Homework 1: Why base 60?

 

Why is 60 convenient? My guess is that 60 is larger than 10, and a large number can be expressed in base 60 with significantly less digits in a base 60 system compared to a base 10 system. Since much of the math was done on clay with wedge shaped markings, putting less markings was likely easier to do and less prone to error. Just like it would be painful to use base 2 in everyday life by placing many 1s and 0s, using wedge shaped markings to put down many digits was likely a difficult task. Also, the number 60 has many divisors, namely 2,3,4,5,6,10,12,15,20,30. Compared to 10, which is only divisible by 2 and 5,it might be easier to group things in multiples of the above divisors to quickly count.

We still see 60s in some measurement units. The first one that comes to mind is time in seconds and minutes and hours. 1 minute is 60 seconds and 1 hour is 60 seconds. This aligns perfectly with a base 60 system, since 1 hour is 3600 seconds which might be hard to remember but as 1,0,0 in base 60 system that’s quite natural. I am not sure if the imperial system implanted with multiples of 12 and 3 were related to base 60 as they are factors of 60. Refresh rates on monitors and TVs are often multiples of 12 and/or 60. Common devices refresh at 60Hz while higher end products refresh at 120hz, often going up to 144hz or as low as 24hz for cinematic effects. Electricity in Canada is delivered at 120V at 60Hz, which makes electronics that are connected to the power source a good time tracker at 60hz.

Source: https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/

Upon some research, it seems like deciphering Babylonian tablets with math on it is no trivial task, and some remain a mystery to us. There are many proposed reasons as to why base 60 is used, and I am quite convinced that base 60 is used because they inherited it from the people before them and they just find 60 beautiful. It has many divisors and can accommodate dividing into groups of 3 or 12 or 5 which come handy. Just like in number theory, some mathematicians are in search of perfect numbers and co-primes or Mersenne primes, it does not surprise me that elegance is simply the answer, even if there is no immediately available applications. Encryption using computers and primes and modular arithmetic was probably not the Greek’s concern when they were posing difficult number theory problems that seem basic to us, and I guess it is not too unreasonable to say that 60 is chosen simply because it looked like a good choice.

Comments

  1. Susan Gerofsky23 September 2020 at 13:15

    Great post, Ian! Your argument from elegance is a convincing one. I had never really thought about the 60 Hz AC electrical system, or connected it to the 24 frames per second of cinematography (or 30 fps for TV), and that is a fascinating thing to look into! Good argument too about the inconvenience of binary and convenience of using a larger base for everyday calculations. I'm pretty sure that the Imperial system, with its 12s , 4s and 3s, was either connected with base 60 or used for the same reasons of convenience of fractions with so many factors....

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