Subtraction without borrowing

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Contents 1 Background 2 Algorithm 3 Descriptive Example 4 Another Example 5 Notes 6 Footnotes 7 References

[edit] Background

A common subtraction method is known as dynamic subtraction¹. To perform a - b, using this method, the subtrahend b is written below the minuend a 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 ten² from the digit of the minuend that is immediately to the left of the current digit.³ Then the value of the “lending” digit is reduced by 1. Anyone who performs subtraction with borrowing must do so from right to left, starting at the lowest-value digits and proceed toward the highest-value digits. It is also necessary to remember to reduce the value of each “lending” digit by 1 when subtracting the corresponding digit of the subtrahend from it. Because of these requirements, learning subtraction with borrowing is difficult for some students.^[1]

An alternative to dynamic subtraction with borrowing is dynamic subtraction without borrowing or simply 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 suppose 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.

[edit] Algorithm

The basic algorithm is:

1. Change the minuend:
   1. Subtract 1 from the rightmost consecutive digit that is larger than the corresponding digit of the subtrahend;
   2. Change each digit to the right of the digit you reduced by 1 (Step #1) to 9;
2. Remember the difference is the number you changed to a series of 9's plus 1;
3. Carry out the subtraction using the new minuend;
4. Add to the result the difference you got in step #2. This is the final answer.

[edit] Descriptive Example

Described here, subtraction without borrowing employs dynamic addition. Description by example:

         65432      (minuend)
       - 27894      (subtrahend)

Change the minuend to 59999.
The difference between the original minuend and this 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

         59999
       - 27894
         32105

Now add the result to the difference

         32105
        + 5433
         37538
         =====

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

[edit] Another Example

         65432      (minuend)
       - 23894      (subtrahend)

Change the minuend to 64999.
The difference between the original minuend and this one is 433. Now perform the revised subtraction

         64999
       - 23894
         41105

Now add the result to the difference

         41105
        +  433
         41538
         =====

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

[edit] Notes

1. 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). ^[2]

[edit] Footnotes

¹ Only a brief, informal discussion is included here until a formal entry of dynamic subtraction is provided.

² This brief discussion of dynamic subtraction 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.

³ Also this discussion of dynamic subtraction does not address more complex situations such as when the “lending” digit is 0 (zero) and, therefore, nothing can be borrowed from it.

[edit] References

1. ^ Subtraction Without Borrowing on MathVentures
2. ^ Vedic Maths, Tutorial 1 on VedicMaths.org