# 0.999...=1: Another Rationality Litmus Test

Peo­ple seemed to like my post from yes­ter­day about in­finite sum­ma­tions and how to ra­tio­nally re­act to a math­e­mat­i­cal ar­gu­ment you’re not equipped to val­i­date, so here’s an­other in the same vein that high­lights a differ­ent way your rea­son­ing can go.

(It’s prob­a­bly not quite as juicy of an ex­am­ple as yes­ter­day’s, but it is one that I’m equipped to write about to­day so I figure it’s worth it.)

This ex­am­ple is some­what more widely known and a bit more el­e­men­tary. I won’t be sur­prised if most peo­ple already know the ‘solu­tion’. But the point of writ­ing about it is not to ex­plain the math—it’s to talk about “how you should feel” about the prob­lem, and how to ra­tio­nally ap­proach rec­tify­ing it with your ex­ist­ing men­tal model. If you already know the solu­tion, try to pre­tend or think back to when you didn’t. I think it was ini­tially sur­pris­ing to most peo­ple, when­ever you learned it.

The claim: that 1 = 0.999… re­peat­ing (in­finite 9s). (I haven’t found an easy way to put a bar over the last 9, so I’m us­ing el­lipses through­out.)

The ques­tion­able proof:

x = 0.9999...
10x = 9.9999… (ev­ery­one knows mul­ti­ply­ing by ten moves the dec­i­mal over one place)
10x-x = 9.9999… − 0.9999....
9x = 9
x = 1

Peo­ple’s re­sponse when they first see this is usu­ally: wait, what? an in­finite se­ries of 9s equals 1? no way, they’re ob­vi­ously differ­ent.

The lit­mus test is this: what do you think a ra­tio­nal per­son should do when con­fronted with this ar­gu­ment? How do you ap­proach it? Should you ac­cept the seem­ingly plau­si­ble ar­gu­ment, or re­ject it (as with yes­ter­day’s ex­am­ple) as “no way, it’s more likely that we’re some­how talk­ing about differ­ent ob­jects and it’s hid­den in­side the no­ta­tion”?

Or are there other ways you can pro­ceed to get more in­for­ma­tion on your own?

One of the things I want to high­light here is re­lated to the na­ture of math­e­mat­ics.

I think peo­ple have a ten­dency to think that, if they are not well-trained stu­dents of math­e­mat­ics (at least at the col­le­giate level), then rigor or pre­ci­sion in­volv­ing num­bers is out of their reach. I think this is definitely not the case: you should not be afraid to at­tempt to be pre­cise with num­bers even if you only know high school alge­bra, and you should es­pe­cially not be afraid to de­mand pre­ci­sion, even if you don’t know the cor­rect way to im­ple­ment it.

Par­tic­u­larly, I’d like to em­pha­size that math­e­mat­ics as a men­tal dis­ci­pline (as op­posed to an aca­demic field), ba­si­cally con­sists of “the art of mak­ing cor­rect state­ments about pat­terns in the world” (where num­bers are one of the pat­terns that ap­pears ev­ery­where you have things you can count, but there are oth­ers). This sounds sus­pi­ciously similar to ra­tio­nal­ity—which, as a prac­tice, might be “about win­ning”, but as a men­tal art is “about be­ing right, and not be­ing wrong, to the best of your abil­ity”. More or less. So math­e­mat­i­cal think­ing and ra­tio­nal think­ing are very similar, ex­cept that we cat­e­go­rize ra­tio­nal­ity as be­ing pri­mar­ily about de­ci­sions and real-world things, and math­e­mat­ics as be­ing pri­mar­ily about ab­stract struc­tures and num­bers.

In many cases in math, you start with a struc­ture that you don’t un­der­stand, or even know how to un­der­stand, pre­cisely, and start try­ing to ‘tease’ pre­cise re­sults out of it. As a layper­son you might have the same ap­proach to ar­gu­ments and state­ments about el­e­men­tary num­bers and alge­braic ma­nipu­la­tions, like in the proof above, and you’re just as in the right to at­tempt to find pre­ci­sion in them as a pro­fes­sional math­e­mat­i­cian is when they perform the same pro­cess on their highly es­o­teric spe­cialty. You also have the bonus that you can go look for the right an­swer to see how you did, af­ter­wards.

All this to say, I think any ra­tio­nal per­son should be will­ing to ‘go un­der the hood’ one or two lev­els when they see a proof like this. It doesn’t have to be rigor­ous. You just need to do some pok­ing around if you see some­thing sur­pris­ing to your in­tu­ition. In­sights are read­ily available if you look, and you’ll be a stronger ra­tio­nal thinker if you do.

There are a few an­gles that I think a ra­tio­nal but un­trained-in-math per­son can think to take straight­away.

You’re shown that 0.9999.. = 1. If this is a sur­prise, that means your model of what these terms mean doesn’t jive with how they be­have in re­la­tion to each other, or that the proof was fal­la­cious. You can im­me­di­ately con­clude that it’s ei­ther:

a) true with­out qual­ifi­ca­tion, in which case your men­tal model of what the sym­bols “0.999...”, “=”, or “1” mean is sus­pect
b) true in a sense, but it’s hid­den be­hind a de­cep­tive ar­gu­ment (like in yes­ter­day’s post), and even if the sense is more tech­ni­cal and pos­si­bly be­yond your in­tu­ition, it should be pos­si­ble to ver­ify if it ex­ists—ei­ther through care­ful in­spec­tion or turn­ing to a more ex­pert source or just ver­ify­ing that op­tions (a) and (c) don’t work
c) false, in which case there should be a log­i­cal in­con­sis­tency in the proof, though it’s not nec­es­sar­ily true that you’re equipped to find it

More­over, (a) is prob­a­bly the de­fault, by Oc­cam’s Ra­zor. It’s more likely that a seem­ingly cor­rect ar­gu­ment is cor­rect than that there is a more com­pli­cated ex­pla­na­tion, such as (b), “there are mys­te­ri­ous forces at work here”, or (c), “this cor­rect-seem­ing ar­gu­ment is ac­tu­ally wrong”, with­out other rea­sons to dis­be­lieve it. The only ev­i­dence against it is ba­si­cally that it’s sur­pris­ing. But how do you test (a)?

