The Mayan and Chinese systems

In Central America the Mayan civilisation made great progress between the 3rd and 10th centuries CE. Among its achievements is a place value system worked out completely independently of anything in Asia, and, like the later Mesopotamians, they used a placeholder for zero. Theirs looked like a seashell.

They wrote 1 as a dot and 5 as a bar, and combined them for 1 to 19. The groups were stacked vertically: the lowest counted the 1s, the one above the 20s, the one above that the 360s, and so on upwards. The shell matters: 361 is one 360, no 20s (a shell) and one 1. Leave out the shell and it would read as one 20 and one 1, which is 21.

And there's the oddity. If the second landmark is 20, the third should be 20 × 20 = 400. Theirs was 360 (only 18 × 20). Some scholars think this had to do with their calendars.

That one irregularity costs them a lot. Because the Mayan system isn't a true base-20 system, it loses the calculating advantages a real base gives: products of landmarks are no longer landmarks. Their place value notation and their zero are still an important advance. (And base 20 lives on oddly elsewhere: you can still see it in the number names of some European languages.)

The Chinese ran two systems side by side: a written one for recording amounts, and a rod one for calculating. The rod numerals were more efficient both to write and to calculate with. They developed by at least the 3rd century AD and were used until the 17th.

The rod system was decimal (base-10), with symbols for 1 to 9 made of laid-out rods. To keep the positions apart, they alternated two forms:

So a numeral reading heng-2, zong-6, heng-3, zong-4 is (2) × 10³ + (6) × 10² + (3) × 10 + (4) × 1 = 2634.

Like the Mesopotamians, they left a blank where a place was empty. But because their symbols for one to nine were all about the same size, the blanks were easier to spot, and the alternating zong and heng forms made it even harder to misread a position.

So the Chinese rod system came very close. As the book says: 'The Chinese system, with a symbol for zero, would be a fully developed place value system.' Everything was there except the zero.