I have always enjoyed learning about the history of math and how ancient systems still influence the math we use today. There are so many things in our daily lives that we never question that at some point had to be established. Here is an example: why are there 60 minutes in an hour? 60 seconds in a minute? Why not 50? 100? Something else?
To find the answer to these questions, we have to go way back in time …
It is estimated that systems of numbers are approximately 10,000 years old. When our ancestors were hunters and gatherers, they had little need for math. But once people started settling down, growing their own crops and domesticating animals, they needed numbers for counting and measuring: Yesterday I had 120 sheep, now there are only 118. Two are missing. I marked the boundaries of my land with stones. It looks smaller today – did someone move my stones? People needed to track seasons to know when to plant. If there is a surplus of crops, how much extra is there to trade?
The ancient Romans used pebbles called calculi to help with counting. This is where our term calculation is derived from. The Romans used numbers we are all familiar with. The year 2023 can be written in Roman numerals as MMXXIII. The Roman number system is what we call an additive number system. To get the number sense of MMXXIII, you must add M + M + X + X + I + I + I. But additive number systems become too cumbersome when we try to do calculations. Any middle-schooler can easily multiply 2023 by 19. Imagine trying to do that with Roman numerals: MMXXIII times XIX. While some short-cuts were used, a Roman would essentially have to add MMXXIII nineteen times to get the answer.
The number system we use is a positional or place-value number system. When we see the number 2023, we understand this to be 2 thousands, no hundreds, 2 tens, and 3 ones. Until the concept of the number zero existed, a positional number system was not possible.
The importance of zero and a base
A positional or place-value number system needs a base. You just have to look at how many fingers are on your two hands to deduce why the base for our number system is ten. Some cultures, such as the Mayans and the Celts, likely included their toes because they developed a base 20 system. A child working with base 10 has ten symbols to memorize (0 to 9) and 45 unique addition facts to memorize starting with 1 + 1, 1 + 2, and so on up to 9 + 9. A child working with base 20, however, has 20 symbols to memorize and 190 unique addition facts to memorize, making this one of the reasons a base 20 system fell out of use. We can still see traces of the use of base 20 number systems. In French, the number 80 is quatre-vingts which translates to “four twenties.”
The ancient Egyptians never made the transition from an additive system to a positional number system. However, the Babylonians eventually did. It’s worth noting that all the mathematics of the various peoples inhabiting the Mesopotamia region, including Sumerians, Babylonians, Assyrians, Amorites, Chaldeans, Hittites, Scythians, Medes, and Persians, is grouped together as mathematics of the “Babylonians.” In the Babylonian number system, they did not use the number zero but left an empty space instead.
In our base 10 counting system, when we are counting and reach the tenth 1, we notate that as 10 (1 in the tens place and 0 in the ones place). When we reach ten 10s, we notate that as 100 (1 in the hundreds place, and 0s in the tens and ones places), and so on. The base the Babylonians used alternated between a base of 10 and a base of 6. When they reached the tenth 1, they notated that (in their own way) as 10, just like we do. However, when they reached SIX 10s, they notated it as 100. When they reached ten 100s, they notated that as 1000. But it only took six 1000s to make 10,000. The “carrying” process alternated between 10 and 6. This system is referred to as a sexagesimal or base 60 system.
While we are certain the Babylonians used a base 60 system because of cuneiforms that have been found, what we don’t know is why a base 60 system was used. This remains one of the greatest unsolved mysteries in the history of math. While there have been many theories proposed, the one which makes the most sense to me is found in Peter S. Rudman’s book How Mathematics Happened: The First 50,000 Years (Amherst: Prometheus, 2007).
Besides needing numbers for counting, they were needed for measurement. If you are in a room with no tape measure and want an idea of the width of the room, you might get an estimate by pacing the room and counting how many of your feet it took to get from one end to the other. Body parts were a starting point for measurements. People used their fingers, hands, feet, etc. In Egypt, a “hand” was 4 “fingers.” A “foot” was 4 “hands.” A “cubit” was the length from the elbow to the fingertip or 6 “hands.”
Counting and measuring systems developed separately. At some point a number system that is compatible with both counting and measuring is needed. There are several options to consider.
One option is to let the counting system and measurement systems be different. That is what the English system we are used to does. We count in base 10. However, in our measurement system there are 12 inches in a foot, 4 quarts in a gallon, and 16 ounces in a pound.
Another option is to keep the counting system and use an unnatural measurement system to match the counting system. This is what was done when the metric system was created in the 1790s, in which all measurements use base 10. There are 10 millimeters in a centimeter, 1000 milliliters in a liter, 1000 grams in a kilogram. In fact, when France adopted the metric system, it decimalized time and made use of that mandatory in 1794. There were 10 hours in a day, 100 minutes in an hour and 100 seconds in a minute. The decimalization of time was abandoned after six months. People just could not adapt to the change.
A third option is to keep the measurement system and force the counting system to mimic the measurement system. This is what Rudman theorizes the Babylonians did. The smallest unit of land measure was a sar. 10 sars made an eshe. But then six eshe made an USH. And so on, alternating back and forth between 10 and 6.
The Egyptians are credited with dividing the day into 12 hours based on their sun dials. The Egyptians, followed by the Greeks, based their time counting methods on the Babylonians. A sexagesimal system divides a base unit (the hour) into 60 parts. Finally, we reach the answer to the question of why there are 60 minutes in an hour and 60 seconds in a minute. In fact, the word “second” derives its name from the “sexagesimal” system. It’s all about that base!
