Written: March 19, 1997
Once upon a time there was a genius who invented Zero. So when one had to multiply XIX by CIX all one had to do is write it as 19x109, and perform long multiplication. Hence one hundred and nine was coded into a vector of digits of length [log(109)]+1 (decimal logarithms.)
This algorithm is fairly efficient, but is nowadays obsolete, since we have Fast Fourier Transform, and besides, we can use a calculator. Nevertheless, we torture our poor fourth- and fifth-graders, by programming them into performing these boring and uninspiring tasks. Of course, neither students nor teacher have any clue why the method works.
If we want to teach our kids the ideas behind multiplication, we would be much better off going back to the sparse, rather than dense, notation. In particular, get rid of this unintuitive and overly abstract concept of zero. If there is nothing to write, don't write it!
So let T:=ten, and write XIX as T+9 and CIX as TxT+9, then to multiply them, let them do: (T+9)(TT+9)= TTT+9T+9TT+(8T+1)=TTT+9TT+(9+8)T+1=TTT+9TT+(T+7)T+1= TTT+9TT+TT+7T+1=TTT+(9+1)TT+7T+1=TTT+TTT+7T+1=2TTT+7T+1.
This way, they would be able to understand much better the ideas behind the positional system. In fact, they should only do it in base 2, but again, without using zero.