The traditional names for the parts of the formula
- c − b = a
are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage, while "difference" is very common.
Subtraction is used to model several closely related processes:
- From a given collection, take away (subtract) a given number of objects.
- Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal.
- Compare two objects to find their difference. For example, the difference between $800 and $600 is $800 − $600 = $200.
In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative— in fact, it is anticommutative— but addition of signed numbers is both.
Basic 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:
- 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.
Now, imagine a line segment labelled 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, ...). 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, …). From 3, it takes 4 steps to the left to get to −1, so
- 3 − 4 = −1.
- Subtraction Game at cut-the-knot
- Subtraction on a Japanese abacus selected from Abacus: Mystery of the Beadbr:Lamadur
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