Note there are plenty of other ‘math para­doxes’ that fall un­der (c) in­stead: for ex­am­ple, those ones that se­cretly di­vide by 0 and de­rive non­sense af­ter­wards. (a=b ; a^2=ab ; a^2-b^2=ab-b^2 ; (a+b)(a-b)=b(a-b) ; a+b=b ; 2a = a ; 2=1). But the differ­ence is that their con­clu­sions are ob­vi­ously false, whereas this one is only sur­pris­ing and coun­ter­in­tu­itive. 1=2 in­volves two con­cepts we know very well. 0.999...=1 in­volves one we know well, but one that likely has a feel­ing of sketch­i­ness about it; we’re not used to hav­ing to think care­fully about what a con­struc­tion like 0.999… means, and we should im­me­di­ately re­al­ize that when doubt­ing the con­clu­sion.

Here are a few an­gles you can take to test­ing (a):

1. The “make it more pre­cise” ap­proach: Drill down into what you mean by each sym­bol. In par­tic­u­lar, it seems very likely that the mys­tery is hid­ing in­side what “0.999...” means, be­cause that’s the one that it’s seems com­pli­cated and li­able to be mi­s­un­der­stood.

What does 0.999… in­finitely re­peat­ing ac­tu­ally mean? It seems like it’s “the limit of the se­ries of finite num­bers of 9s”, if you know what a limit is. It seems like it might be “the num­ber larger than ev­ery num­ber of the form 0.abcd..., con­sist­ing of in­finitely many digits (op­tion­ally, all 0 af­ter a point)”. That’s awfully similar to 1, also, though.

A very good ques­tion is “what kinds of ob­jects are these, any­way?” The rules of ar­ith­metic gen­er­ally as­sume we’re work­ing with real num­bers, and the proof seems to hold for those in our cus­tom­ary rule­set. So what’s the ‘true’ defi­ni­tion of a real num­ber?

Well, we can look it up, and find that it’s fairly com­pli­cated and in­volves iden­ti­fy­ing re­als with sets of ra­tio­nals in one or an­other spe­cific way. If you can parse the defi­ni­tions, you’ll find that one defi­ni­tion is “a real num­ber is a Dedekind cut of the ra­tio­nal num­bers”, that is, “a par­ti­tion of the ra­tio­nal num­bers into two sets A and B such that A is nonempty and closed down­wards, B is nonempty and closed up­wards, and A con­tains no great­est el­e­ment”, and from that it Can Be Seen (tm) that the two sym­bols “1” and “0.999...” both re­fer to the same par­ti­tion of Q, and there­fore are equiv­a­lent as real num­bers.

2. The “func­tional” ap­proach: if 0.999...=1, then it should be­have the same as 1 in all cir­cum­stances. Is that some­thing we can ver­ify? Does it sur­vive ob­vi­ous tests, like other ar­gu­ments of the same form?

Does 0.999.. always act the same was that 1 does? It ap­pears to act the same in the alge­braic ma­nipu­la­tions that we saw, of course. What are some other things to try?
We might think to try: 1-0.9999… = 1-1 = 0, but also seems to equal 0.000....0001, if that’s valid: an ‘in­finite dec­i­mal that ends in a 1’. So those must be equiv­a­lent also, if that’s a valid con­cept. We can’t find any­thing to mul­ti­ply 0.000...0001 by to ‘move the dec­i­mal’ all the way into the finite dec­i­mal po­si­tions, seem­ingly, be­cause we would have to mul­ti­ply by in­finity and that wouldn’t prove any­thing be­cause we already know such op­er­a­tions are sus­pect.
I, at least, can­not see any rea­son when do­ing math that the two shouldn’t be the same. It’s not proof, but it’s ev­i­dence that the con­clu­sion is prob­a­bly OK.

3. The “ar­gu­ment from con­tra­dic­tion” ap­proach: what would be true if the claim were false?

If 0.999… isn’t equal to 1, what does that en­tail? Well, let a=0.999… and b=1. We can, ac­cord­ing to our fa­mil­iar rules of alge­bra, con­struct the num­ber halfway be­tween them: (a+b)/​2, al­ter­na­tively writ­ten as a+(b-a)/​2. But our in­tu­ition for dec­i­mals doesn’t seem to let there be a num­ber be­tween the two. What would it be -- 0.999...9995? “cap­ping” the dec­i­mal with a 5? (yes, we capped a dec­i­mal with a 1 ear­lier, but we didn’t know if that was valid ei­ther). What does that mean im­ply 0.999 − 0.999...9995 should be? 0.000...0004? Does that equal 4*0.000...0001? None of this math seems to be work­ing ei­ther.
As long as we’re not be­ing rigor­ous, this isn’t “proof”, but it is a com­pel­ling rea­son to think the con­clu­sion might be right af­ter all. If it’s not, we get into things that seem con­sid­er­ably more in­sane.

4. The “re­ex­am­ine your sur­prise” ap­proach: how bad is it if this is true? Does that cause me to doubt other be­liefs? Or is it ac­tu­ally just as easy to be­lieve it’s true as not? Per­haps I am just bi­ased against the con­clu­sion for aes­thetic rea­sons?

How bad is it if 0.999...=1? Well, it’s not like yes­ter­day’s ex­am­ple with 1+2+3+4+5… = −1/​12. It doesn’t ut­terly defy our in­tu­ition for what ar­ith­metic is. It says that one ob­ject we never use is equiv­a­lent to an­other ob­ject we’re fa­mil­iar with. I think that, since we prob­a­bly have no rea­son to strongly be­lieve any­thing about what an in­finite sum of 910 + 9100 + 9/​1000 + … should equal, it’s perfectly palat­able that it might equal 1, de­spite our ini­tial reser­va­tions.

