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

Example of the Complement Algorithm

Let’s consider subtracting \(19\) from \(26,\) or

using the Complement (or Complementary) Algorithm. To perform the subtraction \(26-19\) using this algorithm, one gives \(10\) to the minuend and \(10\) to the subtrahend, as in the Equal Additions Algorithm. Where the Complement Algorithm differs from the Equal Additions Algorithm is, not surprisingly, in its use of the complement with respect to \(10.\) Instead of computing \(16-9=7,\) one computes \((10-9)+6=7,\) where \(10-9\) is the complement with respect to \(10.\) Then, one computes \(2-2=0.\)

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

Historical Discussion

The complement, or complementary, algorithm was another widely used algorithm during the 1700s and 1800s. Thomas Dilworth, a well-known author whose arithmetic text went through several editions, advocated the use of this algorithm, writing (Dilworth, 1802, p. 21),

When the lower number is greater than the upper, take the lower number from the number which you borrow, and to that difference add the upper number, carrying one to the next lower place.

As shown in Figure 9, Jonathan Butlar (1785) presented the complement algorithm in his cyphering book as follows:

Put the less number under the greater with units under units tens under tens, then begin at the right hand and take the lower figure from that above it but if it be greater than that above take it from 10 and add the upper figure to the remainder set down the result and carry 1 to the next place and so proceed.

Figure 9. Complement algorithm in the cyphering book of Jonathan Butlar, written in 1785 in Bucks County, Pennsylvania. (This image is reproduced, with permission, from the cyphering book belonging to Nerida F. Ellerton and M. A. (Ken) Clements.)

Note on source: For details about this and many other cyphering books, consult Rewriting the History of School Mathematics in North America 1607–1861, by N. F. Ellerton and M. A. Clements (Springer, 2012). Ellerton and Clements have indicated that photographs of excerpts from the cyphering books in their collection can be reproduced in scholarly papers or presentations or in curriculum materials, provided it is indicated that the cyphering books from which the photographs were taken are in the Ellerton-Clements collection and provided reference is made to the above-mentioned book by Ellerton and Clements (2012).

Although the complement algorithm appeared as early as 1478 in the Treviso Arithmetic (Smith, 1909; Swetz, 1987), 20th and 21st century mathematics educators, at least during the last 100 years or so, have considered the algorithm to be a special case of the equal additions algorithm or have forgotten it entirely. It seems that by the 1930s the “complementary method” was losing popularity. J. T. Johnson (1938) performed a study comparing subtraction algorithms and omitted the complement algorithm from the study.  More recently, Susan Ross and Mary Pratt-Cotter (2000) provided an example of the equal additions algorithm in their article, “Subtraction in the United States: A Historical Perspective,” which is actually an example of the complement method. Willetts (1819, p. 11, as cited in Ross & Pratt-Cotter, 2000) gave the following rule for “subtraction”:

1. Begin at the right hand and take the lower figure from the one above it and set the difference down.
2. Place the lesser number under the greater, with the units under the units, tens under tens, etc.
3. If the figure in the lower line be greater than the one above it, take the lower figure from 10 and add the difference to the upper figure which sum set down.
4. When the lower figure is taken from 10 there must be one added to the next lower figure.

Ross and Pratt-Cotter classified this rule as demonstrating the equal additions algorithm. Upon closer examination of this algorithm, one can see that it is different from the equal additions algorithm in the third step. When executing the subtraction, one is calculating the complement using the digit in the subtrahend and then adding the digit in the minuend. Algebraically, with the complement method, if one is given \[(10a+b)-(10c+d),\quad{d>b,}\] then \[(10a+b)-(10c+d)=((10-d)+b)+10(a-(c+1)).\]

Whereas, with the equal additions algorithm, if one is given \[(10a+b)-(10c+d),\quad{d>b,}\]then \[(10a+b)-(10c+d)=(b+10-d)+10(a-(c+1)).\] To be more concise, the complementary method of subtraction is based on the identity \(a-b=a+(10-b)-10\) (Smith, 1925), whereas the equal additions algorithm is based on the identity \(a-b=((a+10)-b)-10.\) Admittedly, the algorithms are closely related, especially to the modern mathematician’s eye; however, mathematicians and arithmeticians from the 1700s and 1800s viewed the complementary method as a different algorithm from the equal additions algorithm. J. Brookes (1776) mentioned both the equal additions algorithm and the complementary algorithm in his book. He clearly identified both algorithms as different methods of subtraction and clearly stated a preference for the complementary method. Brookes (1776) referred to the equal additions algorithm as a “more methodical” algorithm (p. 25).

Figure 10. Author J. Brookes preferred the complement algorithm, as indicated on page 25 of his A treatise on arithmetic, published in 1776 in New Castle, England. (The image of this page has been reproduced from a Google Book with free access.)