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Logic — the Art of Reasoning

Mathematics — the Art of Studying Patterns Using Logic

Subtraction Without Borrowing — Just Do It!

Uri Geva, MathVentures, a Division of Ten Ninety

Document Version 1.20 (Just Do-It)
Copyright © 2002–04 Uri Geva, Ten Ninety, All Rights Reserved.
This work may be redistributed under the terms of the GNU General Public License
as published by the Free Software Foundation.
This work comes with absolutely no warranty!

Its Simple and Easy

Subtracting without borrowing is very simple and easy. (It does not use either compliment arithmatic or negative numbers.) It can easily be adopted to be your everyday subtracting method. You can learn how to do it by simply following the two examples described below.


A note for the mathematically advanced reader (If you do not think of yourself to be one, skip this note): There are other methods for performing subtraction without borrowing. Some involve the use of negative numbers and others use methods of complements, either 9-Complement or 10-Complement. The technique described here has several advantages over these methods.

Advantages:

This method is very simple.
It requires no more than the mathematical knowledge of a second or third grader. No need to know special concepts as negative numbers and complements.
Often students, who have to perform subtraction, intuitively take a short-cut – they round at least one of the numbers up or down to a number with which they feel more comfortable. This method is nothing more than a general and algorithmic description of this alternative approach.

Examples:
Editorial Note:The text below identifies the numbers involved in a subtraction      operation using the following names:
    9 <==the minuend
– 4 <==the subtrahend
   5 <==the result Notice: For best result viewing the following examples, maximize your browser's window to avoid line wrap.


StepNo.	Action Description	The Numbers' Names	Example 1	Example 2
1)	Example 1. Ordinarily, borrowing is needed right away and in everycolumn except for the last, leftmost one.	The Minuend ==> The Subtrahend ==>	`87,352` `– 49,684`
	Example 2. Ordinarily, borrowing is needed only starting with the third column from the right.	The Minuend ==> The Subtrahend ==>		`675,732` `– 75,921`
2)	Change every digit except for the leftmost one of the Minuend to 9.	The Minuend ==>	*`89,999`*	*`699,999`*
3)	Subtract 1 from the leftmost digit of the minuend in Step #2.	The Minuend ==>	*`79,999`*	*`599,999`*
4)	*Add 1 to the part of the original minuend that we changed to 9's* in Step #2. This sum equals to... the difference between the original** minuend and the result of Step #3.	The changed part of the Minuend ==> *1==>* The Sum ==> Original Minuend ==> The Minuend==> The Difference ==>	*`7,352`* *`+ 1`* *`7,353`* *`87,352`* *`– 79,999`* *`7,353`*	*`75,732`* *`+ 1`* *`75,733`* *`675,732`* *`– 599,999`* *`75,733`*
5)	Remember this number	Remembered Number ==>	*`7,353`*	*`75,733`*
6)	Write the new subtraction, using the new minuend	New Minuend ==> The Subtrahend ==>	*`79,999`* *`– 49,684`*	*`599,999`* *`– 75,921`*
7)	Now the subtraction is simple. Do it in each column in any order you wish.	New Minuend ==> The Subtrahend ==> T`emp Result` ==>	*`79,999`* *`– 49,684`* *`30,315`*	*`599,999`* *`– 75,921`* *`524,078`*
8)	Add to the Temp Result of Step 7 the Remember Number in Step 5.	Remembered Number ==> *Final Result ==>*	*`+ 7,353`* *`37,668`*	*`+ 75,733`* *`599,811`*

That’s it! If it seems complex, it is only because a written description must be explicit and detailed. When Demonstrating it to students in person, they get it right away. [We plan to post here an animated demonstration with graphic and voice to simulate such personal explanation.]. Practice it a few times and you’ll realize how simple and easy it is.

The Trick… If You Want To Understand How It Works

The basic idea is simple too.

Borrowing in necessary because one or more digits in the subtrahend are larger than the corresponding digit in the minuend. Only the leftmost digit of the minuend is larger than the corresponding digit in the subtrahend.

So what we do is this:

Replace all of the digit of the minuend, except this leftmost one, by 9 and reduce the leftmost one by 1. This guarantees us a temporary

minuend of which every digit is equals to or larger than the corresponding digit in the subtrahend.

Now subtraction is very simple.

But we must add to the result the difference between the original minuend and the minuend that we used in the subtraction. Because we used only 9’s in the temporary minuend, this difference equals to the number that is made from the digits that we changed plus 1. So even here we do not need to do any subtraction.

See, the summary of the procedure is quite simple.

Following this subtraction procedure you do

one simple single-digit subtraction, one operation for each column of digits, and
one addition.

instead of subtraction with borrowing.

Now that you've seen it, make your own exercises and practice. You'll master it in no time.

Downloads

You can download the description of subtraction without borrowing in one or both of the following documents.

Subtraction Without Borrowing - Light

This document contains the information described above with some graphics that is aimed at making it easier to understand the examples.

Subtraction Without Borrowing - with Detailed Mathematical Background

This document contains a mathematical discussion and detailed explanation how this procedure works. It also includes more examples.

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