(I’m sure there are other ap­proaches too, but this got long with just four so I stopped look­ing. In real life, if you’re not in­ter­ested in the de­tails there’s always the very le­gi­t­i­mate fifth ap­proach of “see what the ex­perts say and don’t worry about it”, also. I can’t fault you for just not car­ing.)

By the way, the con­clu­sion that 0.999...=1 is com­pletely, un­equiv­o­cally true in the real num­bers, ba­si­cally for the Dedekind cut rea­son given above, which is the com­monly ac­cepted struc­ture we are us­ing when we write out math­e­mat­ics if none is in­di­cated. It is pos­si­ble to find struc­tures where it’s not true, but you prob­a­bly wouldn’t write 0.999… in those struc­tures any­way. It’s not like 1+2+3+4+5...=-1/​12, for which claiming truth is wildly in­ac­cu­rate and out­right de­cep­tive.

But note that none of these ap­proaches are out of reach to a care­ful thinker, even if they’re not a math­e­mat­i­cian. Or even math­e­mat­i­cally-in­clined.

So it’s not re­quired that you have the fi­nesse to work out de­tailed math­e­mat­i­cal ar­gu­ments—cer­tainly the defi­ni­tions of real num­bers are too pre­cise and tech­ni­cal for the av­er­age layper­son to deal with. The ques­tion here is whether you take math state­ments at face value, or dis­be­lieve them au­to­mat­i­cally (you would have done fine yes­ter­day!), or pick the more ra­tio­nal choice—break­ing them down and look­ing for low-hang­ing ways to con­vince your­self one way or the other.

When you read a sur­pris­ing ar­gu­ment like the 0.999...=1 one, does it oc­cur to you to break down ways of in­spect­ing it fur­ther? To look for con­tra­dic­tions, func­tional equiv­alences, sec­ond-guess your sur­prise as be­ing a run-of-the-mill cog­ni­tive bias, or seek out pre­ci­sion to re­al­ign your in­tu­ition with the ap­par­ent sur­prise in ‘re­al­ity’?

I think it should. Though I am pretty bi­ased be­cause I en­joy math and study it for fun. But—if you sub­con­sciously treat math as some­thing that other peo­ple do and you just be­lieve what they say at the end of the day, why? Does this cause you to ne­glect to ra­tio­nally an­a­lyze math­e­mat­i­cal con­clu­sions, at what­ever level you might be com­fortable with? If so, I’ll bet this isn’t op­ti­mal and it’s worth iso­lat­ing in your mind and look­ing more closely at. Pre­cise math­e­mat­i­cal ar­gu­ment is es­sen­tially just ra­tio­nal­ism ap­plied to num­bers, af­ter all. Well—plus a lot of jar­gon.

(Do you think I rep­re­sented the math or the ra­tio­nal ar­gu­ments cor­rectly? is my philos­o­phy le­gi­t­i­mate? Feed­back much ap­pre­ci­ated!)

• 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.)

• My first guess is that this was caused by five be­ing half of ten, and so if we wanted to have the same prop­erty in hex­adec­i­mal we would in­stead be look­ing at the pro­gres­sion based on eight. But that didn’t work, and so now I’m sus­pect­ing that it’s also im­por­tant that five is odd. (It works if you start with three in base six, which makes me guess those are the pri­mary re­quire­ments, but it might also be im­por­tant to be prime. (Look­ing at the pro­gres­sion start­ing with x=nine in base eigh­teen, that looks right, and base four­teen pro­vides more con­fir­ma­tion.)

• Ac­tu­ally, the most im­por­tant num­ber-the­o­retic prop­erty that drives this phe­nomenon is the fact that ten has more than one prime fac­tor. It can’t hap­pen in a prime or prime power base (and these are ba­si­cally the same thing for these pur­poses any­way). The rule de­scribing which ini­tial digits make things work is more com­pli­cated; for starters, try six in base ten, then try base fif­teen.

If you want to look up more, the key­word is p-adic num­bers. Here we’re work­ing in a num­ber sys­tem called the ten-adic num­bers. The ten-adic num­bers form a com­mu­ta­tive ring, which ba­si­cally means you can add and mul­ti­ply them and the usual laws of alge­bra you’re fa­mil­iar with will ap­ply. (This is some­thing you can ver­ify for your­self: that it always makes sense to add and mul­ti­ply in­finite dec­i­mals to the left. You just keep car­ry­ing fur­ther and fur­ther to the left.) But un­like the real num­bers, they don’t form a field, which means you can’t always di­vide by a nonzero ten-adic num­ber.

• 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, 13 is 0.1 and 110 is 0.00220022..., but 110 “isn’t” an in­finitely re­peat­ing dec­i­mal and 13 “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.)

• They must both be pos­i­tive in­te­gers.

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.

• …9990

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.

• I was able to make sense of this ar­gu­ment through the (rather un­so­phis­ti­cated) rea­son­ing that 0.333...= 13, and mul­ti­ply­ing both sides by three gives 0.999...=1. (I’m not sure if this ac­tu­ally adds any­thing, but it was how I made my­self be­lieve the val­idity of the ar­gu­ment)

• But how do you know that 0.333… = 1/​3? (And that mul­ti­ply­ing an in­finite dec­i­mal by 3 cor­re­sponds to mul­ti­ply­ing each of its digits by 3?)

In the spirit of my com­ment, con­sider the analo­gous ar­gu­ment for in­finite dec­i­mals to the left. Let x = …333. Then

3x = …999 = −1 (we es­tab­lished this ear­lier)

so x = −1/​3. Are you satis­fied with that?

