” here, but 0.999… isn’t guaranteed to be an integer a priori.
Aside from that, everything you’ve said is basically correct. But… well, there’s something pretty interesting going on with infinite decimals to the left. For numbers that don’t exist they sure do have a lot of interesting properties. This might be worth a top-level post.
Interesting, I’ve never looked closely at these infinitely-long numbers before.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1. I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12 though I still reject that that’s a fair use of the symbols ‘+’ and ‘=’ without qualification).
Re the second part: interesting. Nothing is immediately coming to mind.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1.
Yes, this is one way of justifying the claim that −1 is the “right” answer, via analytic continuation of the function 9/(1 - x). But there’s another arguably more fun way involving making rigorous sense of infinite decimals going to the left in general.
I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12
Cesaro and Abel summation don’t assign a value to either of these series.
I think a reasonable position is “I personally do not know how to make sense of this notation,” but are you claiming that “nobody knows how to make sense of this notation”? Would you be willing to make a bet to that effect, and at what odds, for how much money?
Thomas, I think you may be misunderstanding what Qiaochu_Yuan is trying to do here, which is not to argue that 0.999… actually might (for all he knows, or for all you know) be something other than 1, nor to argue that any particular other non-standard[1] construction actually might (for all he or you know) have a coherent meaning.
Rather, he is saying: someone who hasn’t come across this stuff before might reasonably not see any important difference between these constructions (if the difference seems obvious to you, it’s only because you have seen it before; it took mathematicians a long time to figure out how to think correctly about these things) and adopt parallel attitudes to them. This would be reasonable, and rational in any sense that doesn’t require something like logical omniscience.
It seems as if you are arguing against the first sort of claim (which I believe QY is not making) rather than the second (which I believe he is making).
[1] In the sense of “not usually used in mathematics”, not that of “model of real analysis with infinities, infinitesimals and a transfer principle”.
What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.
I agree that a careful thinker confronted with this puzzle for the first time should eventually conclude that the crux is what exactly the expression “0.999...” actually means. At this point, if you don’t know enough math to give a rigorous definition, I think a reasonable response is “I thought I knew what it meant to have an infinite number of 9s after the decimal point, but maybe I don’t, and absent me actually learning the requisite math to make sense of that expression I’m just going to be agnostic about its value.”
Here’s an argument in favor of doing that. Consider the following proof, nearly identical to the one you present. Let’s consider the number x = …999; in other words, now we have infinitely many 9s to the left of the decimal point. What is this number? Well,
10x = …9990
x − 10x = 9
-9x = 9
x = −1.
There are a couple of reasonable responses you could have to this argument. Two of them require knowing some math: one is enough math to explain why the expression …999 describes the limit of a sequence of numbers that has no limit, and one is knowing even more math than that, so you can explain in what sense it does have a limit (the details here resemble the details of 1 + 2 + 3 + … but are technically easier). I think in the absence of the requisite math knowledge, seeing this argument side by side with the original one makes a pretty strong case for “stay agnostic about whether this notation is meaningful.”
And on the third hand, I can’t resist saying one more thing about infinite sequences of decimals to the left. Consider the following sequence of computations:
5^2 = 25
25^2 = 625
625^2 = 390625
0625^2 = 390625
90625^2 = 8212890625
890625^2 = 793212890625
It sure looks like there is an infinite decimal going to the left, x, with the property that x^2 = x, and which ends …890625. Do you agree? Can you find, say, 6 more of its digits, assuming it exists? What’s up with that? Is there another x with this property? (Please don’t spoil the answer if you know what’s going on here without some kind of spoiler warning or e.g. rot13.)
Zl svefg thrff vf gung guvf jnf pnhfrq ol svir orvat unys bs gra, naq fb vs jr jnagrq gb unir gur fnzr cebcregl va urknqrpvzny jr jbhyq vafgrnq or ybbxvat ng gur cebterffvba onfrq ba rvtug. Ohg gung qvqa’g jbex, naq fb abj V’z fhfcrpgvat gung vg’f nyfb vzcbegnag gung svir vf bqq. (Vg jbexf vs lbh fgneg jvgu guerr va onfr fvk, juvpu znxrf zr thrff gubfr ner gur cevznel erdhverzragf, ohg vg zvtug nyfb or vzcbegnag gb or cevzr. (Ybbxvat ng gur cebterffvba fgnegvat jvgu k=avar va onfr rvtugrra, gung ybbxf evtug, naq onfr sbhegrra cebivqrf zber pbasvezngvba.)
Actually, gur zbfg vzcbegnag ahzore-gurbergvp cebcregl gung qevirf guvf curabzraba vf gur snpg gung gra unf zber guna bar cevzr snpgbe. Vg pna’g unccra va n cevzr be cevzr cbjre onfr (naq gurfr ner onfvpnyyl gur fnzr guvat sbe gurfr checbfrf naljnl). Gur ehyr qrfpevovat juvpu vavgvny qvtvgf znxr guvatf jbex vf zber pbzcyvpngrq; sbe fgnegref, gel fvk va onfr gra, gura gel onfr svsgrra.
