Positional numeral system

For 5,000 years, humans counted with tokens—small clay objects representing sheep, jars of oil, measures of grain. Temple accountants in Mesopotamia tracked inventories by sealing tokens inside clay envelopes, then pressing the tokens into the wet surface to show contents without breaking the seal. Eventually, the impressions became sufficient. The tokens disappeared, and numbers became marks on clay.

Around 2000 BCE, Babylonian scribes took a step that Egyptians, Greeks, and Romans would never take: they made position meaningful. In additive systems like Roman numerals, each symbol carries fixed value regardless of where it appears—V always means five. In positional notation, a symbol's value depends on its column. The same wedge mark could mean 1, 60, 3,600, or any power of 60, determined entirely by position.

The system was sexagesimal—base 60. This choice, inherited from Sumerians, gave Babylonian scribes enormous computational advantages. Sixty has twelve divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Converting fractions to clean integers became trivial. One-third equals 20 in base 60—no repeating decimals. One-fifth equals 12. Scribes calculating grain allotments, worker wages, or canal construction costs could work in whole numbers where decimal arithmetic would have produced endless remainders.

The clay tablet YBC 7289, a student's rough work from around 1800 BCE, demonstrates the system's power. A square with side marked as 30 shows its diagonal calculated as 1;24,51,10 in sexagesimal notation—equal to 1.414213 in decimal, the square root of 2 accurate to six decimal places. The error is less than one in two million. Plimpton 322, from the same era, lists fifteen rows of Pythagorean triples—integer solutions to a² + b² = c²—more than a millennium before Pythagoras.

Administrative pressure drove adoption. The Old Babylonian Empire under Hammurabi's dynasty (1830-1531 BCE) required calculations for taxation, canal maintenance, army provisioning, and interest on loans. Scribal schools produced over 160 multiplication tables as student exercises. The same cuneiform wedges—just two symbols, one for units and one for tens—could represent any number through position and combination.

The Maya of Mesoamerica arrived at positional notation independently, using base 20 and counting both fingers and toes. Their vigesimal system required only three symbols: a dot for one, a bar for five, and a shell glyph for zero. Complete isolation from the Old World proves this was convergent evolution—two civilizations reaching the same solution under similar mathematical pressure.

China developed positional counting rods by the 5th century BCE, with columns representing increasing powers of ten. Rods alternated between vertical and horizontal orientation to prevent confusion between adjacent positions. Alexander Wylie noted that Chinese mathematicians "were in possession of a notation for numbers more perfect than any which was invented before our own era in Europe."

What Egypt, Greece, and Rome lacked was not mathematical sophistication but positional pressure. Their additive systems—separate symbols for each power of ten—worked well enough for recording fixed quantities on monuments and in records. They simply never encountered the administrative complexity that made position-dependent calculation necessary. When Indian mathematicians finally developed decimal positional notation with true zero (around 500-628 CE), the synthesis of Babylonian structure with decimal base created the system that would eventually reach Europe through Arabic transmission and become universal.