• Thank you for the re­ply! I get that 1/​3= 0.333… from just di­vid­ing 1 by 3, but I do have a lack of un­der­stand­ing for what it means to mul­ti­ply an in­finite dec­i­mal by some in­te­ger. I ap­pre­ci­ate the ex­pla­na­tion with the dec­i­mals to the left!

• This looks like a can­di­date for the not-yet-ex­ist­ing book “The Sim­ple Math of Every­thing”. But the ex­pla­na­tion would have to be the real ex­pla­na­tion, in­volv­ing how we con­struct the “real num­bers” and why we con­struct them that way.

• Cool, an­other one! I’m sup­posed to be sleep­ing now rather than work­ing, so I can en­gage with this.

(b), “there are mys­te­ri­ous forces at work here”

we would have to mul­ti­ply by in­finity and that wouldn’t prove any­thing be­cause we already know such op­er­a­tions are sus­pect.

In­finity is weird, and it makes math weird. I think a fuzzy ver­sion of this be­lief is pretty wide­spread—look what you get when you do an image search for “di­vide by zero”, for ex­am­ple. For me, and I sus­pect for a lot of peo­ple with a very lit­tle gen­eral math knowl­edge, “in­finity” is a stop sign. In­quiry ends, shoulders are shrugged, hands are thrown up. “Of course it doesn’t ap­pear to make sense—it’s got in­finity it it!”.

I don’t re­mem­ber where I got this no­tion but it must have been early, be­cause I re­mem­ber see­ing a ver­sion of the “dis­guise a di­vi­sion by zero > 1=2” trick in a book (Fer­mat’s Last The­o­rem by Si­mon Singh, if any­one’s in­ter­ested) when I was about 14 and be­ing baf­fled by it, and go­ing over and over it try­ing to find the mis­take. When I gave up and read on, and saw the ex­pla­na­tion of how one of the can­celed terms in the equa­tion was zero, I was in­stantly satis­fied. “Oh, of course. It di­vides by zero which is a sneaky way of in­tro­duc­ing in­finity to the mix—so nat­u­rally the re­sult makes no sense.”

This is one of those situ­a­tions where a lit­tle in­com­plete knowl­edge is ac­tu­ally worse than none—a per­son who hadn’t ever heard about the in­finity-makes-ev­ery­thing-weird “rule” could see some­thing like 0.999… = 1 and keep dig­ging, in­stead of say­ing “yeah, that’s in­finity for you, what can you do”.

The idea that in­finity is some sort of mag­i­cal spell that you can cast upon “real” math and turn it into a frog (us­ing real in the ev­ery­day sense, not the math-sense) is ob­vi­ously an ir­ra­tional thought-stop­per. It means you could pre­sent a false state­ment to me and I wouldn’t ques­tion it so long as in­finity was there to point to as the culprit.

(If you’re able to quickly for­mu­late an ex­am­ple of a su­perfi­cially math-y look­ing propo­si­tion in­volv­ing in­finity that’s ac­tu­ally to­tal BS, that would be awe­some—I could use it in fu­ture con­ver­sa­tions about the topic.)

By the way, I’m not talk­ing about some ver­sion of me in the dis­tant past—I re­al­ized that I use “in­finity makes ev­ery­thing weird” as a thought-ter­mi­nat­ing cliche five min­utes ago. I didn’t re­al­ize I was ex­empt­ing math­e­mat­ics from the same sort of bias-ques­tion­ing ra­tio­nal­ity I try to ap­ply to ev­ery­thing else un­til you pointed it out.

So, thanks for that—I still may not un­der­stand why 0.999… = 1, or how di­vid­ing by zero leads to re­sults like 1=2, but at least from now on I won’t let a non-an­swer like “in­finity did it!” kill my cu­ri­os­ity.

• In­fini­ties are okay if they come with a defi­ni­tion of con­ver­gence. For ex­am­ple, we can say that an in­finite se­quence of real num­bers x1, x2, x3… “con­verges” to a real num­ber y if ev­ery in­ter­val of the real line cen­tered around y, no mat­ter how small, con­tains all but finitely many el­e­ments of the se­quence. For ex­am­ple, the se­quence 1, 12, 13, 14… con­verges to 0, be­cause ev­ery in­ter­val cen­tered around 0 con­tains all but finitely many of 1, 12, 13, 14… Some se­quences don’t con­verge to any­thing, like 0, 1, 0, 1..., but it’s an easy ex­er­cise to prove that no se­quence can con­verge to two differ­ent val­ues at once.

Now the only sen­si­ble way to un­der­stand 0.999… is to define it as what­ever value 0.9, 0.99, 0.999… con­verges to. But that’s ob­vi­ously 1 and that’s the end of the story for peo­ple who un­der­stand math.

You can use the same pro­ce­dure for in­finite sums. x1+x2+x3+… can be defined as what­ever value x1, x1+x2, x1+x2+x3… con­verges to. For ex­am­ple, 1+1/​2+1/​4+1/​8+… = 2, be­cause the se­quence of par­tial sums is 2-1, 2-1/​2, 2-1/​4, 2-1/​8, … and con­verges to 2.

By now it should be clear that 1+2+3+4+… doesn’t con­verge to any­thing un­der our defi­ni­tion. But our defi­ni­tion isn’t the only one pos­si­ble. You can make an­other self-con­sis­tent defi­ni­tion of con­ver­gence, where 1+2+3+4+… will in­deed con­verge to −1/​12. But that defi­ni­tion is com­plex, es­o­teric and much less use­ful than the reg­u­lar one, which is why that viral video re­ally shouldn’t have used it with­out re­mark.

Most para­doxes in­volv­ing in­finity are just pul­ling a fast one on you by not spec­i­fy­ing what they mean by con­ver­gence. If you try to use the com­mon sense defi­ni­tion above, or re­ally any self-con­sis­tent way to as­sign val­ues to in­finite ex­pres­sions, the para­doxes usu­ally go away.

