Editing Subtraction (section)

==Of integers and real numbers==
===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}}.

===Natural numbers===
Subtraction of [[natural numbers]] is not [[Closure (mathematics)|closed]]: the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches:
# Conclude that 26 cannot be subtracted from 11; subtraction becomes a [[partial function]].
# Give the answer as an [[integer]] representing a [[negative number]], so the result of subtracting 26 from 11 is −15.

===Real numbers===
The [[Field (mathematics)|field]] of real numbers can be defined specifying only two binary operations, addition and multiplication, together with [[unary operations]] yielding [[Additive inverse|additive]] and [[Multiplicative inverse|multiplicative]] inverses. The subtraction of a real number (the subtrahend) from another (the minuend) can then be defined as the addition of the minuend and the additive inverse of the subtrahend. For example, {{math|1=3 − ''π'' = 3 + (−''π'')}}. Alternatively, instead of requiring these unary operations, the binary operations of subtraction and [[Division (mathematics)|division]] can be taken as basic.