Addendum to "What is a number"

I reread the earlier blog posts in preparation for writing more and I realized something I had left out in the "What is a number?" posting.

I left out the concept of "embedding". I touched upon the related idea of extending but didn't get explicit about it.

One of the important aspects of all the number systems I described in that posting was that in all the newer or weirder ones the old number system still exists, inside the new one. This is what's meant by "embedded". To take the example of natural numbers and real numbers, for instance, there's a collection of real numbers which obey all the rules of natural numbers. Formally, we say the embedding is a structure-preserving one-to-one map from natural numbers to real numbers. "One-to-one" in this case means that the real numbers corresponding to two distinct natural numbers are distinct. "Structure-preserving" means that all the properties we expect of natural numbers are maintained by the map. If we map natural numbers a,b to real numbers A,B then a<b means that A<B, a+b maps to A+B, a*b maps to A*B, etc.

Sometimes the properties of the simpler system hold for all of the new system (like in the naturals to reals case), sometimes they don't. I mentioned in the prior post that Ordinals didn't have a commutative addition, and that ω+1 does not equal 1+ω. Ordinals are modeled on the natural numbers, and there is a "canonical" embedding of natural numbers in the Ordinals -- 1 maps to the first Ordinal, 2 maps to the second, etc onto all the finite Ordinals. Within the finite Ordinals, all works as expected, addition is commutative. It's only when you go into the "unmapped territory" of the transfinite Ordinals do the expected properties start to break down.

The flip-side of embedding is "extending" -- adding things to an existing system to cover gaps in what you are doing. You want to split 4 apples between 3 people; natural numbers won't let you do that, so you extend them to allow arbitrary ratios of natural numbers, filling in the gaps between the old numbers with new numbers. The old numbers still work just fine. You discover that there's still gaps between rational numbers, so you add them to the mess, getting real numbers. The rational numbers are still there (and the natural numbers too). You can't take the square root of a negative number? Add a number defined as the square root of -1 and say that all the other rules still apply and see what it gets you -- voila! Complex numbers. If one square root of -1 is good, how about 2? No dice, it doesn't work, but with three you get eggrollQuaternions! (seven yields Octonians). Natural numbers are good for telling you the number of items in any finite set, but there's not a finite number of natural numbers. How many natural numbers are there? Georg Cantor extended natural numbers to "transfinite Cardinals" to find out.

So all the varied and wonderful number systems out there have embedded in them simpler number systems, and all are extensions of simpler number systems. Well, all except natural numbers. You have to start somewhere.

Things which aren't "numbers", like n by n matrices, tend to fail in that they don't have an obvious useful embedding of a simpler system, or aren't a clear extension of a simpler system. The closest embedding I can think of for reals into an n by n matrix is simple multiples of the identity matrix, which isn't very interesting or clear.