Subtraction without ‘’ Borrowing’’

Using the properties of Real Numbers to simplify the subtraction of whole numbers

The main purpose of this article is to use the properties of real numbers to provide another subtraction algorithm for whole numbers. For children who might have problems using the standard algorithm for subtraction of whole numbers, which involves

 ‘’borrowing’’ may find this method easier to cope with.

First, we will state the property of real numbers which allows us to develop another method of subtracting whole numbers without ‘’borrowing’’.

Subtraction Property of Real Numbers

Let a, b, and c be any real numbers. Then

a – b = a – b + c – c = (a – c) – (b – c) = (a + c) – (b + c)   …………………….. (1)

That is, adding or subtracting the same number c, from the minuend and the subtrahend does not affect the difference of two numbers.

Some definitions:

If a number b, is subtracted from a number a, to give a number c, then a, is called the minuend, b is called the subtrahend and c is called the difference. See diagram.

Algorithm:    A systematic and structured step-by-step procedure for solving  a problem is called an algorithm.

Note:

 If a – b = c, then b + c = a. For example, 43 – 17 = 26. Therefore 17 + 26 = 43

We will now use equations (1) to illustrate how to solve subtraction problems without using the standard algorithm of borrowing.

Examples:

1           48       2)    9111         3)       403      

-29              -999                  -218

           -------           --------                --------

     4)    124          5)    756

            - 45                -389

             -------             --------

 Solutions:

1         48               49

    - 29          =   - 30   (adding 1 to each number)

                          =     19    (subtracting 30 from 49)

2         9111           9112

-999       = -1000  (adding 1 to each number)

                         =    8112   (subtracting 1000 from 9112)

3             403              405

            -218        =    - 220    (adding 2 to each number)

                            =     485

                                -  300  (adding 80 to each number)

                            =     185   (subtracting 300 from 485)

4         123             128

-45       =     - 50   (adding 5 to each number)

              =    178

                   -100   (adding 50 to each number)

            =      78     (subtracting 100 from 178)

5             756              757

             - 389     =     -390   (adding 1 to each number)

                            =     767

                                 - 400  (adding 10 to each number)

                            =     367   (subtracting 400 from 767)