• Here’s how di­vid­ing by zero leads to re­sults like 1=2:

You may have heard that func­tions must be well-defined, which means x=y ⇒ f(x)=f(y). This prop­erty of func­tions is what al­lows you to ap­ply any func­tion to both sides of an equa­tion and pre­serve truth do­ing it. If the func­tion is one-to-one (ie x=y ⇔ f(x)=f(y)), truth is pre­served both ways and you can un-ap­ply a func­tion from both sides of an equa­tion as well. Mul­ti­pli­ca­tion by a fac­tor c is one-to-one iff c isn’t 0. There­fore, un-ap­ply­ing mul­ti­pli­ca­tion by 0 is not in gen­eral truth-pre­serv­ing.

• Slightly off topic to the main point of the ar­ti­cle, which is how to deal with not un­der­stand­ing some­thing.

For any­one who wants to fully un­der­stand this, you need to read “Baby Rudin”: Rudin “Prin­ci­ples of Math­e­mat­i­cal Anal­y­sis”.

Real num­bers are defined as equiv­alence classes of con­ver­gent se­ries. The se­ries 1,1,1,1,1 and 0.9, 0.999 … are in the same equiv­alence class and so are the same real num­ber. Peo­ple of­ten get caught up with the as­sump­tion that two differ­ent se­ries (or two differ­ent dec­i­mal rep­re­sen­ta­tions) must be differ­ent num­bers.

• The OP states:

A very good ques­tion is “what kinds of ob­jects are these, any­way?” Since we have an in­finite dec­i­mal they can’t be ra­tio­nal num­bers.

This is just wrong. A ra­tio­nal num­ber is a num­ber that can be writ­ten as a frac­tion of two in­te­gers. Lots of in­finite dec­i­mals are ra­tio­nal num­bers. 13 = .3333333..., 19 = .1111111.… 17 = .142857142857142857… etc.

• Ah, of course, my mis­take. I was try­ing to hand-wave an ar­gu­ment that we should be look­ing at re­als in­stead of ra­tio­nals (which isn’t in­her­ently true once you already know that 0.999...=1, but seems like it should be be­fore you’ve de­ter­mined that). I fool­ishly didn’t think twice about what I had writ­ten to see if it made sense.

I still think it’s true that “0.999...” com­pels you to look at the defi­ni­tion of real num­bers, not ra­tio­nals. Just need to figure out a plau­si­ble sound­ing jus­tifi­ca­tion for that.

• I think the point is that you’re writ­ing down “0.999...” and as­sum­ing that that must define a num­ber at all. If you’re as­sum­ing that ev­ery dec­i­mal ex­pres­sion gives a num­ber then you must be work­ing with the re­als.

• I sup­pose you might be right for some peo­ple. For me, the fact that re­peat­ing in­finite dec­i­mal ex­pan­sions are ra­tio­nal is deeply deeply in­grained. Since your post is es­sen­tially how to square your feel­ings with what turns out to be math­e­mat­i­cally true, you have a lot of room for dis­agree­ment as there is no con­tra­dic­tion in differ­ent peo­ple feel­ing differ­ent ways about the same facts.

For me the most fun thing about 0.9999.… is that 19 = .11111… and there­fore 9x1/​9 = 9x.111111..… and this last ex­pres­sion ob­vi­ously = .99999...

You should also do a search on “right” in your post and edit it, you use “right” one time where you re­ally need “write” I think it is “right down” in­stead of “write down” but I’ll let you do the look­ing.

• Fixed the typo. Also changed the ar­gu­ment there en­tirely: I think that the easy rea­son to as­sume we’re talk­ing about real num­bers in­stead of ra­tio­nals is just that that’s the de­fault when do­ing math, not be­cause 0.999… looks like a real num­ber due to the dec­i­mal rep­re­sen­ta­tion. Skips the prob­lem en­tirely.

• Part 3. “the ar­gu­ment from con­tra­dic­tion” ap­proach dikd his­tor­i­cally ac­ti­vate for me. Ex­cept I found a way where the op­er­a­tions make sense: I ap­pri­ci­ate that it needs to make sense with your cur­rent un­der­t­stand­ing level. But ar­gu­ment from lack of imag­i­na­tion is a pretty lousy one. One could say that “x^2 = −1″ is ab­surd but con­sid­er­ing what world would look like if it could be made true can be in­ter­est­ing and use­ful. By similar logic one could ar­gue that nega­tive num­bers are “un­real”. I ended up recogn­ins­ing how the stan­dard for­mu­la­tion is trans­finite hos­tile. In­stead of whether a reulst is pos­si­ble or not you end up ask­ing whether the rules are in­evitable or not.

• When I en­coun­tered this re­sult in school for the first time, in the con­text of learn­ing the al­gorithm for con­vert­ing a re­peat­ing dec­i­mal into a frac­tion, I even­tu­ally rea­soned “If 1 and 0.999… are differ­ent num­bers, there ought to be a num­ber be­tween them, but there isn’t. So it must re­ally be true that they’re the same.”

• Of all the differ­ent ex­pla­na­tions and in­ter­pre­ta­tions peo­ple have been giv­ing in this thread this is the most satis­fy­ing to my math­e­mat­i­cally illiter­ate brain. It’s trou­ble­some for me to grasp how 0.999… isn’t always just a bit smaller than 1 be­cause my brain wants to think that even an in­finitely tiny differ­ence is still a differ­ence. But when you put it like that—there’s nowhere be­tween the two where you can draw a line be­tween them—it seems to click in. 0.999… hugs 1 so tight that you can’t mean­ingfully sep­a­rate them.

• It’s in­struc­tive to set out the proof you give for 0.999...=1 in num­ber bases other than ten. For ex­am­ple base eleven, in which the max­i­mum value sin­gle digit is con­ven­tion­ally rep­re­sented as A and amounts to 10 (base ten). 10 (base eleven) amounts to 11 (base ten). So

Let x = 0.AAA...

