I don’t think I understand how important the concept of zero is. As Petzold notes, because the Hindu-Arabic numeral system had a zero they did not need a special symbol to represent ten (unlike, for example, the Roman “X”). The zero allowed them to use numbers positionally, with each position being a power of ten:
1439 = (1 x 103) + (4 x 102) + (3 x 101) + (9 x 100)
“10” itself being not a single symbol, but (1 x 101) + (0 x 100)
It’s as if the zero is a placeholder that allows you to keep the notion of 9, carry it over to a new position, add it to the next number in the sequence (in this case, 1—giving “10”), then use that as a base to continue counting from 1 – 9 again. This repeats until we reach 99, at which point we move over to a new position (100), using now both zeros to keep what we have achieved and open up a new space for further counting.
And yet, as amazing as it is, it’s unfortunate that we don’t emphasize the arbitrary nature of base 10 in education. We simply teach the mechanical manipulation of numbers, and even then rather poorly. As a child, I was taught to memorize the multiplication table up to 12, but was never once shown what other counting systems would look like. We force our children to figure out the underlying assumptions and philosophy on their own—if they ever do. Since there is little incentive to it is no wonder most students find math both boring and tedious, when a calculator or cash register can do the job just fine. And thus is so much beauty—so much insight into the human mind—lost.