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 9⁄10 + 9⁄100 + 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!)