10x = A.AAA...

10x—x = A

Ax = A

x = 1

0.AAA… = 1

But 0. A (base eleven) = 1011 (base ten) which is big­ger than 0.9 (base ten) = 910 (base ten). So shouldn’t that in­equal­ity ap­ply to 0.AAA… (base eleven) and 0.999… (base ten) as well? (A de­bat­able point maybe). If so, then they can’t both equal 1, un­less we say some­thing like 0.999...=1 and 0.AAA...=1 are both valid but base de­pen­dent equa­tions, as in­deed any such equa­tion would be when us­ing the top val­ued sin­gle digit of its base. This would mean 0.111...=1 in bi­nary.

• f(x)=2/​x

g(x)=1/​x

f(x) > g(x) for all x but lim f(x) = lim g(x) = 0. Just be­cuause f gets there “later” does not mean it gets any less deep.

Re­peat­ing dec­i­mals are far enough re­moved from dec­i­mals its like mix­ing ra­tio­nals and in­te­gers.

• I think I see your first point.

0.A{base11} = 1011

0.9 = 910

0.A − 0.9 = 0.0_09...

0.AA = 1011 + 10121

0.99 = 910 + 9100

0.AA − 0.99 = 0.00_1735537190082644628099...

Does this mean that be­cause the differ­ence or “late­ness” gets smaller tend­ing to zero each time a sin­gle iden­ti­cal digit is added to 0.A and 0.9 re­spec­tively, then 0.A… = 0.9...?

(Whereas the differ­ence we get when we do this to say 0.8 and 0.9 gets larger each time so we can’t say 0.8… = 0.9...)

• No I be­lieve you are reach­ing a differ­ent con­cept. It is true that the differ­ence squashes to­wards 0 but that would be differ­ent line of think­ing. In a con­tex where in­finides­i­mal are al­lowed (ie non-real) we might as­so­ci­ate the se­ries to differ­ent amounts and in­deed find that they differ by a “minus­cule amount”. But as we nor­mally op­er­ate on re­als we only get a “real pre­ci­sion” re­sult. For ex­am­ple if you had to say whether 34, 1 and 54 name which in­te­gers probalby your best bet would be that all of them name the same in­te­ger 1, if you are only re­stricted to in­te­ger pre­ci­sion. In the same way you might have 1 and 1-ep­silon to be differnt num­bers when in­finides­i­mal ac­cu­racy is al­lowed but a real + any­thing in­finides­i­mal is go­ing to be the same real re­gard­less of the in­finides­i­mal (1 and 1-ep­silon are the same real in real pre­ci­sion)

What I was ac­tu­ally go­ing fo is that, for any r < 1 you can ask how many terms you need to get up to that level and both se­ries will give a finite an­swer. Ie to get to the same “depth” as 0.999999… gets with 6 digits you might need a bit less with 0.AAAAA… .It’s a “hori­zon­tal” differ­ence in­stead of a “ver­ti­cal” one. How­ever there is no num­ber that one of the se­ries could reach but the other does not (and the num­ber that both se­ries fails to reach is 1, it might be helpful to re­mem­ber that an suprenum is the small­est up­per limit). if one se­ries reaches a sum with 10 terms and other reaches the same sum in 10000 terms it’s equally good, we are only in­ter­ested what hap­pens “even­tu­ally” or af­ter all terms have been ac­counted for. The way we have come up what the re­peat­ing digit sign means refers to limits and it’s pretty guaran­teed to pro­duce re­als.

• So shouldn’t that in­equal­ity ap­ply to 0.AAA… (base eleven) and 0.999… (base ten) as well? (A de­bat­able point maybe).

Not de­bat­able, just false. For­mally, the fact that for all does not im­ply that .

If I were to poke a hole in the (pro­posed) ar­gu­ment that 0.[k 9s]{base 10} < 0.[k As]{base 11} (0.9<0.A; 0.99<0.AA;...), I’d point out that 0.[2*k 9s]{base 10} > 0.[k As]{base 11} (0.99>0.A; 0.9999>0.AA;...), and that this gives the op­po­site re­sult when you take (in the stan­dard sense of those terms). I won’t demon­strate it rigor­ously here, but the faulty link here (un­der the stan­dard mean­ings of real num­bers and in­fini­ties) is that car­ry­ing the in­equal­ity through the limit just doesn’t cre­ate a nec­es­sar­ily-true state­ment.

0.111...{bi­nary} is 1, ba­si­cally for the Dedekind cut rea­son in the OP, which is not base-de­pen­dent (or rep­re­sen­ta­tion-de­pen­dent at all) -- you can define and iden­tify real num­bers with­out us­ing Ara­bic nu­mer­als or place value at all, and if you do that, then 0.999...=1 is as clear as not(not(true))=true.

• 0.9{base10}<0.99{base10} but 0.9...{base10}=0.99...{base10}

0.9{base10}<0.A{base11} but 0.9...{base10}=0.A...{base11}

0.8{base10}<0.9{base10} and 0.8...{base10}<0.9...({ase10}

0.9{base10}<0.A{base11} and 0.9...{base10}<0.A...{base11}

I’m not try­ing to prove “0.999...{base10}=1 “is false, nor that “0.111...(base2)=1” is ei­ther—in fact it’s an even more fas­ci­nat­ing re­sult.

Also “not(not(true))=true” is good enough for me as well.

• You are as­sum­ing that there is a link be­tween the per-term value and the whole se­ries value. The con­nec­tion just isn’t there and if you think it would be it would be im­por­tant to show why.

I could have two small finite se­ries of A=10 and B=2+3+5 and com­pare that 2<10, 3<10 and 5<10 and then be sur­prised when A=B. When the term amount is not finite it’s harder to ver­ify th­jat you haven’t made this kind of er­ror.

