## Tuesday, 22 February 2011

### "Borrowing" in Subtraction, A Brief History

Subtract joins two easy to understand roots, the sub which commonly means under or below, and the tract from words like tractor and traction meaning to pull or carry away. Subtraction then, literally means to carry away the bottom part. The "-" symbol for subtraction was first used in Germany as markings on barrels to indicate those that were underfilled. Around the 1500's it began to be used as an operational symbol and it became common in English after it was used by Robert Recorde in The Whetstone of Witte in 1557.

The term subduction was often used in older English books up until about 1800. John Wallis uses the term in his "Treatise on Arithmetic", 1685 in describing subtraction... "Supposing a man to have advanced or moved forward, (from A to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how much he had advanced (upon the whole march) when at C? Or how many yards he is now forwarder than when he was at A? I find (by subducting 2 from 5,) that he is advanced 3 yards. Samuel Johnson's 1768 dictionary defines both terms, but includes "substraction" as part of the definition of subduction.

In the same dictionary, Johnson defines both subtract and substract, but for subtraction, the reader is referred to "see substraction" so I assume that was the more common term.

The method of subtraction commonly called Borrowing or decomposition seems to go back at least to the 1200s. In The Art of Nombrynge by John of Hollywood (Sacrobosco) subtraction was taught "entirley like the method of today, 'borrowing' and all." [E. R. Slight from "The Craft of Nombrynge, Mathematics Teacher, Oct. 1939].  Robert Recorde used the term in The Ground of Arts (1543) and Denniss and Smith (Robert Recorde, the life and Times of a Tudor Mathematician) describe the term as "a term that had already gained some currency".   The word, "borrow" may not have been commonly used until around 1600 as the earliest listing in the OED associated with subtraction is "[1594 BLUNDEVIL Exerc. I. (ed. 7) 91] Take 6 out of nothing, which will not bee, wherefore you must borrow 60." The borrowing of 60 suggests the exercise may have been about time. Here is a link with an image of a page from the Arithmetic of John Ayres in 1695 in which the word borrow is used in subtraction, although the method is more like what Ross and Pratt-Cotter (below) call the "equal additions" method.

Susan Ross and Mary Pratt-Cotter [Subtraction in the United States: An Historical Perspective, from The Mathematics Educator] show that prior to about 1940 in the US there were three common approaches to subtraction in arithmetic texts. The Borrow method, a method they call "equal additions" which also seems to date back to the 15th century (and is probably a logic based alternative to the borrowing approach), and a method of subtracting by adding on from the subtrahend, which is sometimes called the Austrian algorithm. (images of all three types can be found in the link above to Ross and Pratt-Cotter's document)
The research by Ross and Pratt-Cotter indicated that before 1937 there were few illustrations in American textbooks that show any physical "marking through or numbers being rewritten". Their work states that almost overnight, after a study by William Brownell, "most textbooks used the decomposition (borrow) method for describing borrowing in subtraction, and the use of the crutch described by Brownell became very popular. Today this method of subtraction is used in most textbooks that teach subtraction." The study also states,
Only one example was found, from a text published in 1857(Ray's Practical Arithmetic), where markings were used to keep track of the renaming process. This was done in only one problem in the text, with all other problems worked without any markings. Brownell was not aware, however, of any textbook employing this technique.
This statement, which I assume to be true, and the existence of a clear example of the "borrow" with markings in an 1898 copy of Gill's Oxford and Cambridge Practical Arithmetic , shown at right, make me suspect that the borrowing crutch first appeared in England and then made it's way to America. {if you have knowledge of any earlier appearance in textbooks in ANY country, please write and if you include a digital image, I will sacrifice a student in your honor)

I posted a request for information about texts or other sources of the use of the "crutch" and received the following from Ralph Raimi of the University of Rochester: