In the prior post in this series, we looked at chains of operators for which the net effect was to do nothing. We started by comparing the picture on the left with the picture on the right, below, and noticing that both have the same net effect on the number coming in:

So the net effect of boxes b and c together is to do nothing at all. We then looked at trains of two cars, trains of one car, and trains of no cars at all, all of which netted out as doing nothing at all. We can call those identity trains, as the number coming out is identical to the one that came in. Before we look more closely at two car trains (where one car undoes the effect of the other), let’s expand this idea to longer trains.

On the left we show a train of three cars, and the net effect of first adding 4, then adding 1, and then taking away 5 is to have done nothing at all to the number coming in. On the right, the same cars are shown in a circular arrangement. A number starting out at the “in” label can go around the circle once, twice, forever, without changing. This is kind of obvious, but what isn’t quite so obvious is that it means we can break up such a circle at any point, and get a new identity train.

The figure above shows the (linear) train that results from breaking up the circle between car p and car q. And, indeed, it is an identity train, too. Looking at the circular arrangement, we can see that cars q and r, together, precisely undo the effect of car p. Cars r and p, together, precisely undo the effect of car q. Cars p and q, together, precisely undo the effect of car r.

Now let’s take our promised closer look at two-car identity trains, trains where one car undoes the effect of the other. We will start out showing them as circular arrangements, to emphasize the symmetry of the situation:

In the figure on the left, we have “subtract 1” undoing the effect of “add 1”, but equally, we have “add 1” undoing the effect of “subtract 1”. Subtracting something undoes the effect of adding the same “something”. Because of this, it is often said that subtraction undoes addition, and the formal – but somewhat unprecise – way of saying it is that addition and subtraction are inverse operations.

Note that there is a difference between the add/subtract pairs above, and the situation below:

Here, “add 1” isn’t paired with “subtract 1”, but with “add negative 1”, etc. Here we’d say that to undo adding something, you add the negative of that something. The difference between the two sets above is fairly subtle, though.

We can also find do/undo pairs with multiplication and division:

Here, “divide by 4” undoes “multiply by 4”, and also: “multiply by 4” undoes “divide by 4”. Dividing by something undoes multiplying by the same “something”. Again, you often hear people say that multiplication and division are each other’s inverses. Let’s just agree that “multiplying by 4” and “dividing by 4” are each other’s inverses. The word “inverse” is used here in the same sense as we’ve been using “undo”, as a way to get back to where you were before.

In the next post, we’ll look for some uses of the idea of do/undo pairs.

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