• So would you say that 0.999...(base10) = 0.AAA...(base11) = 0.111...(base2)= 1?

• Yes, it hap­pens to be that way.

• Still not en­tirely con­vinced. If 0.A > 0.9 then surely0.A… > 0.9...?

Or does the fact this is true only when we halt at an equal num­ber of digits af­ter the point make a differ­ence? 0.A = 1011 and 0.9 = 910, so 0.A > 0.9, but 0.A < 0.99.

• I think you are still treat­ing in­finite des­i­mals with some ap­prox­i­ma­tion when the ques­tion you are pur­su­ing re­lies on the more finer de­tails.

**Ap­peal to graph­i­cal asymp­totes**

Make a plot of the value of the se­ries af­ter x terms so that one plot F is 0.9, 0.99,0.999,… and an­other G is 0.A, 0.AA, 0.AAA,.… Now it is true that all of Gs have a F be­low them and that F never crosses “over” above G. Now con­sider the asymp­totes of F and G (ie draw the line that F and G ap­proach to). Now my claim is that the asymp­totes of F and G are the same line. It is not the case that G has a line higher than F. They are of ex­actly the same height which hap­pens to be 1. The mean­ing of in­finite dec­i­mals is more closely con­nected to the asymp­tote rather than what hap­pens “to the right” in the graph. There is a pos­si­bly sur­pris­ing “tak­ing of limit” which might not be to­tally nat­u­ral.

**con­s­tus­truc­tion of wedges that don’t break limit**

It might be illu­mi­nate­ing to take the re­verse ap­proach. Have an asymp­tote of 1 and ask what all se­ries have it as it’s asym­tote. Note that among the can­di­dates some might be strictly greater than oth­ers. If per term value dom­i­na­tion forced a differ­ent limit that would push such “wedg­ings” to have a differ­ent limit. But given some se­ries that has 1 as limit it’s always pos­si­ble to have an­other se­ries that fits be­tween 1 and the origi­nal se­ries and the new se­ries limit will be 1. Thus there should be se­ries whose are per item-dom­i­nat­ing but end up sum­ming to the same thing.

**Rate mis­match be­tween ac­cu­racy and digits**

If you have 0.9 and 0.99 the lat­ter is more pre­cise. This is also true with 0.A and 0.AA. How­ever be­tween 0.9 and 0.A, 0.A is a bit more pre­cise. In gen­eral if the bases are not nice mul­ti­ples of each other the level of ac­cu­racy won’t be the same. How­ever there are crit­i­cal num­ber of digits where the ac­cu­racy ends up be­ing ex­actly the same. If you write out the sums as frac­tions and want to have a com­mon de­nom­i­na­tor one lazy way to guaran­tee a com­mon de­mon­i­na­tor is to mul­ti­ply all differ­ent de­mo­ni­a­tors to­gether. This means that a frac­tion in a dec­i­mal num­ber mul­ti­plied by 11 and a frac­tion in un­dec­i­mal mul­ti­plied by 10 will have the same de­nom­i­na­tors. This means that 0.99999999999 and 0.AAAAAAAAAA are of same pre­ci­sion and have the same value but one has 11 digits and the other has 10. If we go by pure digits to digits com­par­i­son we end up com­par­ing two 11 digit num­bers when the equal value is ex­pressed by a 10 and 11 digit num­bers. At this level of ac­cu­racy it’s fair to give dec­i­mals 11 digits and un­dec­i­mals 10 digits. If we go blindly by digit num­bers we are un­fair to the amount of digits available for the level of ac­cu­racy de­manded. Sure for most level of ac­cu­racy there is no nice nat­u­ral num­ber of digits that would be fair to both at the same time.

**Graph­i­cal rate mis­match**

One can high­light the rate mis­match in graph­i­cal terms too. Have a nice x=y graph and then have a dec­i­mal scale and un­dec­i­mal salce on the x axis. Mark ev­ery point of the x=y that cor­re­sponds to a scale mark on both scales. Com­par­ing digit to digit cor­re­sponds to firt go­ing to 9/​10th marker on dec­i­mal scale and 10/​11th mark on the un­dec­i­mal scale and then go­ing 9th sub­di­vi­son on the dec­i­mal scale and 10th sub­di­vi­sion on the un­dec­i­mal scale. If we step so it’s true that on each step the un­dec­i­mal “rest­ing place” is to the right and up to the dec­i­mal rest­ing place. But it should also be clear that each time we take a step we keep within the origi­nal com­part­ment and we end up in the high part of the orginal de­part­ment and that right side of the com­par­ment will always be limited by (x=1,y=1). By ev­ery 11 dec­i­mal steps we land in a lo­ca­tion that was landed in by the un­dec­i­mal se­ries and by ev­ery 10 un­dec­i­mal steps we land in a lo­ca­tion that will be vis­ited by the dec­i­mal steps. This gives a nice in­ter­pre­ta­tion for hav­ing a finite num­ber of digits. What you do when you want to take in­finite steps? One way is to say you can’t take in­finite steps but you can talk about the limit of the finite steps. For ev­ery real num­ber less than 1 both step­pings will at some finite step cross over that num­ber. 1 is the first real num­ber for which this doesn’t hap­pen. Thus 1 is the “des­ti­na­tion of in­finite steps”.

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• 0.999… doesn’t map into 1 di­vided by {{0,1,2,3,4,5...}|} (=ep­silon)

How­ever those that dis­agree are ob­vi­ously think­ing some­thing akin to 1-ep­silon which is not equal to 1. How­ever you can’t re­fer to it with a dec­i­mal sys­tem (atleast the stan­dard one). Ar­gu­ments that re­fer to (spe­sific) dec­i­mal places are there­fore in­ap­pli­ca­ble. Reals are archi­me­dian but sur­re­als are not. For sur­re­als there are el­e­ments a,b a>b so taht there is no N so that a<b*N (ar­bitarar­ily large finite mul­ti­ples of b are not guaran­teed to by­pass a).

