Numbers Across Cultures and Time

About 4000 years ago, scribes in Mesopotamia (in present-day Iraq) pressed numbers into clay tablets with a reed. They used only two symbols: a vertical wedge for 1 and a corner wedge for 10, repeated as needed to make any digit from 1 to 59. Those digits were then placed in base 60: the places were worth 1, 60, $60^2=3600$, and so on. One reason base 60 is handy: 60 has twelve divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), so halves, thirds, quarters, fifths and sixths of 60 are all whole numbers. That is why a third of an hour is exactly 20 minutes. For a long time the Babylonians had no zero symbol and simply left a gap, which could be misread. By about 300 BCE scribes were using a placeholder sign (two slanted wedges) to mark an empty place inside a number, but it was a placeholder only, not a number you could add or subtract.

Cuneiform Numbers - Numberphile

The Maya of Mesoamerica (present-day southern Mexico, Guatemala and Belize) wrote numbers with three symbols: a dot for 1, a bar for 5, and a shell for 0. Dots and bars make any digit from 0 to 19 (for example, 13 is two bars and three dots). Digits are stacked vertically, with the ones at the bottom and the twenties above, so the system is base 20. In their Long Count calendar the third place was worth $18\times 20=360$ days rather than 400, so read calendar dates with that rule. The Long Count, a place-value count of days that needs a zero, was already carved on monuments in the region in the first century BCE, and Maya monuments clearly show a zero sign by the 300s CE (stelae at Uaxactun, Guatemala, dated 357 CE, use a flower-shaped zero; the shell is the form drawn in later Maya books). That makes the Maya one of the few cultures known to have invented a zero independently. Base 20 is still alive: in 1994, middle-school students in Kaktovik, Alaska, noticed that their Iñupiaq language counts in base 20 and designed new digits to match. The Kaktovik numerals were added to the Unicode standard in 2022, so they can now be typed on computers and phones.

The Mayan Number System: Writing Mayan Number in Base 10

A placeholder says "nothing in this place." Treating zero as a number in its own right, something you can add, subtract and multiply, was a separate step. In 628 CE the Indian mathematician and astronomer Brahmagupta wrote rules for calculating with zero and with negative numbers, which he described as "fortunes" (positives) and "debts" (negatives). Most of his rules match ours: a number minus itself is zero, a number plus zero is unchanged, and the product of two debts is a fortune (a negative times a negative is positive). He also tried to define division by zero, which mathematicians today leave undefined. The Indian base 10 digits with zero spread through scholars writing in Arabic, including al-Khwarizmi in Baghdad in the 800s CE. They reached Europe gradually, and Fibonacci's book Liber Abaci (1202) did much to popularize them there. That is why they are called Hindu–Arabic numerals.

Even earlier, Chinese mathematicians calculated with negative numbers using bamboo counting rods laid out on a counting board. The collection The Nine Chapters on the Mathematical Art, assembled over several centuries and in its known form by around the first century CE, used negative quantities while solving problems with several unknowns. In the commentary written by Liu Hui in 263 CE, red rods stand for positive numbers and black rods for negative numbers. Notice that this is the reverse of the modern business phrase "in the red," which means losing money. A colour is just a code: what matters is that the rule is agreed on and used consistently.

Around 300 BCE the Greek mathematician Euclid described how to cut a line segment into two parts so that the whole is to the longer part as the longer part is to the shorter part. That ratio is now called the golden ratio, written $\phi$ (the Greek letter phi): $\phi =\frac{1+\sqrt{5}}{2}=1.6180339\dots$. Because $\sqrt{5}$ is irrational, so is $\phi$. It connects to the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …, where each term is the sum of the two before it: the ratios of neighbouring terms ($\frac{8}{5}=1.6$, $\frac{13}{8}=1.625$, $\frac{21}{13}\approx 1.615$) get closer and closer to $\phi$, the same kind of limit you met in the previous lesson. The counts of spirals in sunflower heads and pine cones are often neighbouring Fibonacci numbers. Be wary of popular claims that famous buildings or paintings were designed around $\phi$: many of those claims rest on loose measurements.