michael_nielsen

(I've glossed over 0 in various ways in this account. And over the fact that place-number system pre-dated the romans. If I was more conscientious I'd have talked more about these - they're incredibly deep ideas - but this thread is already long, so I'll keep glossing.)

At this point we have a new numeral system. It's nice in a couple of ways when compared to roman numerals - it doesn't need new symbols to represent larger numbers, and it's extremely compact.

Still, those things perhaps don't seem that important. Certainly not worth replacing an entire piece of intellectual infrastructure with!

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.