Yes, Virginia, 0.999999… = 1

Algebra 1=0.999
Image by sparr0 via Flickr. Thought about trying this one with him, but didn't have time to attach the whiteboard to his ceiling...

I love my kids.  Even though they don’t like math as much as I do (give them time), they are still willing to talk about it and ask why various statements are true as opposed to taking their truth as necessary facts.  And for mathematical thinking–and for thinking, at all–that is great.

Earlier this week, Boy #2 was grappling with an idea that was keeping him from calming down his brain and falling to sleep.  After praying together, as I was about to leave, he let me have it: “Dad, why is ‘point nine repeating’ the same as one?”

I’ve written out his question in the words he used, but that would normally be written, “Why is it true that 0.999999… = 1?”

For those who don’t know, the notation “…” means that the sequence of 9’s goes on forever with no stopping point.  The more familiar case for most of us is the fact that one-third written as a decimal is 0.333333…, and infinite sequence of 3’s.  (You discover this when you divide 1 by 3 using long division.)  This is sometimes just written as “0.3 with a ‘bar’ over the 3.”  LaTeX for those whose browsers will show it:

0.333333... = 0.\overline{3}

(Facebookers, I suspect that FB will not show that equation.  You might need to go to the original post)

And for those who did know that, it is also true, as my son said, that the number 0.9999999… is exactly the same as the number 1.

But how do you convince an eleven-year-old in five minutes, so that you do not fall into his trap and let him stay up another half-hour?  I tried a few different arguments…

  • If two numbers are different, then we can always find a new number between those two numbers.  But what number can fit between 0.999999… and 1?  None!  Therefore, they must be the same number.
  • Well, you agree that 0.333333… = one-third, right?  And 3 times one-third is one, right?  So doesn’t that mean that 3 times 0.333333… is one?  And 3 times 0.333333… would also equal 0.999999…  Therefore, 0.999999… must equal one.
  • We know that 0.111111… is one-ninth by dividing it out.  And, similarly, 0.222222… is two-ninths, 0.333333… is three-ninths (or one-third), 0.444444… is four-ninths, etc.  So, what is 0.777777…?  Seven-ninths, right!  And 0.888888…?  Eight-ninths, great!  Finally, 0.999999…?  That would have to be nine-ninths.  But nine-ninths is the same as one.  So, 0.999999… = 1.

We discussed it a bit more, and I will admit that I don’t know if he was entirely convinced.  Though it did give me an opportunity to mention that will is an inherent aspect of belief.  (See Romans 1:28.)  We will ultimately believe nothing that we do not have the will to believe, and we will believe everything that we do not have the will to disbelieve.  It’s why believers in self-proclaimed “prophets” go on believing even when the prophet’s “prophecies” completely fail (in violation of Deuteronomy 18:22), and it’s why commodian (I’m sorry, did I misspell that? I meant “comedian”–sorry!) Bill Maher may never believe in God until he is forced by that same God to reconsider certain facts.  The role that will plays in our belief is generally ignored by most people.  But it’s there.

It’s even true of the fact that 0.999999… = 1.  I knew a wonderful old math teacher when I was teaching high school in (what feels like) a previous life.  And he told me once: “I know that 0.999999… has to equal one, by the design of mathematics and the meaning of the symbols.  But I just can’t believe it.  I just can’t believe that 0.999999… and one are the same!”

His honest confession has given me insight into Jeremiah 17:9 and the nature of belief that I have appreciated for a long time.

And, no, I did not take that long with Boy #2, nor did I discuss it with him in that sort of detail.  After all, talking about math too much at bedtime will get someone so terribly excited that he’ll never fall asleep, right?  Right?  FYI: I imagine you all saying, “Right!” at this point.

Perhaps, you do not believe it either.  However, for me and my house, 0.999999… = 1.  (Apologies to Joshua.)

[P.S.  For those who’d like more exciting, keep you awake with glee, frequently asked math questions, you might want to check out my “Why is a negative times a negative a positive?” post.  The truth is out there…]

[UPDATE, 10/15/2010. My thanks to Lyndell for this suggestion: The Wikipedia article “0.999…” has more than you may ever want to know on this subject. Good stuff! I especially appreciated the references to studies on students acceptance or denial of the 0.999… = 1 identity.]

16 thoughts on “Yes, Virginia, 0.999999… = 1

  1. Nice mathematical reasoning, Mr. Smith. (“Commodian”? Why not just call Bill Maher “Commodore” and be done with it? 😛 )

    Yes, I can well believe 0.9999… = 1. And with a little more willpower, I can (along with some T-shirts) maintain that “2 + 2 = 5 (for very large values of 2)”. 😀 I suppose there’s a mathematical “proof” behind that statement somewhere (perhaps like the set of equations which “proves” that 1 = 2, involving a hidden division by zero).

  2. Ed Ewert

    “I knew a wonderful old math teacher … he told me once: “I know that 0.999999… has to equal one, by the design of mathematics and the meaning of the symbols. But I just can’t believe it. I just can’t believe that 0.999999… and one are the same!””

    I believe that 0.999999… = 1, but not with the certainty that I believe that 1+1=2. At the back of my mind, there’s this idea lurking that some day, a clever mathematician will come along and prove it to be an error!

  3. Norbert

    I can understand the symbolic meaning of the numbers are equal. 0.999999… = 3/3 = 1. However here’s an interesting question, why does a number with no end equal a number with an end? By all appearences the two are distinctly different.

