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Why Base 60 for a Number System?

Date: 15 Jan 1995 20:13:12 -0500
From: Roy P. Sachs
Subject: What were they thinking?

I recently sent you a question regarding the origin of the concept of a 
circle containing 360 degrees.  Your answer has led to another question.  
Contained in the answer was the statement that the Babylonians used a 
base 60 number system.  What were they thinking when they chose that 
base?  Why 60, and not 10 (fingers), or 20 (fingers and toes), or 5 
(fingers on one hand).  What did they have 60 of?
Thanks in advance.

Roy Sachs

Date: 16 Jan 1995 15:00:06 -0500
From: Dr. Sydney
Subject: Re: What were they thinking?

Dear Roy, 

        Great question!  It does seem like 60 is an odd choice for a base,
doesn't it?  I think there are a couple of reasons why the Babylonians used
base 60, and right now I can think of only one of them.  So, if there are
more, either one of the other math doctors or I will write back to you.  

        One reason that 60 was used for a base was that 60 has so many
divisors.  Its divisors include: 2, 3, 4, 5, 6, 10, etc.  It is the smallest
number for which 2, 3, 4, 5, and 6 are divisors.  That makes it a nice base
to work with.

        If we come up with other reasons or more on this, we'll write back.  
I hope this helps.  We are glad you wrote again, and feel free to write with
more questions.


Date: 17 Jan 1995 19:26:54 GMT
From: Dr. Math
Subject: Re: What were they thinking?

Hi Roy,  
   That same question bothered me too.  Why in the world did the
Babylonians use a base 60 system?  Here is an answer from John
Conway that I thought was really insightful.

From: "John Conway"
Date: Thu, 12 Jan 95 01:18:29 EST

   It is generally believed that the reason the Babylonians divided
a full circle into 360 degrees is that the Sun goes around the Earth in
about 360 days.   (365 would be too awkward a number).   I think
the standard ruler and compass construction of the hexagon gives
quite a nice explanation of their partition of it into 6 blocks of
size 60.

    It has often amused me that there are electronic calculators
that still use the division of the degree into 60 minutes of 60
seconds each.

    The reason is ultimately this.  The trigonometric tables of
Ptolemy (which were probably copied from Hipparcus, who live a few
hundred years earlier), had all the numbers written in base 60,
following the Babylonian tradition.  [Indeed probably they had
Babylonian precursors.]

    As the centuries passed, more or less all the other information
was translated into more modern notations.  But the work of changing
the scale in the trig tables was just too much to be contemplated.
So nobody ever did it!   As a consequence, trigonometric tables are
STILL most often written in the Babylonian scale, and even the most
modern technological gadgetry is likely to use it!

            John Conway

Hope that helps,
      Ethan Doctor On Call
Associated Topics:
Middle School History/Biography

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