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:
(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.]