Editing Subtraction (section)

===Integers===
[[File:Line Segment jaredwf.svg|left| ]]
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]]:
:''a'' + ''b'' = ''c''.

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

[[File:Subtraction line segment.svg|left| ]]
Now, a line segment labeled with the numbers {{num|1}}, {{num|2}}, and {{num|3}}. From position 3, it takes no steps to the left to stay at 3, so {{nowrap|1=3 − 0 = 3}}. It takes 2 steps to the left to get to position 1, so {{nowrap|1=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 number]]s, 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 {{nowrap|1=3 − 3 = 0}}. But {{nowrap|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:
:{{nowrap|1=3 − 4 = −1}}.