Why Base 60?

A little exploration of Babylonian numbers

Presently, people are most familiar with using a completely base ten system (not saying "10" here, because that could mean anything depending on the base you choose... just kidding). It's great - we have 10 fingers, 10 toes, and are all 10 out of 10 individuals, I think.

However, the Babylonians had a different idea - a base 60. The places went something along the lines of (right to left, starting from the decimal comma):

• 1, 0 to 59 (the unit place; note, within this place there are two places for two different types of wedges)
• 60, 0 to 59 (multiply by 60)
• 3600, 0 to 59 (multiply by 60)
• ...and so on (multiply by 60 for the next place)

If we used our Arabic numerals, then something like 337 using this base system would be:

(3 x 3600) + (3 x 60) + (7 x 1) = 10987 in our regular base 10. Interestingly, the Babylonian system would use far fewer digits than ours.

So let's ask: why would they use 60 for their places? When do we use 60?

First, we will speculate. There were two trains of thought that I had: the first being its divisibility, and the second being its relation to the human form. Regarding the former, 60 is a highly composite number that can be divided by 2, 3, 4, 5, 6, 10, 12, 15, and 30. Fantastic! There was a point brought up in class as well regarding clocks which use 60 for minutes and seconds, and I find it particularly convenient due to its divisible nature. Mathematically, this would make sense. From the factors, we can also make connections with the number 12, which is the number of months or Chinese zodiac animals, or the number of astrology signs. Meanwhile, the number 10 is strictly divisible by only 2 and 5. Not as flexible!

On the other hand, we have 5 fingers. Well, the first hand probably also had 5 fingers, but let's not get into those details. In any case, these fingers can be convenient for counting. Children learn to count to ten on their hands as they have 10 fingers, whereas computer scientists get clever and count to 1023 using a binary system (fingers can be one of two states, after all). Maybe they'd get extra creative and use their tongue to represent the 1024s place, and count up to 2047. But I digress. In the case of the Babylonians, I suspect that they used their digits on one hand to tally up the 10s in the sub-10s place. Both hands would be used if they wanted to work on the units sub-place.

Other speculations could be that 60 represented something special for them. The people of that time and place were really into the moon, and a moon cycle is about 30 days which is half of 60. Maybe a Babylonian ruler really liked the number 60, but at this point the theories are becoming outlandish.

After some quick research, what have I found out about the significance of 60?

Upon doing a quick search on the significance of 60, I found that I was on to a lot of things (such as the divisibility and time aspects) but also omitted various others! The key takeaway for me is that this seemingly arbitrary number actually has a solid grounding in our civilization, from ancient to contemporary times.

Before the sexagesimal system was conceived, the prevailing theory started with two peoples with different base systems eventually merging to become one group. The original base systems were 5 and 12, so the marriage of the peoples led to the marriage of their base systems of 60. From there, the ancient peoples made advancements in measuring time using the number 60, which had ramifications on today's time-telling. Do you see those modern SI units where every unit is some power of 10 up or down from another? No one is casually calling a thousand seconds a kilosecond unless they are in a specific discipline that requires it. On the topic of keeping track of time, there are also ties to Indian and Chinese calendars, with a 60-day unit of time being particularly popular. Amongst the Chinese, their beliefs in the five elements and twelve zodiacs have parallels with the marriage depicted above, as the Chinese also believe in a full 60-year cycle that features all possible pairings of elements and zodiacs.

Other significances of 60 that can be found in modern times include: degrees as a unit of angle (360 degrees makes a full round, and 360 is divisible by 60, which is further divisible by all the factors above - very helpful especially when dealing with trigonometric functions and the unit circle), electrical applications (for example, monitors and programs running at "60 frames per second or hertz" - as someone who plays fast-paced video games, the subdivision of 60 frames in a minute is of personal significance when comparing action speeds and input delay issues), and even the latitudes and longitudes that act as navigational grids (180 of each, divisible by 60; smaller units also abide by a sexagesimal system). The former two items are particularly applicable to me, given my time spent working with math and gaming, so I have to give a shoutout to the Babylonians for having the right idea. However, I imagine the people like the navigators of the world are also thankful for the divisibility of 60 as they interpret their coordinates.

Sources of information and inspiration:

• https://thoughtco.com/why-we-still-use-babylonian-mathematics-116679
• https://ridingthebeast.com/numbers/nu60.php
• https://en.wikipedia.org/wiki/60_(number); particularly the ideas under "In other fields"