In the prior post of this series, we looked at pairs of operators in which one of the pair cancels out the effect of the other. We call one of these the inverse operator of the other one. For example, “adding two” is the inverse operator of “subtracting two”.

If we look once more at a long train of operators chained together, for example as below:

we can pair up this train with another, quite easily, to make a do-nothing (identity) train:

You can probably see how this was accomplished: each car in the original train is matched with a car directly below it that undoes its effect. The train consisting of cars e, f, g and h precisely undoes the effect of cars a, b, c and d.

Now imagine that the train of cars a, b, c and d is the device we’ve been exploring, and we now ask the question: what number would have to appear on the input “in” for the output to be a particular number, say, 99. In other words, what number needs to go into car a so that the output of car d is 99? It turns out that the identity train a,b,c,d,e,f,g,h gives us everything we need: we can see what happens if 99 is presented on the output of d, and then becomes the input of car e: the output of car e will be 102, and this is now the input to car f. The input to car g will be the output of car f, which is half of 102 or 51. The output to car g (and the input to car h) will be 50, and the output to car h will be 45. We conclude that presenting 45 on the input to car a will produce 99 on the output of car d!

Inverse trains help us decide what inputs generate particular desired outputs. Isn’t this neat? Without necessarily realizing this, we’ve built machinery for solving certain types of equations.

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