Place value

The Mesopotamian system started with different symbols for different landmarks, but later became a base-60 system, the sexagesimal system, with a remarkably efficient way of writing numbers.

Why 60? Nobody is sure. Theories point to the length of their lunar month of 30 days and other periods; to how easy it is to write fractions in base 60; and to their older landmark sequence 1, 10, 60, 600, 3600, 36000 being trimmed down to just the powers of 60.

Whatever the reason, it's still with you: 1 hour = 60 minutes and 1 minute = 60 seconds come straight from ancient Mesopotamia.

Their landmarks are 1, 60, 60² = 3600, 60³ = 216000, and so on, with symbols for 1 and for 10 used to build the numbers 1 to 59.

Take 640. In landmarks: 640 = (10) × 60 + 40. The Egyptian way would be to draw ten 60-signs and four 10-signs. But there's a shorter way: write 'ten', then 'forty', and read it as ten 60s and 40 ones, exactly as the equation says.

Another: 7530 = (2) × 3600 + (5) × 60 + 30, written as three groups of symbols. No power of 60 can appear 60 or more times, because sixty of them would bundle into the next power up.

Now the leap. What if we drop the signs for the powers of 60 altogether, and let position do that job? The rightmost group counts the 1s, the group to its left the 60s, the next the 3600s. That's exactly what the Mesopotamians did, leaving a blank space wherever a power didn't occur.

A system with a base that uses the position of a symbol to show which landmark it counts is a positional or place value system. It's the high point of this whole story: a handful of symbols, and every number can be written.

But their version wasn't complete. Blank spaces are hard to size consistently, so 12, 62 and 3602 could look alike except for the gaps. Later Mesopotamians found a brilliant fix: a placeholder symbol standing for a blank. That's our 0. Zero, the symbol for nothing, turns out to be indispensable. Even so, some confusion remained, because they used the placeholder mostly in the middle of a number and not at the end.