I think the best chance with a young kid would be:
“Well, if two numbers are different, then there must be another number between them, right? [At this point you can point out that even numbers next to each other like 3 and 4 have numbers between them, like 3.5 etc] Can you think of a number between 0.999… and 1?”
If the kid is a bit older and has done some math, this is pretty intuitive as well:
x = 0.999…
10x = 9.999…
9x = 9.999… – 0.999…
9x = 9
x = 1
This doesn’t *exactly* answer the question, but I discovered this pattern as a kid playing with a calculator:
1/9 = 0.1111…
2/9 = 0.2222…
3/9 = 0.3333…
4/9 = 0.4444…
5/9 = 0.5555…
6/9 = 0.6666…
7/9 = 0.7777…
8/9 = 0.8888…
Cool, right? So, by that pattern, you’d expect that 9/9 would equal 0.9999… But remember your math: any number divided by itself is 1, so 9/9 = 1. So if the pattern holds true, then 0.9999… = 1
I understood it to be true but struggled with it for a while. How does the decimal .333… so easily equal 1/3 yet the decimal .999… equaling *exactly* 3/3 or 1.000 prove so hard to rationalize? Turns out I was focusing on precision and not truly understanding the application of infinity, like many of the comments here. Here’s what finally clicked for me:
Let’s begin with a pattern.
1 – .9 = .1
1 – .99 = .01
1 – .999 = .001
1 – .9999 = .0001
1 – .99999 = .00001
As a matter of precision, however far you take this pattern, the difference between 1 and a bunch of 9s will be a bunch of 0s ending with a 1. As we do this thousands and billions of times, and infinitely, the difference keeps getting smaller but never 0, right? You can always sample with greater precision and find a difference?
Wrong.
The leap with infinity — the 9s repeating *forever* — is the 9s *never* stop, which means the 0s *never* stop and, most importantly, the 1 *never* exists.
So 1 – .999… = .000… which is, hopefully, more digestible. That is what needs to click. Balance the equation, and maybe it will become easy to trust that .999… = 1
Depending on the kid, there might be different things to help give them the aha. Someone else suggested effectively proving it by contradiction, i.e., Well tell me a number between them if they are different. I like that.
Another one that might work is to try to explain that we haven’t really written a number down, have we? No matter how many 9s you write down, you haven’t really written down the number. When you have …, you’re hinting at where it’s going, but you haven’t written it down. So, where’s it hinting at going? As we write down more 9s, what are we getting close to?
Then you can kind of combine that with the above argument, i.e., if they say “a million 9s!” then you say OK, but at some point we’ll go past that, right? But we never ever go past 1. And we go past everything else less than 1, eventually. So the … is hinting at… 1.
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