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!)