• For ev­ery ep­silon greater than zero, the differ­ence 1-0.99999999… is even smaller. Smaller than any pos­i­tive num­ber.

Then, if it’s not nega­tive, then it’s zero. This differ­ence is zero.

This is the most cor­rect way to put it, I be­lieve.

• Yes. Still this is the con­cept of limits and it is a sig­nifi­cant step for most peo­ple. I think the most com­mon first re­ac­tion is “Huh?”.

But peo­ple will make the effort if you ex­plain this is a solu­tion to the mys­te­ri­ous­ness of “in­finites­i­mals”.

• This ar­gu­ment more or less as­sumes its con­clu­sion; af­ter all, if it weren’t the case that 1 − 0.999… were zero, then it would be some pos­i­tive num­ber x, so you could pick ep­silon = x.

• And in cer­tain con­struc­tions, ep­silon is a dis­tinct num­ber—so it’s ac­tu­ally fal­la­cious with­out go­ing back to the defi­ni­tions!

• No, it does not!

What­ever ep­silon you might choose, you can eas­ily take enough 9s (nines) af­ter the 0. - to have the differ­ence smaller than this ep­silon of yours.

• Again, that’s as­sum­ing the con­clu­sion; what if 1 − 0.999… weren’t zero, and you picked that as ep­silon? You’re skip­ping steps. It’s worth writ­ing down ex­actly what you think is hap­pen­ing more care­fully.

(To be clear, I’m not claiming that you’ve as­serted any false state­ments, but I think there’s an im­por­tant sense in which you aren’t tak­ing se­ri­ously the hy­po­thet­i­cal world in which 1 − 0.999… isn’t zero, and what that world might look like. There’s some­thing to learn from do­ing this, I think.)

• If I may, let me agree with you in di­alogue form:

Alice: 1 = 0.999...
Bob: No, they’re differ­ent.
Alice: Okay, if they’re differ­ent then why do you get zero if you sub­tract one from the other?
Bob: You don’t, you get 0.000...0001.
Alice: How many ze­ros are there?
Bob: An in­finite num­ber of them. Then af­ter the last zero, there’s a one.

Alice is right (as far as real num­bers go) but at this point in the dis­cus­sion she has not yet proved her case; she needs to ar­gue to Bob that he shouldn’t use the con­cept “the last thing in an in­finite se­quence” (or that if he does use it he needs to define it more rigor­ously).

• An in­finite num­ber of them. Then af­ter the last zero,

There is no “af­ter the last” zero.

• you aren’t tak­ing se­ri­ously the hy­po­thet­i­cal world in which 1 − 0.999… isn’t zero

In this (math) world it is zero only be­cause for ev­ery nonzero pos­i­tive ep­silon, you can pick a FINITE num­ber of 9s, such that 1-0.999999...99999 (a FINITE num­ber of 9s) is already SMALLER than that ep­silon.

For EVERY real num­ber greater than zero, you have a FINITE num­ber of 9s, such that this differ­ence is smaller.

There­fore the differ­ence can­not by a num­ber greater then 0.

I have always been em­pa­thetic to the ar­gu­ment, from peo­ple first pre­sented with this, that they are differ­ent. Un­der­stand­ing how math deals with in­finity ba­si­cally re­quires hav­ing the math­e­mat­i­cal struc­ture sup­port­ing it already known. I’m not par­tic­u­larly gifted at math, but the first 4 weeks of real anal­y­sis re­ally changed the way I think, be­cause it was ba­si­cally a con­densed rapid up­load of cen­turies of col­lab­o­ra­tive work from some of the smartest men to ever ex­ist right into my brain.

Other­wise, at least in my ex­pe­rience, we op­er­ate in a dis­crete world that moves through time. So, what I pre­dict is hap­pen­ing, is that when you ask that ques­tion to peo­ple their best ap­prox­i­ma­tion is a dis­crete world tick­ing through time.

Is 0.999...=1? Well, each tick of time an­other set of [0.0...9]’s is added, when the ques­tion is fi­nally an­swered the time stops. You’re then left with some finite num­ber [0.0..01]. In their mind it’s a dis­crete algo run­ning through time.

The re­al­ity that it’s a limit that op­er­ates ab­sent of time, in­stan­ta­neously, is hard to grasp, be­cause it took brilli­ant men cen­turies to figure out this profoundly un­in­tu­itive re­sult. We un­der­stand it be­cause we learned it.

• Its a sim­ple ar­gu­ment that tries ot be rigor­ous. If I don¨t agree with it I must dis­agree with some part of it. When I go step by step over it there is a sus­pi­cious step.

The proof as­sumes/​states that 9.99999… −0.999999 = 9. I am un­con­fi­dent with op­er­a­tions on in­finite dec­i­mal place dec­i­mals that I am not sure that I agree. 9.99999… −0.9999 could also be 8.00...009. In par­tic­u­lar I don’t know whether you get the same ob­ject if you muliti­ply 0.9999… by ten or if you set the first zero equal to 9.

Un­der­stand­ing to agree with how the proof han­dles is to be profi­cient on what re­als are and the tech­ni­cal­ities and to un­der­stand that re­als are what is meant.

Hav­inga stan­dard that things are not real if they can’t be re­al­ised in re­als would make i and com­plex num­bers to be “un­in­tel­lig­ble”.

• You left out a pos­si­bil­ity; true de­pen­dent on some­thing out­side your realm of knowl­edge. In this case, it’s true for real num­bers, but false for sur­real num­bers.

• No, be­cause it’s not a pos­si­bil­ity that when you thought you were do­ing math in the re­als this whole time, you were ac­tu­ally do­ing math in the sur­re­als. Us­ing a sys­tem other than the nor­mal one would need to be stated ex­plic­itly.