Why not Babylonian numerals?

Numbers and systems for writing them have a very long and varied history. Not every number system used the same base, in fact, some used base 5, base 12, base 20, or even base 60!

But wait, you might ask, what is a base?

Positional number systems (like the customary Arabic numerals) represent numbers as multiples of a base and powers of it. For example, in our base ten Arabic numerals, this is what we mean when we write the number 1559.37:

Number 1 5 5 9 .3 7
Power of 10 103 102 101 100 10-1 10-2
Meaning One group of 1000 Five groups of 100 Five groups of 10 Nine ones Three groups of one tenth Seven groups of one hundredth

Now, we don’t have to 10 as our base as we did there. We could have used any number other than zero or one. Actually, base one number systems exist: tally marks use base one. However, they are not truly positional number systems.

The Babylonians used 60. In their base 60 number system, this would have been the way they thought about 1559.37:

Number 25 59 .22 12
Groups of this power of 10 601
(60)
600
(1)
60-1
(1/60)
60-2
(1/3600)
Meaning 25 groups of 60 59 groups of 1 22 groups of 1/60 12 groups of 1/3600

Of course, the Babylonians didn’t write it like that. They would have written this:




The Babylonian number system has separate symbols for each number from one to 59:

Table of Babylonian numerals. Image: Josell7 via Wikimedia Commons

I will not write out any more Babylonian numerals with their notation. Instead I will use parenthesis around normal base 10 numbers. For example, I will write 60 like (1)(0).

Babylonian numbers were written in a fashion similar to ours. The first numeral on the left was the most significant, or the one representing the largest value. The second one was the second most significant and so on. At first, a blank space was used to mean what we would use zero for. That would have sometimes been problematic, as it might not always be clear how many blank spaces had been left. Around 311 BC a placeholder was added, . It was not the same as our zero. It was only used between other numerals as a placeholder. It never was used at the end of a number in the way we use zero in numbers like 10 or 200. The Babylonian system did’t include a direct equivalent to our decimal point. That is, if you wrote the numeral for one it wasn’t totally clear if you meant 1 or 60. The reader had to know something about what you were writing to be able to figure that out. On the other hand, it is usually pretty clear if a number should be 1, 60, or 3600. You wouldn’t wonder whether you were looking at one ox or 60 oxen!

This was a significant advance compared to previous numeral systems such as the non-positional Egyptian one. Before Babylonian numerals, most systems had a different symbol for each power of the base: a symbol for 10, another for 100, and so on. That meant that it was not possible to write numbers larger than a certain amount in those systems.

Base 60 numbers have significant advantages over base 10 numbers. 60 is a very nice number. It can be divided evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, or 30. Ten, on the other hand can only be divided evenly by 2 or 5. This makes a difference in how easy it is to write fractions. In base 10, such a simple fraction as 1/3 does not terminate: 0.33333… In base 60, it is easy to write since 1/3 is 20/60. It would be written as 0.(20) where (20) is a single numeral. In general, it is much easier to write a fraction if it’s denominator is a factor of the base, hence more numbers can be expressed simply with base 60 numbers.

Fun facts:

  • One drawback to base 60 numbers is that if you wanted to memorize a base 60 multiplication table, you would have to memorize 1830 entries! In base 10 there are only 55.
  • The angle composing 1/360 of the circle is a Babylonian invention. In base 60 numbers, it made sense to define it this way as 360 in Babylonian numerals would be (4)(0). Furthermore, the degree is divided into 60 minutes and each minute is divided into 60 seconds.
  • Our 60 minute hour also came from the Babylonian preference for base 60 numbers.
  • It has been suggested that base 60 arose from a finger counting system. On one hand, each of the 12 finger bones represented a unit. To represent multiples of 12, the thumb or a finger from the other hand was placed between two different fingers.
  • The Babylonians knew enough astronomy to realize that there are 365 days in a year. In base 60, the 365 is (4)(5). Writing it that way seems a bit tidier.

References

NRICH by the University of Cambridge

Babylonian Numerals on Wikipedia

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