— the Art of Reasoning
— the Art of Studying Patterns and Relationships Using Logic
Subtraction Without Borrowing
Uri Geva, MathVentures,
a Division of Ten Ninety
Document Version 2.0
Copyright © 2002–06 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
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
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.
It's that simple!
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
You can download the description
of subtraction without borrowing in one or both of the following
above example is displayed as an image, which you can save to your computer.
Without Borrowing - Light
This document contains the information
described above with some graphics that is aimed at making it easier to
understand the examples.
Without Borrowing - with Detailed Mathematical Background
This document contains a mathematical
discussion and detailed explanation how this procedure works. It also includes
© 1993-2008 Ten Ninety, All Rights Reserved
Last Update: Sep. 1, 2008