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Babylonian Numerals

Babylonian numeral: Credit: https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg

In astronomy, we tend to care a lot about really small angles. Astronomers like to use units called arc-minutes and arc-seconds which subdivisions of a degree when viewing the sky.  It’s actually pretty convenient.  1 arc-minute is 1/60th of a degree. 1 arc-second is 1/60th of an arc-minute. For example, the moon and the sun have apparent diameters of 30 arc-minutes diameter each, and Mars has an apparent angular diameter of 25 arc-seconds.  Stars are much smaller. Betelgeuse, the red star in Orion, is one of the largest stars by angular diameter and has an apparent diameter of 50 thousands of an arc-second (or 50 milli-arcseconds) – which is the approximate resolution achieved by the Hubble Space Telescope. So, it’s really hard to get high enough resolution to get a star to spread across multiple pixels.  Lastly – the supermassive black hole recently imaged at the center of M87 had a diameter of about 50 micro-arcseconds using a technique resulting in unprecedented resolution. Imagine what treasures we might see when we get our resolution down to nano-arcseconds!  (see gravitational lensing).  Back to the point of the post – fractions of arc-seconds is the way astronomers and astronomy fans use to keep measurements of the Universe in context.

EHT image of black hole at the center of M87 – the first image of its kind. The apparent diameter is almost 1 million times smaller than that of Mars. Source: ESO https://www.eso.org/public/images/eso1907a/

But as engineer, building a metric telescope with steel measured in metric, often have to convert form radians. I find myself doing the conversion so often I’ve considered getting the conversion 1″ = 4.848 urad tattooed to the back of my hand.   I’ve often wondered, why the heck is the circled divided up in 360 degrees anyway? And why is a day divided in 24 hours, but then hours by 60 minutes, etc…

Today I stumbled upon a math puzzle in Babylonian numerals.  It turns out the Babylonians used a base 60 number system on which our measurements of time and angles are based.  I was perplexed – this is so interesting, even more than Roman numerals which are everywhere – why hadn’t I heard about this before?    I can’t do the full history justice, but you can read more here: https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/

Here are some fast facts on Babylonian numerals:

  • Base 60 is called sexagesimal.
  • The way we measure time, and angles is actually mixed radix – in which different positions in the number measure at different bases.
  • Originally developed by the Sumerians around 2000BC!
  • All numbers can be represented with only 3 symbols! One for 10s, one for 1s, and one for 0s.
  • We know about the system because of about 400 hundred clay tablets.
  • One of these tablets show a computation of square root of 2 (used to find the diagonal of a square).
Tablet discovered that shows the square root of 2 in Babylonian. (1+25/60+51/60^2+10/60^3). Credit: Bill Casselman (mentioning http://www.math.ubc.ca/~cass/Euclid/ybc/ybc.html) and the Yale Babylonian Collection.

It’s amazing to think of how different the world was from today, but there was still an interest in beauty of mass.

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