  4. Why is an imaginary number times an imaginary number a real number?

    If I imagine that I am imagining something, does it become real?

    If Boy #2 pretends he is prentending something, does it come into existence?

    Back to the scientific method! Ask your son to try it, and see if it works. But if it does, what kind of world might that lead to?

  5. If I understand the relevant research properly, what we call “belief” is directly connected to that cognitive process which deals with personal and (interestingly) universal values. It is a decision-making process: “rational” in Jung-speak, which means we have much control over it (as opposed to the “irrational” information-gathering processes, which happen with little or no control on our parts). And as I noted in my LCG Commentary “Our values, God’s values”, this is the one cognitive process which has a truly non-negotiable element to it. People will decide to believe what they want to believe (i.e., according to what they personally value), unless they are transformed by universal values – which ultimately seems to require God’s Spirit and its “renewing of the mind”, starting right there in that part of our minds.

  6. Lyndell: Thanks for the Wikipedia comment. I’ve added a link in the text – lots of good stuff in there!

    Richard: Ha! OK, that’s funny.

    Norbert: The problem is in saying “no end” and “all appearances.” 1 does not necessarily have an end if you think of it as 1.000000… The difference is in thinking of 0.999… as some sort of process versus a whole object. (The Wikipedia article discusses this rather well, actually.) It may not have a “final digit” but 0.999… is not a process – it is simply a number with an odd name. 🙂 And as for appearances, the fact is that all numbers have multiple ways of being named. The fraction 1/3, can be written as 2/6, 3/9, 4/12, etc. Even (-37)/(-111), 42 ÷ 126, and 2 × (1/6). And, of course, 0.333…, as mentioned above. Multiple ways of indicating the same number shouldn’t bother us. With numbers as in life, appearances can be deceiving. I’ll leave applications of 1 Sam. 16:7 to the real numbers to the mathematical theologians (or theological mathematicians) out there. 🙂

    Ray Schaefer: I imagine it would be interesting (but I don’t imagine myself imagining it). 🙂

    rakkav: Here’s a link to that commentary: “Our values, God’s values.” And here’s to the day when those universal values become more personal for everyone!

  7. Mr. Smith,

    The idea of a curve asymptotically (sp.?) approaching a vertical line comes to mind here. The farther you go on the curve toward infinity (x-values along the way: 0.9, 0.99, 0.999, 0.9999, etc.), the closer you get to 1.0, and you finally reach it at infinity.

    Am I on the right track? Are we dealing with the mathematics of limits here? I never got quite that far in my own high school and early collegiate studies.

  8. Yes, Mr. Wheeler, you’re in the ballpark. The key is that 0.999… is not a process or a sequence, it’s a number. Creating the sequence 0.9, 0.99, 0.999, etc. creates a growing list of numbers, each of which takes you closer to the number that is 0.999…, but the number 0.999… is not the same as the sequence. Those who think of 0.999… as a process are confusing the number with that sequence that approaches it.

    The analogous situation is the sequence 1.1, 1.01, 1.001, etc., which gets closer and closer to 1 (or 1.000…) as we extend it further and further. But the number to which it approaches (1) is not the same as the sequence.

    Your reference to an asymptote is akin. The graph that approaches the asymptote is like the sequence, and the asymptote, itself, is like the number. We tend to confuse the process of approaching 1 “from below” (0.9, 0.99, 0.999, etc.) with the number, itself.

    Oddly, as the Wikipedia article points out, individuals not infrequently have this problem with 0.999… = 1 but not so much with 0.333… = 1/3, though the two identities are, really, identical in principle. Perhaps this is because 1 is so familiar to us, while 1/3 has always been a bit shady and weird, what with that whole 33 1/3 % thing and all. 🙂

  9. Paul Kendall

    I try to always have a familiar starting point when thinking of math questions, and my starting point is always pie; cold lemon meringue pie (although a good ice cream pie works well too if it stays frozen). I prefer not to use apple or cherry pie because they’re too runny and that presents even more problems when slicing it into ultra-thin pieces.

    So in this case, if you could cut a lemon pie into an infinite number of slices, then you still have one pie. The problem here is that in our dimension, you’re not allowed to slice a pie into an infinite number of pieces, because eventually the slices are so thin that they aren’t worth eating and they all get stuck together. This means that we don’t really have any use for infinitely small numbers because they get so close to the number 1 that you can’t tell them apart. So Unless you live in another dimension, which you don’t, it’s best just to stop slicing and start eating.

    Hope this helps.


  10. Thomas

    I know that 0.999999… = 1 (just part of the mental furniture, now).

    Not willing to believe that Boy#2 = “Virginia”, though.

  11. Mr. Smith et al:

    Here is a short, easy proof that .999999… = 1:

    Let x = .999999…

    Multiply both sides by 10

    10x = 9.999999…

    The decimal still runs out infinitely.

    10x – x = 9.999999… – .999999…

    9x = 9

    x = 1

    This works because it is the nature of an unending decimal. None of us can really grasp infinity, but we can mathematize about it.

    Larry Bruce

  12. Thanks, Mr. Bruce. That one’s related to the one in the picture above, and is one of my favorites–in fact, the very same one I always showed my own math students when I was teaching (though I think I tended to use s as the variable for some reason). With my son it seemed as though the other reasons would help him accept the conclusion more than a proof. Thanks, again!

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