Saturday 24 August 2024

Simon Stevin's Non-fraction method of Decimals

I mentioned the name of Simon Stevin to a young math teacher in talking about decimal fractions the other day and the response I got led me to believe he had never heard of him. I didn't ask, so maybe I was wrong, but it made me think I wanted to post this older blog just to bring his name to some new teachers out there who may not yet have heard of this amazing guy:
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A while back, John Cook at the Endeavour Blog  posted about a comment by Keith Kendig in his book, Conics.

It happened when I started learning about decimals in school. I knew then that ten has one zero, a hundred has two, a thousand three, and so on. And then this teacher starts saying that tenth doesn’t have any zero, a hundredth has only one, a thousandth has only two, and so on. … Only much later did I have enough perspective to put my finger on the problem: The decimal point is always misplaced!

John demonstrates the proposed solution as well.

The proposed solution is to put the decimal point above the units position rather than after it. Then the notation would be symmetric. For example, 1000 and 1/1000 would look like this:
Of course decimal notation isn’t likely to change, but the author makes an interesting point.

I then commented on John's blog that, in fact, Simon Stevin, who probably is more responsible than anyone else for introducing decimal numbers into the west had used a very similar approach as the one suggested by Keith Kendig. The image below is from De Thiende, which translated into English appeared as Decimal arithmetic. In fact, Stevin didn't think of his method as using fractions at all. In fact the English Translation in the full, self-advertising manner of books of the period, was 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. (My emphasis)

So, as I mentioned at John's blog, "He seems to have viewed the values as integers, much as we now think of minutes and seconds as integers. Few people consciously think of 3 minutes as 3/60 of an hour in regular computations. This was the view that Stevin took. He did not even use fraction names for the place values, but referred to them as prime, second, third, etc. (It has been often suggested that the use of ', ", etc for the minute and second in time, 12 23' 13", date back to the Greeks measure for angles of arc, but Cajori finds no evidence of their use prior to the 16th Century. The names minute and second came from the Latin for "minor part" which gave us minute, and the "second minor part" which gives us seconds.)

Decimal fractional numeration and the

decimal point in 15th-century Italy

The notation, in spite of the objections of folks like Mr. Kendig, didn't seem very useful, and so in 1612 Bartholomaeus Pitiscus opted for the decimal point we use today, and when it was used by John Napier, well here we are.
His decimal points first appear in his 1608 edition of Trigonometria in the added trigonometric tables and can also be found in the 1612 edition. (The word 'trigonometry' is due to Pitiscus and first occurs in the title of his work Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus first published in Heidelberg in 1595 as the final section of A Scultetus's Sphaericorum libri tres methodicé conscripti et utilibus scholiis expositi. In 1600 a revised version of Pitiscus's work was published in Augsburg as Trigonometriae sive de dimensione triangulorum libri quinque. )

Now if all Simon Stevin had done was introduce a really nice book on decimal fractions, you could give him a pat on the back and a big
"ataboy" and send him on his way, but:
He was big on hydrostatics, handy if you are Dutch, and in fact he made many improvements in the Dutch windmill pumping system. He also figured out that water pressure depended on height, and not on the shape of the container, and he was the first to explain the tides as the effect of the moon.
And for a walk on the wild side, he invented those land yachts you see racing up and down beaches. The one he made for the Prince of Orange was said to outrun the horses (with 26 passengers!).
he was one of the first to write about the equal temperament musical scale related to the twelfth root of two, which he seems to have gotten from Galileo's father.
And in math, he was the first in the west to write about a general solution to the quadratic equation; which alone should make him a name known to high school students and teachers.

In the Stevin plaza on the statue of Stevin is an image that many find curious.  A nice explanation below comes from the Futility Closet of Greg Ross.
"Drape a chain of evenly spaced weights over a pair of (frictionless) inclined planes like this. What will happen? There’s more mass on the left side, but the slope on the right side is steeper. Simon Stevin (1548-1620) realized that in fact the chain won’t move at all — if it did, we could link the ends as shown and produce a perpetual motion machine."

This is remembered as the “Epitaph of Stevinus.” Richard Feynman wrote, “If you get an epitaph like that on your gravestone, you are doing fine.”

the statue in the middle of Simon Stevinplein in Bruges.

It is important to distinguish between the (re)invention of the decimal point, and the creation of decimal fractions.
In a mercantile context, Leonardo of Pisa had come close to a pure representation of decimal
fractions in his 1202 Liber abaci by extending from metrological considerations, where (for instance) he represented the number

71.7579463519 as  9  1  5   3   6   4    9   7   5   7      71.
10 10 10 10 10 10 10 10 10 10

Note the fraction is read from right to left and if you read the value reading only one upper digit at a time the first approximation is 7/10.  When you add in the five you multiply all the tens below and right of it to get 5/100 more or 75/100.