If you want to look up more, gur xrljbeq vf c-nqvp ahzoref. Urer jr’er jbexvat va n ahzore flfgrz pnyyrq gur gra-nqvp ahzoref. Gur gra-nqvp ahzoref sbez n pbzzhgngvir evat, juvpu onfvpnyyl zrnaf lbh pna nqq naq zhygvcyl gurz naq gur hfhny ynjf bs nytroen lbh’er snzvyvne jvgu jvyy nccyl. (Guvf vf fbzrguvat lbh pna irevsl sbe lbhefrys: gung vg nyjnlf znxrf frafr gb nqq naq zhygvcyl vasvavgr qrpvznyf gb gur yrsg. Lbh whfg xrrc pneelvat shegure naq shegure gb gur yrsg.) Ohg hayvxr gur erny ahzoref, gurl qba’g sbez n svryq, juvpu zrnaf lbh pna’g nyjnlf qvivqr ol n abamreb gra-nqvp ahzore.
My gut response (I can’t reasonably claim to know math above basic algebra) is:
Infinite sequences of numbers to the right of the decimal point are in some circumstances an artifact of the base. In base 3, 1⁄3 is 0.1 and 1⁄10 is 0.00220022..., but 1⁄10 “isn’t” an infinitely repeating decimal and 1⁄3 “is”—in base 10, which is what we’re used to. So, heuristically, we should expect that some infinitely repeating representations of numbers are equal to some representations that aren’t infinitely repeating.
If 0.999… and 1 are different numbers, there’s nothing between 0.999… and 1, which doesn’t jive with my intuitive understanding of what numbers are.
The integers don’t run on a computer processor. Positive integers can’t wrap around to negative integers. Adding a positive integer to a positive integer will always give a positive integer.
0.999… is 0.9 + 0.09 + 0.009 etc, whereas …999.0 is 9 + 90 + 900 etc. They must both be positive
integers.There is no finite number larger than …999.0. A finite number must have a finite number of digits, so you can compute …999.0 to that many digits and one more. So there’s nothing ‘between’ …999.0 and infinity.
Infinity is not the same thing as negative one.
All I have to do to accept that 0.999… is the same thing 1 is accept that some numbers can be represented in multiple ways. If I don’t accept this, I have to reject the premise that two numbers with nothing ‘between’ them are equal—that is, if 0.999… != 1, it’s not the case that for any x and y where x != y, x is either greater than or less than y.
But if I accept that …999.0 is equal to −1, I have to accept that adding together some positive numbers can give a negative number, and if I reject it, I just have to say that multiplying an infinite number by ten doesn’t make sense. (This feels like it’s wrong but I don’t know why.)
I think you mean “they must both be positive” here, but 0.999… isn’t guaranteed to be an integer a priori.
Aside from that, everything you’ve said is basically correct. But… well, there’s something pretty interesting going on with infinite decimals to the left. For numbers that don’t exist they sure do have a lot of interesting properties. This might be worth a top-level post.
Interesting, I’ve never looked closely at these infinitely-long numbers before.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1. I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12 though I still reject that that’s a fair use of the symbols ‘+’ and ‘=’ without qualification).
Re the second part: interesting. Nothing is immediately coming to mind.
Yes, this is one way of justifying the claim that −1 is the “right” answer, via analytic continuation of the function 9/(1 - x). But there’s another arguably more fun way involving making rigorous sense of infinite decimals going to the left in general.
Cesaro and Abel summation don’t assign a value to either of these series.
(answer: http://www.numericana.com/answer/p-adic.htm#integers)
This has no sense, really.
I think a reasonable position is “I personally do not know how to make sense of this notation,” but are you claiming that “nobody knows how to make sense of this notation”? Would you be willing to make a bet to that effect, and at what odds, for how much money?
I am saying you cannot write …9990 - the decimal point, then an infinite number of 9s and then the last zero!
Okay, perhaps you can in some other axiomatic system. But not for the ordinary real numbers.
Sure. What is different about the situation with 0.999...? How do you know that that is a sensible name for a real number?
0.999… is the limit of 9/10+9/100+9/1000+...
...9990 is what?
Thomas, I think you may be misunderstanding what Qiaochu_Yuan is trying to do here, which is not to argue that 0.999… actually might (for all he knows, or for all you know) be something other than 1, nor to argue that any particular other non-standard[1] construction actually might (for all he or you know) have a coherent meaning.
Rather, he is saying: someone who hasn’t come across this stuff before might reasonably not see any important difference between these constructions (if the difference seems obvious to you, it’s only because you have seen it before; it took mathematicians a long time to figure out how to think correctly about these things) and adopt parallel attitudes to them. This would be reasonable, and rational in any sense that doesn’t require something like logical omniscience.
It seems as if you are arguing against the first sort of claim (which I believe QY is not making) rather than the second (which I believe he is making).
[1] In the sense of “not usually used in mathematics”, not that of “model of real analysis with infinities, infinitesimals and a transfer principle”.
I see your angle now. Perhaps his angle, too.
What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.