Subtraction without borrowing

A commonly known subtraction method is the "borrowing method" [Only a brief, informal discussion is included here until a formal entry of "dynamic subtraction" is provided.] . To perform "a - b" using this method, "b", the subtrahend, is written below "a", the minuend, such that the digits of the two numbers are aligned in columns. When a digit of the minuend is smaller than the corresponding digit of the subtrahend below it, the procedure calls for "borrowing" one power of 10 [ This brief discussion of the "borrowing method" refers to the decimal number system and decimal place values. The basic procedures described here can be applied to other situations. Only the boundary values and the borrowed magnitudes depend on the number system and the place values. For example, while performing time subtraction, if borrowing is necessary for the tens digit of the seconds, the borrowed minute adds 60 seconds to the number of seconds specified by the current digit, not 10.] from the digit of the minuend that is immediately to the left of the current digit. [ Also this discussion of the "borrowing method" does not address more complex situations such as when the “lending” digit is 0 (zero) and, therefore, nothing can be borrowed from it.] Then the value of the "lending" digit is reduced by 1. (More formally this is the "decomposition algorithm" because the "borrowed" power of 10 does not have to be and is never returned. [ Subtraction in the United States: An Historical Perspective, Susan Ross, Mary Pratt-Cotter, "The Mathematics Educator", Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.)http://math.coe.uga.edu/TME/Issues/v10n2/5ross.pdf] ) Anyone who subtracts using the borrowing method must do so from right to left, starting at the lowest-value digit and proceeding toward the highest-value digit. It is also necessary to remember to reduce the value of each “lending” digit by 1 before subtracting the corresponding digit of the subtrahend from it. Because of these requirements, learning subtraction with borrowing is difficult for some students. [ [http://www.mathventures.com/mathed/Arithmetic/SubtractionWithoutBorrowingLite.htm Subtraction Without Borrowing] on [http://www.mathventures.com/ MathVentures] ]

An alternative to the borrowing method is "subtraction without borrowing". There are several methods for subtraction without borrowing. However most of them employ other computational methods that are as complex as the methods they aim to replace or they introduce mathematical concepts for which the students who learn subtraction with borrowing are not ready. Among these are methods that use negative numbers and some variations of the method of complements.

Note. The subtraction methods mentioned here assume that the minuend is larger than the subtrahend. If the minuend is smaller than the subtrahend, the roles of the numbers are reversed and any subtraction method can be used followed by setting the result to be negative. If the two numbers are equal the result is zero and no method is needed.

Algorithm

The basic algorithm is:

# Change the minuend:
## Subtract 1 from the rightmost consecutive digit that is larger than the corresponding digit of the subtrahend;
## Change each digit to the right of the digit you reduced by 1 (Step #1.1) to 9 -- this is your new minuend;
# Add 1 to the number whose digits you changed to 9's -- this sum is the difference between the original minuend and the new one;
# Carry out the subtraction using the new minuend;
# Add the difference you got in Step #2 to the result of Step #3. This is the final answer.

Descriptive example

As demonstrated by the example below, subtraction without borrowing employs single-digit subtraction in each column of digits and then addition.

65432 (minuend) - 27894 (subtrahend)

*Change the minuend to 59999.
*The difference between the original minuend and this new one is 5433.

It is necessary to remember this difference. This is easy because it equals to the minuend without its leftmost digit plus 1:

5432 + 1 5433

*Now perform the revised subtraction, using the new minuend 59999 - 27894 32105
*Now add the difference, which we noted above, to the result

32105 + 5433 37538 =

This is the final answer (65432 - 27894 = 37538).

Example 2 - the Simplified Procedure

A small change in writing the numbers simplifies this procedure significantly, especially during the learning process. [ [http://www.mathventures.com/mathed/Arithmetic/SubtractionWithoutBorrowingSimple.htm A descriptive example of the simplified procedure] (at [http://www.mathventures.com MathVentures] )]

*We start by writing the two numbers with a blank line between them:

65432 (minuend) - 23894 (subtrahend)

Since we notice that, in this case, each of the two most-significant digits of the minuend is larger than the corresponding digit of the subtrahend, we...

*Reduce by 1 the second most significant digit, not the first, and

*Change only the last three digits of the minuend to 9.

*Write the new minuend -- 64999 -- on the blank line under the original minuend: 65432 (original minuend) 64999 (new minuend) - 23894
*Note that the difference between the original minuend and the new one equals the number consisting of the digits that we changed to 9 plus 1: 432 + 1 = 433

*Now perform the revised subtraction (from the new minuend).

We continue to write the original minuend for reference only, because it is important to remember the original task; we no longer use it.

65432 (original minuend) 64999 (new minuend) - 23894 41105
*Finally add the difference, which we noted above, to the result 41105 + 433 41538 =

This is the final answer (65432 - 23894 = 41538).

After it is completed, the operation may appear on the worksheet like so: 65432 (original minuend) 64999 (new minuend) - 23894 41105 + 433 41538 =
* Note that the original minuend is not crossed. When kids are taught to cross text they often overdo it to a point the original text can no longer be read. It is important to know the original minuend and be able to reconstruct the original subtraction assignment.

Notes

#This is a simplified, informal algorithm. In some cases a different procedure is called for or simpler one can be used.
#This is a generalization of a common practice of many people, including early-grade students. For example, when facing a subtraction problem such as 1001 - 567, teachers have reported that some third-grade students had solved it by doing 999 - 567 + 2. Indeed, an ancient Hindu system of mathematics, known as Vedic mathematics includes this special case as one of its sixteen sutras (principles). [ [http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp#tutorial1 Vedic Maths, Tutorial 1] on [http://www.vedicmaths.org VedicMaths.org] ]

Footnotes and References