Last week in Fundamental Understanding of Mathematics XLVIII we took a look at rate problems. This week I'd like to start a little foray into abstract math, with a toy mathematical system that uses a four color spinner. I'm playing around a bit with this, so I thought I'd write about it.
Build a Spinner
The basic spinner looks like this:
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49blue.png)
It's colored on both sides, if you turn it over it looks like this:
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49back.png)
The colors wrap around from one side to the other.
Read a Spinner
The way to "read" the spinner is simple: it's the color on top. In both the above images, the spinner is in position Blue.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49green.png)
This is Green
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49yellow.png)
This is Yellow
and this
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49red.png)
is Red
So our spinner defines a set with four elements in it: Blue, Red, Green and Yellow. In set notation, we would write
Spinner Set = {Blue, Red, Green, Yellow}
Elements in the set are listed, in no particular order, inside the curly brackets, separated by commas.
The name of the set "Spinner Set" is shown as equal to the list of set elements.
In order to turn this into a toy arithmetic, we need some kind of operation. There are two general types of operations, unary operations, that take one set element as input, and give some output, and binary operations, that take two set elements and give some output. In regular arithmetic, unary operations are things like taking square roots, or factorials.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49a.png)
Binary operations are things like addition or multiplication
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49b.png)
Interesting unary operations tend to be complicated little gadgets, often relying on binary operations in their definitions. Binary operations are much more straightforward, so we will begin by defining a binary operation.
The Rotation Operation
We'll call it "Rotation." It will take two colors from the Spinner Set as input, and will give some color from the spinner set as output.
Here's how it works:
Suppose we want to figure out Blue rotate Green. We start with the spinner set to read "Blue"
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49blue.png)
and we rotate it clockwise so the Blue section moves to where the Green section is.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49c.png)
and the result is this
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49red.png)
or, Red.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49i.png)
It doesn't matter which direction the spinner is rotated, but it should not be flipped over so the back side is facing up.
Notice that Blue rotate Red = Green can be done by rotating the spinner 90 degrees counterclockwise or 270 degrees clockwise, while Blue rotate Yellow = Yellow needs a 180 degree rotation in either direction.
We can continue to play around with our spinner, starting at various colors, and rotating by colors. One thing we notice is that the rotation operation is closed for our Spinner Set. What ever colors from the set we use for input, we will get something from the set as output.
If we record our results in an organized table, it might look something like this.
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49d.png)
In this table, the color in the column on the left is rotated by the color in the row on top. In other words,
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49e.png)
this represents
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49f.png)
not Yellow rotate Green = Blue. In fact, Yellow rotate Green is not Blue,
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49g.png)
So this is an interesting observation:
![title=](http://i69.photobucket.com/albums/i64/Orinoco_guy/FUM%20Daily%20Kos/FUM49h.png)
The rotation operation does not seem to be commutative. So, unlike addition, where a + b = b + a, or multiplication, where ab = ba, this spinner arithmetic does not obey the commutative law. Which is why it's not the commutative law, it's the commutative property. Our new rotation operation doesn't have this property.
This may be a new way to think about the commutative property. In learning arithmetic and algebra, it's taken for granted that addition and multiplication are commutative. It's hard to imagine how they could be anything else, given the way addition and multiplication are defined. But here we have defined a simple binary operation, rotation of a four color spinner, and it turns out that it does not have the commutative property. We now have the option to think about the commutative property (and, by extension, other mathematical properties) as a thing in and of itself, and not as something automatically attached to addition and multiplication.
Have fun in the comments.