# A Friendly Chat About Whether 0.999... = 1

Does .999… = 1? The question invites the curiosity of students and the ire of pedants. A famous joke illustrates my point:

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/

My Calculus prof proved this for us in class.

Let N = 0.999…

Assume N = 1, now multiply both sides by 10
10N = 10, now subtract 9 from both sides
N = 1

I think that’s how he did it.

Why “0.999 = 1” is counter-intuitive:

If you have “0.99” instead of “0.9”, it means that you are one step closer to 1, as close as the 10-digit-notation allows in one step, but without reaching 1. If you add another 9 and arrive at “0.999”, you have again stepped as close to 1 as you could in one step, but without reaching 1.

Even if you would do this an infinite amount of times, every step would have the same rule: “… without reaching 1”.

Last year I wrote about it (albeit in Italian) at
http://xmau.com/mate/art/0-999999a.html and http://xmau.com/mate/art/0-999999b.html . My line of reasoning is more or less like yours (oh yeah, there’s also http://xmau.com/mate/art/0-999999c.html where I wonder about the difference between 1.00 and 1, and ramble about the measuration errors!)

Ever head of this guy Cauchy? He might wanna have a word with you.

i really like your explanations of tricky math concepts. do keep posting more good stuff like this. looking forward to your post on approximating functions

also from what i understand, dark matter is different from anti-matter. When matter meets anti-matter they annihilate and release energy. Anti-matter is well understood while dark matter is not. It is simply conjectured to exist to explain the speeding up of the expansion of the universe when it should really be slowing down. So you can make that (infinitesimal) correction into your excellent post

I think you meant anti-matter where you said "Dark matter destroys regular mass when they come in contact […])

The hyperreal case is a little bit more subtle than that.

See, the object 0.999…, as you understand it, doesn’t quite exist there. The hyperreals add infinitesimals to the real line by also adding infinitely large numbers, including infinitely large integers. And since {0, 0.9, 0.99, 0.999, …} is a sequence on the positive integers, it gets a lot more terms when it gets embedded into the hyperreal system; it becomes a hyper-sequence, for lack of a better term. Canonically it corresponds to a hyper-sequence whose length is unbounded even in the hyperreals, and still has 1 as a limit.

Now, we could also look at the hyper-sequence which started the same way, but stopped getting bigger at some hyper-integer w. The difference between 1 and that limit would be 10-w, which is positive in our system. (This is what the arXiv paper you cited is talking about.) There are many sequences which are increasing like 0.999… on the standard integers, then take a constant value x on most of the nonstandard integers, but x could be anything.

So the real problem is that 0.999… isn’t well-defined in the hyperreals—it doesn’t really equal anything. I know of no context where 0.999… has a clear resolution and it isn’t 1.

(And for the record, we aren’t rigorous because we like to be. We’re rigorous because the subject demands rigor. Intuition fails a lot.)

sorry, 10-w should be 10^(-w)

some bla guy — I think you’re right about why 0.999… = 1 is a counter-intuitive fact, but there’s an easy counter. Each finite term 0.99…9 is larger than the previous term; so the “infinite term” 0.999… is larger than all the finite terms. So the fact that 1 is larger than 0.99…9 isn’t an obstacle; in fact, it’s a requirement.

Oh, since I’m here anyway:

Aaron — Any proof that assumes N = 1 to prove N = 1 is dead in the water. The usual “proof” goes like this: If (1) N = 0.999…, then (2) 10N = 9.999…; subtract (1) from (2) to get (3) 9N = 9, from which N = 1. I say “proof” because first you have to establish that arithmetic with infinite expansions makes sense, and it’s usually easier to do some other proof instead.

ram — Your broad point is correct, but what you’ve described is “dark energy”. Dark matter is mass that we know must exist, because of its gravitational effect on visible objects near it, but can’t see, because it doesn’t interact with the electromagnetic field. I’m pretty sure it would actually slow the expansion of the universe, but don’t quote me on that.

I always use this quick explanation to the layman:
1/3 (one third) can be represented by 0.333…
If you take each thirds and add them up (0.333… + 0.333… + 0.333…) they add up to 1.0, not 0.999…

A Friendly Chat About Whether 0.999… = 1…

Via reddit I stumbled upon this site which talks about something called hyperreal numbers and claims that in this theory, the equation 0.999… = 1 is false.For those who follow the internet, the question of whether 0.999…=1 has come up a quadbrazill…

A very simple way to solve this:

Take two numbers 3 and 5; now, to see if they are equal we can try to find a number between them. Well, a number like 4 or 3.75 is between.

So, now let’s take 0.999… and 1.0. Can you find a number that is between an infinitely repeating set of 9’s before it goes to 1? No, there is no number between 0.999… and 1. If you truly figure 0.999… as an infinite string of 9’s then there is nothing before you would have to round up to 1.

However, for practical purposes we have to round to a finite number. A finite string of 0.9999’s is only equal to one because humans can’t work/comprehend an infinite string of 9’s.

You object that, when asked whether .99999…=1, we view .99999… within “our system”, whatever that means. Well, of course we do! How else could we possibly interpret the question or try to answer it?

