In the previous post in this series, we broached the subject of equivalence classes. We did so in the context of looking at an addition table and noticing how each value in the table occurred multiple times. We drew a blue line to connect all the entries that had the same value in it, and this blue line now marks a particular equivalence class. What all the entries in that equivalence class has in common is the value of the sum. Same sum, same equivalence class; different sum, different equivalence class. The value of the sum is the property that links the entries in the equivalence class together.
In this post, I’m going to develop a similar theme with multiplication and multiplication tables. In the diagram on the left, I’ve drawn a simplified multiplication table, only going up to 5. In the middle, I’ve repeated all that information but this time drew some contour lines connecting entries with the same value for the multiplication result (the product). Compared to the addition case in the prior post, the contour lines are no longer straight. Only some of the contour lines are shown, but you can get a good idea about the others, especially since 1.5, 2.5, 3.5 and 4.5 are marked also. Finding the product associated with any point in that middle diagram is directly analogous with how you find the altitude on a topographical map such as the one shown on the right.
In such a topographical map, the contour lines represent locations of equal altitude. You can see contour lines marked for elevations of 400 ft and 350 ft (this is a US Geologic Survey map) with 4 contour lines separating them. From this you can conclude that adjacent contour lines are 10 ft in latitude apart. The altitude of any point on this map can be estimated fairly accurately by looking at contour lines below and above.
It’s the same with the product. In the middle diagram, the contour lines are drawn 2.5 apart, from 0 to 25, though the contour line for 0 is not strictly on the diagram itself, but rather just beyond the top left corner. You could draw contour lines 1 apart instead, and this certainly would remove the need for estimating if what is being multiplied is two whole numbers (from 1 to 5). Of course, the need for estimating would rear its head in the same way, when you wanted to use the map for multiplying numbers that weren’t whole numbers. And you could. The middle diagram is a simple (and somewhat crude) example of a class of computational devices called nomographs, of which the Wikipedia entry provides some interesting examples. Now largely replaced by electronic devices, for centuries nomographs represented the most sophisticated computing devices available. (A particularly versatile and useful one, for centuries, is the slide rule. I still have the slide rule I bought for college, in those days obligatory for math/science/engineering students. The real geeks of the day had them hanging from their belt.)