To recap: we’ve now defined the decadic metric on integers by
where is not divisible by 10, and also . According to this metric, two numbers are close when their difference is decadically small. So, for example, and are at a distance of , but and are closer, with a distance of . and are closer still, with a difference of . Put simply: the more final digits two integers share, the closer they are. This is a reversal of the usual situation in which integers are close when they share a lot of initial digits (like and ). Mark James notes that it is almost as if we were “reversing” the numbers. Jonathan notes that this is not quite right, negative and positive integers can be close without sharing digits—for example, and are apart. But in this case the negative number ends with digits that are “dual” to the digits the positive number ends with.
Let’s keep thinking about negative numbers a bit. , so and are apart. and are apart. and are closer still, . In fact, I hope you can see that in general, is very close to : specifically, and are always apart. So the more 9s we add, the closer we get to .
What if we let this process go on “forever”? After all, we do this with the normal decimal numbers: , , , etc. get closer and closer to , so we define the infinite sequence . We can do the same thing here: we define the infinite decadic number
to be equal to . Weird, huh? But it gets weirder (and cooler). The really neat thing is that arithmetic still “works” in the same way as it does for normal integers. What do I mean? Well, if we add and we had better get , right? Let’s try it:
Adding 9 and 1 gives us 10, so we write 0 and carry the 1… giving us 10 again, and so on forever.
And if we add to itself we ought to get something that behaves like , right?
Well, adding the last two 9’s gives us 18, so the last digit of the answer is 8… we carry the 1, so the next addition gives us 19; we write down 9 and carry the 1… and this pattern is now obviously going to repeat forever, so
which I claim represents : you should have no trouble verifying that .
Here are some parting exercises for you to work on before my next post:
- What number is represented by ? How do you know?
- How about ?
- Can you find an infinite decadic number which represents ? How about ? Or ?