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An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - The Decomposition Algorithm

Nicole M. Wessman-Enzinger (Illinois State University)
Example of the Decomposition Algorithm

Consider again the same subtraction problem as on the preceding page, \(940-586,\) or 

  \(9\) \(4\) \(0\)
\(-\) \(5\) \(8\) \(6\)

Nearly all students in the United States today utilize a method that begins with “borrowing” one group of ten from the 4 in the minuend, exchanging it for ten units, and giving these ten units to the 0 in the minuend. This is called the Decomposition Algorithm because, to subtract, students initially decompose the number \(940\) to \(900+30+10.\) One would need to borrow again from the 9 in the minuend. Thus, the decomposition is written as \((800+130+10)-(500+80+6).\)

See the Decomposition Algorithm in action using the example \(940-586\):

Historical Discussion

The decomposition algorithm is the method of subtraction used predominantly today in the United States, and is referred to as the “standard subtraction algorithm” in many current texts. This algorithm is quite old, dating back to Spain in the thirteenth century, Italy in the Middle Ages, and India even earlier (Smith, 1909). In fact, there is evidence of the decomposition algorithm present in the writings of Rabbi ben Ezra (1140, cited in Smith, 1925).

Prior to being called decomposition, this algorithm was also referred to as “simple borrowing” (Smith, 1909). The “borrowing” in this algorithm is different than the “borrowing” utilized in equal additions. In the equal additions algorithm, ten is added to the minuend and then “repaid” by adding it to the subtrahend. On the contrary, with the decomposition algorithm no numbers are added to the subtrahend or minuend; rather, the subtrahend and minuend are simply “decomposed” and the decomposition of the minuend re-arranged. For example, \(41-29\) can be thought of as \((30+11)-(20+9).\) Then, the units are subtracted and the tens are subtracted, as follows: \[(30+11)-(20+9)=(30-20)+(11-9)=12.\] Chauncey Lee, a popular author in the 1790s, advocated the decomposition algorithm, writing (Lee, 1797, p. xi):

This (decomposition method) I conceive to be a more simple, natural and easy mode, especially in whole numbers.

Warren Colburn (1824) also encouraged the use of the decomposition algorithm. Colburn’s explanation, in Figure 7, is distinct from all the other texts because he actually demonstrated the “decomposing.”

Figure 7. Decomposition algorithm on p. 150 of First lessons in arithmetic (2nd ed., 1824), by Warren Colburn. (This image has been reproduced from a Google Book with free access.)

Probably influenced by Colburn, Daniel Adams (1830) also supported the decomposition algorithm. Adams’ explanation, shown in Figure 8, was typical of the time period, with no markings and no decomposition illustrated.  Also of special interest in the passage in Figure 8 is that Adams presented the equal additions algorithm at the end of this excerpt as an alternate method.

Figure 8. Typical explanation of decomposition algorithm, with the equal additions algorithm suggested at the end as an alternate method, on p. 22 of Adam’s [sic] New Arithmetic, by Daniel Adams, published in Keene, New Hampshire, in 1830. (This image has been reproduced, with permission, from the book belonging to Nerida F. Ellerton and M. A. (Ken) Clements.)

Nicole M. Wessman-Enzinger (Illinois State University), "An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - The Decomposition Algorithm," Convergence (January 2014)