michael_nielsen

But where this number system really shines is in simplifying certain other things you might want to do. For instance, consider addition of the numbers wx and yz.

We have wx + yz = (w*10+x) + (y*10+z) = (w+y)*10 + (x+z). Fiddle with this for a while - it's more work than I want to go through here - and you eventually recover the grade school algorithm for adding two-digit numbers. Do some more work, & you get the n-digit algorithm.

There's a miracle going on here, one we don't notice b/c it's so familiar: I pointed out above that the numerals have very different meanings, depending on their location. But despite this, in the grade-school algorithm we use the _same rules_ for addition, regardless of place!

E.g., in computing 27+38 at some point in the computation we'll use 2+3 = 5; in computing 72+83 at some point we'll also use 2+3=5. That's despite the fact that the 2 and the 3 in the first sum have a very different meaning than in the second sum!

What's responsible for this astonishing fact? If you look back at the reasoning above, you see it's a consequence of associativity, commutativity, and distributivity. That's a pretty huge set of things! And it makes addition _really_ nice in this representation.

In particular, it means that although the numerals have different meanings in different locations, in many ways they continue to _behave_ as though they are in some sense "the same". They're _not_ the same. But they retain many of the same affordances, in modern interface lingo.

You can go through the same kind of thing, trying to figure out how to multiply two numbers: wx*yz = (w*10+x)*(y*10+z) = x*y*10*10 + ... I won't go through it, but you can guess what happens: you get the grade-school algorithm for multiplication.

And, again, even though the numerals have very different meanings in different positions, they have a lot of the same affordances, and so behave in some ways as the "same" object.