A World Without A (Decimal) Point

genius

I did not realize until reading Michael J. Bradley’s book, Age Of Genius: 1300 to 1800, just how recently the Base-60 Numbering System of the Egyptians remained in at least partial use in mathematics.

Thanks to the Indians and the Arabs, the world long ago replaced the okay number system of the Romans with the superior base-ten place-value system we use today for whole numbers.  However, as recently as Shakespeare’s time, mathematicians were still using the Base-60 system for fractions.

Today, fractional values are most scientifically rendered using the Decimal system– which as the name implies– is a Base-10 system.  As the Decimals system also utilizes place-value, it integrates perfectly with our whole number system.  Truly, it’s a beautiful thing, and mathematics surely ranks as one of the most sublime inventions of mankind.

Base-60 fractions were commonly in use before being supplanted by the decimal,  and the system was called the “Sexagesimal” system (or so I’m told, although I wonder how many people using it actually called it that at the time; I’ve never to my recollection heard the term come up before in my historical or epistolary readings).

In a recent post, I talked of the intricate connection between Astronomy and the development of Mathematics, so I won’t go into detail here about how the Base-60 system of fractions ultimately derives from the fact that the year-measurement used by Egyptians divided into 360 days, which is a multiple of 60.  This is also where the (otherwise arbitrary) idea came from that a circle is comprised of 360 degrees.

As I said in that same recent post, many of the developments in Mathematics originated from work being done with circles.  That means… when Mathematicians were calculating, they were often dealing with degrees.  Therefore, when they went to express a fractional amount, they were speaking of partial-degrees.  By tradition, degrees had long been divided into “minutes,” with each degree containing sixty minutes (and I’m sure with each minute containing sixty seconds since this is precisely where we get our clock measurements of today).

Knowing this background, it becomes clearer why mathematicians would have ever bothered with a Base-60 Fractional System.

To use Bradley’s example from Age Of Genius, if a mathematician in the days of the Sexagesimal system were to write the decimal fraction…

0.017452406437283571

.

…he likely would have expressed it as:

0: 1, 2, 49, 43, 11, 14, 44, 16, 20, 17

.

This was a shortcut way of writing:  1/60 + 2/60^2  + 49/60^3 + 43/60^4… and so on.

One of the earliest exponents (no math pun intended) of the Base-10 system over the Sexagesimal system was al-Kashi, who lived 1380 to 1429.  He used fractions with denominators of powers of 10, not 60, which means he was basically using a proto-decimal system.

Nearly a hundred years later, Francois Viete [vee ET] (1540- 1603) was still trying to convince stubborn mathematicians to switch to decimal fractions.

It wasn’t until the 1590s that G.A. Magini and Christoph Clavius introduced the “decimal point.”

And it was the Scotsman John Napier [NAY pee yur]  (1550 – 1617), a slightly older contemporary of Shakespeare’s, who finally convinced the world to begin adopting the decimal system with his book, Constructio.

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