I remember as a child studying fractions, being told that the top was called the numerator and that the bottom was called the denominator, for reasons that were not made clear to me at the time. In retrospect, it’s possible that I was told and that it just didn’t make any sense to me anyway, but it seems more likely that I was just told to call them that and be happy and move on with my life.
Now, I suppose that, given I was probably in elementary school at the time, it made some sense that my teachers wouldn’t clutter my head with the mathematical theory behind those names. However, as the years went by, it seems to me that it might have been useful at some point for someone to fill in the gap.
Here’s the gap. Now that I understand it, I have a deeper respect for fractions than I did before.
The bottom part is the trickier part, but I think it’s important to start there. Fractions are really division, and division returns the same value as multiplication of the inverse. Which is to say, for example: Two-thirds is really two times a unit broken into three pieces. Multiplication is the collection of n units, while division is breaking a unit into n pieces. In the grand scheme of things.
When we think of the word “denominator”, we might think of “denomination”, which is a word we use to describe coins and bills. Mostly bills. It’s strange to think of a dime as having a “denomination” of 0.10 of a dollar, but in the case of fractions, that’s exactly what you should think. the denominator specifies how many pieces the theoretical unit should be broken into.
In a way, the denominator is like a unit; a unit of a unit, if you will. That’s the heady part that they probably think (and probably rightly so) that most elementary schoolchildren would get confused about.
The top part is a lot easier to see: “Numerator” is related to “enumerate” and to “number”. It tells you to count how many of those pieces. See, all the words I’ve been using are pretty much the words they used when they taught me fractions, and that they still use: The bottom breaks it up into so many pieces and the top tells you how many chunks to take.
So the denominator names the size of the unit-chunks you’re going to use, and the numerator enumerates how many you need.
Which leads me to two beefs I have about fractions. The first is the idea of “improper fractions”. A proper fraction, we’re told is one where the top is less than the bottom. It turns out it’s impossible to show any rational non-integer greater than one as a proper fraction. 1.5 is either 1 1/2 or 3/2, neither of which is proper.
But if we’re really taking the bottom as a specification of the chunk size (with the default being the unit, which doesn’t need to be specified), then any fraction consisting of an integer over an integer is in its ideal form.
And here’s the other beef: Mixed fractions set students up to be confused in algebra. 1 1/2 means 1 PLUS 1/2. x 1/2 means x TIMES 1/2. There’s no need for that.
Short. Sweet. Simple. When I run the world, I will fix this.