The Subtraction reference article from the English Wikipedia on 24-Jul-2004
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Subtraction

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In mathematics, subtraction is one of the four basic arithmetic operations. It is usually denoted by an infix minus sign.

The traditional names for the terms of the subtraction

cb = a
are difference (a), minuend (c) and subtrahend (b).

Basic subtraction

right
Imagine a straight line of length b painted on the ground with the left end labeled a and the right end labeled c.

Starting from position a, it takes you b steps to the right to reach position c. This movement to the right, called addition, can be stated as:

a + b = c

From position c, it takes you b steps to the left to get back to a. This movement to the left, called subtraction, can be stated as:
cb = a

right
Now, imagine a line labelled with the numbers 1, 2, and 3.

From position 3, it takes no steps to the left to stay at position 3, so

3 − 0 = 3

From position 3, it only takes 1 step to the left to get to position 2, so
3 − 1 = 2

From position 3, it takes you 2 steps to the left to get to position 1, so
3 − 2 = 1

What would happen if you continued the process by going 3 steps to the left of position 3? For our example, you would walk off the end of the line which is not allowed. So, for this operation to be valid, the line must be extended.

For subtraction of natural numbers, the line would have every natural number (0, 1, 2, 3, 4, ...) on it.

Using the natural number line, from position 3, it takes you 3 steps to the left to get to position 0, so

3 − 3 = 0

But, for natural numbers, 3 − 4 is invalid since it leaves the line. So, for this operation to be valid, the line must be extended.

Using the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …), from position 3, it takes you 4 steps to the left to get to position −1, so

3 − 4 = −1

See also