Decimal fractions were introduced to Europe by Simon Stevin in his book *De Thiende, leerende door ongehoorde lichticheyt allen rekeningen afveerdighen door heele ghetalen zonder ghebrokenen* (‘The tenths’), first published in Dutch in 1585. The English version, *Decimal arithmetic: Teaching how to perform all computations whatsoever by whole numbers without fractions, by the four principles of common arithmetic: namely, addition, subtraction, multiplication, and division *was translated by Robert Norton and printed in 1608.

Stevin felt that decimal fractions was such a great innovation he declared the universal introduction of decimal coinage, measures and weights to be merely a matter of time. Stevin either independently derived, or more likely, learnt the concept from the Arabic works such a as those of Abul Hassan Al-Uqlidisi (written in AD952-3). Al-Uqlidisi considered the problem of successively halving 19 five times, and gave the answer as

The vertical mark on the 0 represents the zero point on an axis and shows that the decimal fraction part of the number starts with the digit to the right.

The notation propounded by Stevin the dropped the arabic zero and used numbers in small circles to indicate the power of 10 that the number should be multiplied by.

In the example above, his notation represents 184 times 10^{0} plus 5 times 10^{1} plus 4 times 10^{2} plus… you get the idea. The modern dot notation implies the same thing, but the powers are understood from the position of the digit a particular number of places to the left or right of the decimal point. In Stevin’s notation, it is possible to skip straight from 10^{0} to 10^{5} if the intervening digits are all zeros.

The earliest published notation that uses the decimal point that we’d recognise was in the 1612 *Trigonometrical Tables* of Bartholomaeus Pitiscus. Scottish gentleman mathematician John Napier, 8th Laird of Merchistoun, then used the modern dot notation in his *Mirifici Logarithmorum Canonis Descriptio* (1614) and other works. Napier is credited with introducing the dot notation to the English-speaking world and popularising its use.

More on Napier in a later post. So many of the ideas were taken from Arabic scholars, I probably should call the series: *Intelectual Property Theft of the Seventeenth Century*.

## Further reading

*Bartholomaeus Pitiscus* on Wikipedia, http://en.wikipedia.org/wiki/Bartholomaeus_Pitiscus

*Decimal Arithmetic* at Muslim Heritage http://www.muslimheritage.com/topics/default.cfm?ArticleID=542 accessed 1 May 2010

*John Napier*, Wikipedia, http://en.wikipedia.org/wiki/John_Napier, accessed 22 April 2010

Napier, J., *Mirifici logarithmorum canonis descriptio*, 1614, also published in English as

*A Description of the Admirable Table of Logarithmes* by Edward Wright, 1619.

Napier, J., *Mirifici logarithmorum canonis constructio*, 1619 http://books.google.com/books?id=VukHAQAAIAAJ

*The story of Simon Stevin*, http://mathsforeurope.digibel.be/Stevin.htm, accessed 1 May 2010

*Simon Stevin* at Wikipedia, http://en.wikipedia.org/wiki/Simon_Stevin#Decimal_fractions, accessed 1 May 2010

Stevin, S., *De Thiende*, 1585. There’s a copy at http://home.planet.nl/~hopfam/Thiende.html which Google translator recognises as Dutch, but but don’t expect Google to translate it. It’s much worse than my translation of the Oostend Leather Bucket.

Of course I immediately had to go to google translator…No voorwaer, but one trade gantsch so bad, was undoubtedly datse nau particular weather and wear, because a coarse ghelijck byghevalle Man does one find sizes Schadt, without eenighe const there in ghelegen to be, here ist oock toegheghaen. That cleared things up doesn’t it!

It’s easy to forget that so many of the conventions we use in daily life had to be invented (?) at some point. Thanks for the mathematical reminder, Rev.

By:

internationalroutieron May 4, 2010at 9:30 am

Shouldn’t the explanation of Stevin’s notation be “184 times 1 plus 5 divided by 10 plus 4 divided by 100…” (the formatting for superscript doesn’t seem to hold for a comment)?

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mimbleson May 5, 2010at 9:23 am

The minus signs have dropped out of the published example somewhere in the process. I’ll use the convention ** to represent raising a base to a power, so 10**2 is 10 squared =100. Similarly 10**-2 is 10 to the power of -2 or 0.01. The explanation should be 1 x 10**2 plus 8 x 10**1, plus 4 x 10**0 plus 5 x 10**-1, plus 4 x 10**-2 plus 2 x 10**-3… giving

100 + 80 + 4 + 0.5 + 0.04 + 0.002 …

By:

Leatherworking Reverendon May 5, 2010at 8:18 pm

Thought it might be something like that :-)

By:

mimbleson May 5, 2010at 8:22 pm

[…] Point Erratum I must apologise to the reader for a technical problem in the Technical Innovations of the Seventeenth Century – The Decimal Point post. During the process of transmitting the text from my brain to the keyboard, all the negative […]

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Decimal Point Erratum « International Routier-the Blogon May 12, 2010at 8:25 am