In the two previous posts in this series, we looked at decidedly different models that each could explain the same simple external behavior. To recap, we have been looking at internal models for a device that looks like this:
The first model we showed was essentially that of an analog clock face (remember analog clocks?), but with ten markings instead of twelve, which would advance one position whenever the red button is pushed. The second model we showed was essentially a counter, where we make sure that only the rightmost digit of the full counter value is shown in the window.
The third model, dealt with in this post, starts from the observation that mostly, the next digit shown is one more than the current digit. If the current digit is , then in virtually all cases, the next digit shown is . We can now do something that is fairly common in mathematics, but almost unheard of in school math: we bluff. We introduce a new notation, reminiscent of the old one, as follows: , and claim that this new operation does what we want it do without exception. This new notation is sufficiently evocative in that it evokes something similar to addition, and the circle around the “+” it could be taken to mean that somethings is behaving in a cyclical way. And because I’m the one who is bluffing here, I get to say how this new operation works! And how I say it is going to work is somewhat arbitrary, but not totally arbitrary. I want this new operation to share as many properties with ordinary addition as I can manage. In particular, I want the new operation to be commutative and associative. These are well-established technical words for properties of addition that most of us take for granted – if you aren’t familiar with those terms, you won’t lose much here. We all know that when we write we get the same answer no matter of the order in which we perform the additions. I can add 6 + 8 first, and then add 4 to the intermediate result, but I can also add 6 + 4 first, getting 10, and then adding the remaining 8 to that, getting 18. I could also add 4 + 8 and then add 6 to that result. We have come to expect that when we add numbers, order and sequencing just aren’t important to the final result. For our new operation we will strive to have the same thing be true.
If you look at the diagram below, and focus on the table on the left:
you see the beginnings of something that looks like an addition or multiplication table, but for our newly introduced operation . Since we wanted to correspond to what the red button appears to do, we can fill in the first column of the table accordingly. It does look a lot like straight-forward addition, except when we come to the row for 9, where we propose that is 0 (rather than 10 as it would be in ‘real’ addition).
Notice that from the table on the left, we see that . It turns out that this simple fact is enough to fix all the remaining values in the table on the right. For example, to fill in the entry for we can see that this must be the same as , at least if we are really serious in wanting to be able, as with ordinary addition, to get the result of in any order we choose. Since is 6 (from the table on the left), and is 7 (also from the table on the left), we wouldn’t want to have be different from that. Using similar reasoning, we can fill out the table on the right, one column after another.
You may wonder why we bother filling out the entire table if the only part of the table we need to model the workings of the red button is the piece of it.
In the table above, on the left, we add even more columns, and we can then compare that table with the one on the right, which contains the results of ordinary addition. You can see how much the tables look alike, and that you can get the left one from the right one by systematically throwing away all the tens in the result. So, where , we get . We could say, though it is a bit sloppy, that tens mean nothing in the left table. And we could make this real by imagining we’re playing with dimes and pennies, and systematically always trade out our pennies for dimes whenever we have 10 pennies. The table on the left would show us how many pennies (not cents) we have after adding some number of pennies with another number of pennies. For example, the entry would show us that if we start with 7 pennies, and then get 8 pennies more, we would end up with 15 pennies, but then trade ten of them for a dime, so we have 5 pennies left.
We’ll leave it for a future post to compare this model with the previous two, and also to compare our jury-rigged notation with more established mathematical notation.