Subtraction
Integers

Imagine a line segment of length b with the left end labeled a and the right end labeled c.Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition:

From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:

Now, a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.

To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6,...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction.

The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3,...). This way, it takes 4 steps to the left from 3 to get to −1:
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