One of the marvelous inventions that we have inherited from generations before us is the decimal number system. There is a lot that could be said and has been said about this system; in this post I will look at it as a representation: as a way to represent quantity.
How does a number like 56.37 represent a particular quantity? Is it a lot or is it a little? How much is it?
In a sense, as adults we’ve become so familiar with this particular system of representation that it isn’t necessarily obvious what is even meant by the question “how does it represent a quantity?” Many of my middle school kids would respond something like this: “What do you mean? 56.37 is the answer.” I assume that what they mean by that is that 56.37 is the end point of all their work, they consider that they are done now. It is a measure of the success of the decimal number system that, to us, 56.37, looks like the answer.
In contrast, if I ask middle schoolers how XII represents a quantity, they will readily show me that the X represents a ten, the I represents a one, and the other I represents another one, and that you are supposed to add these all together, and you end up with the quantity twelve. And when I show them IX they will show me that the X represents a ten, the I represents a one, and because the I is shown before the X, you subtract them, and you end up with the quantity nine. In other words, they can recognize the system of Roman numerals as a system of representation of quantity.
What middle schools can do with Roman numerals, they can also do with decimal numbers, once you call their attention to it. Typically, they have no trouble in seeing that 56.37 has a fifty piece in it, and a six piece. Often, they can see that there are 37 hundredths there, and fewer will offer on their own that the 3 represents three tenths and that the 7 represents seven hundredths. These pieces, fifty, six, three tenths, and seven hundredths, make up the quantity represented by 56.37. How do they make up that quantity? Well, they are added together. If I ask how do you add these pieces together, it appears like a trick question. “What do you mean how do you add them together? They are already added together: 56.37!”
What is hidden is that some questions about a representation can only be answered meaningfully from another representation. You can show how 56.37 represents a quantity by using coins, or using a number line, or using a meter stick with the right kind of markings, or using base-ten blocks.
The decimal number system is an extraordinarily successful compromise. The representation supports a certain number of important activities very well: we can compare two numbers fairly easily, we can count with it easily, we can add and subract two numbers fairly easily, and even multiplication isn’t too terrible. It isn’t often appreciated that it is the representation that results in addition being easy and multiplication being hard. Other representations are possible in which multiplication is easy and in which addition is hard. People who remember slide rules know exactly what I mean by this. I intend to show other examples in later posts – and there are many such representations. The representations in which multiplication is really easy have never found widespread use, and for understandable reasons – those tend to be special purpose representations and not general purpose representations.