When someone asks whether .999… equals 1, they are most certainly asking within the context of the real numbers. Switching gears and trying to interpret the question within the hyperreals is as arbitrary and evasive as choosing to instead view it in the p-adics, where another different (but equally valid) answer could be given.

This is silly. This “argument” really needs to be put to rest. Mathematical rigor exists precisely for this reason.

Ah, I knew this would this would invite the curiosity of students and the ire of pedants!

@Aaron: Great question. These types of proofs make assumptions about how addition and subtraction would work with these infinite decimals (does 9 * 0.999 = 8.999…?), but they do work for the regular number system (see http://math.fau.edu/Richman/HTML/999.htm).

@some bla guy: I’d love to have a word with Cauchy – I bet he’d be interested in learning about new number systems that can rigorously approach the same problems differently!

@mau: Neat – I’ll have to see how well Google translate does at math :).

@ram: Whoops, thanks for the correction! Yes, I meant anti-matter :).

@Chad: Great points, thanks for the discussion! I guess it depends on the meaning of 0.999…, which is indeed ambiguous. I think the better phrasing may be “The hyperreal number uH=0.999…;…999000… with H-infinitely many 9s, for some infinite hyperinteger H, satisfies a strict inequality uH < 1” (from Wikipedia).

I think the higher meta-point is figuring out the question behind the question – the layman isn’t asking about 0.999… as constructed in the real number system. They want to know what happens when one number gets “infinitely close” to another – can this be represented? 0.999… is the most convenient form of this question (also see 1/infinity – does this equal 0? Yes, if you take the limit approach, no if you take the hyperreal).

@haileris: Does 1/3 = .333… exactly, or is 0.333… different at the infinitely small level?

@Adam: Great point – because our current number system cannot represent the difference between 0.999… and 1 (there’s no number in-between), in our current system they are equal. However, other systems allow it, so I take the approach of “it depends”.

@Jeff: Thanks for dropping in, but I disagree that it needs to be put to rest. Transform the problem: if it’s 1600 and someone asks 1600s Jeff what does sqrt(-1) mean, what do you say? That they are asking this question in the context of the real numbers, and the answer is undefined? How else do you answer it?

The alternative is to explore a new number system (complex numbers, hyperreal numbers) and see if it has interesting properties. You can’t take the question at face value, it’s really about exploring the nature of infinitely small numbers.

Hailis has the answer and yes, Kalid, 1/3 does equal 0.333… The numerator is really 1.000… and the division continues ad infinitum. To say that 0.999… does not equal 1 is to say that neither 3/3 nor 9/9 equal 1. I would love to hear your explanation as to why the rules for reducing 9/9 are different than those for reducing 8/8 or, for that matter, 1/1.

Additionally, I don’t think there is much assumption involved when considering how addition might work with these particular infinitely repeating decimals. Try adding 1/3 and 1/7. When expressed as a decimal, each has an infinitely repeating sequence; yet we can identify a very specific and uncontroversial answer: 10/21.

If we were discussing 0.12341234… it might be a different story; that number is not rational. 0.989898… comes close to 1 but never touches, which makes it an interesting candidate for the tolerance and accuracy portions of this discussion. But nobody is proposing that 0.989898… equals 1.

0.9… is indeed a special case, but it is not the number-line equivalent to infinitely-close-but-not-touching (like the way my fingers don’t actually touch my keyboard as I type - there is a tiny gap between the atoms). 0.9… is 9 * 1/9. It’s a concept we can imagine and denote, but it doesn’t really exist as a unique number. It is, in truth, 1.

About the “ire of pedants” … There is no point discussing infinity unless you are being pedantic and rigorous.

Since we’re wondering out loud, I will say that this kind of number theory issue makes me wonder whether complex number are more real than real numbers.

@Ogre_Kev: I agree that in the current real number system, .333… = 1/3. But what this means is this:

“The infinite sequence (.3, .33, .333, .3333…) converges to the limit 1/3”, which is another way of saying “We can make an element of (.3, .33, .333…) as close to 1/3 as we wish”.

You might want to check out http://math.fau.edu/Richman/HTML/999.htm:

“Perhaps the situation is that some real numbers can only be approximated, like the square root of 2, whereas others, like 1, can be written exactly, but can also be approximated. So 0.999… is a series that approximates the exact number 1. Of course this dichotomy depends on what we allow for approximations. For some purposes we might allow any rational number, but for our present discussion the terminating decimals—the decimal fractions—are the natural candidates. These can only approximate 1/3, for example, so we don’t have an exact expression for 1/3”.

So, as long as we stay in the real number system, 1/3 is the limit of .333… [which is fine, but we don’t have to stay in the real number system; others can capture the idea of what we mean when we say infinitely close].

As a side comment: if 3/10 is not 1/3, and 33/100 is not 1/3, at what point does another digit make it exactly 1/3? This is a bit like Zeno’s paradoxes, which have not been fully resolved :). The meta-point is that we can make that sequence as close to 1/3 as we need, which in the real number system means they are equal in the limit.

@Igor: I think it’s possible to sketch out ideas intuitively and return with rigor to cement the foundation – Calculus developed this way, did it not?

@Michael: Great question – I think all numbers may be equal abstractions of the mind. The real number system may be “less real” because it’s more limited than others.