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

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

Consider the subtraction problem, \(56-19,\) or

  \(5\) \(6\)
\(-\) \(1\) \(9\)

The Austrian Algorithm for subtraction is quite intuitive, taking advantage of subtraction as the inverse of addition. In this example, one would first consider what whole number should be added to \(9\) to obtain \(6.\) Since this is not possible with only whole numbers, one would then consider the whole number that could be added to \(9\) to obtain \(16,\) which is \(7.\) To compensate for the \(10,\) one would now look at the \(1\) as a \(2.\) Using similar logic, \(2\) plus a whole number would need to give \(5,\) which gives \(3\) in the tens place and a difference of \(37.\)

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

Historical Discussion

Due to efforts to introduce this algorithm into Austrian schools and subsequently German schools, the algorithm acquired the name of “Austrian method.” The algorithm has also been called the “addition” or “making change” method (Smith, 1909) because it utilizes addition. One must think of what must be added to the subtrahend to get the minuend. This algorithm is again illustrated in the following excerpt (McClellan & Ames, 1902, as cited in Johnson, 1938, p. 23):

From 94,275 take 67,492:


Thus: 2 and 3 are 5; 9 and 8 are 17, carry 1 to 4 as in addition, making it 5, 5 and 7 are 12; carry 1 to 7 making it 8; 8 and 6 are 14; carry 1 to 6 making it 7; 7 and 2 are 9.

Only one German text from 1700 to 1900 in the Ellerton-Clements collection of arithmetic books used the Austrian method of subtraction, that of Albert Braune, published in Leipzig, Germany, in 1882. This book, written in German, may have been utilized by German-speakers in the United States. J. T. Johnson (1938) wrote that the excerpt given above was the earliest use of the Austrian method in the United States, but the existence of the Braune book may disprove this claim. Johnson also wrote (1938, p. 23),

The strictly addition procedure in subtraction is mentioned in the Handbuch der Mathematik by [Adam] Bittner, published in Prague in 1821; this method is explained in [Joseph] Salomon’s (1849) Lehrbuch der Arithmetik und Algebra.

Based purely on inference from Johnson’s research and the books examined in this research, the Austrian method, while possibly prevalent in books in Germany and Austria and potentially within some books in the United States, was not often used as an algorithm in the United States between 1700 and 1900.

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