Subtraction
Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of
addition. Subtraction is denoted by a
minus sign in infix notation.
The traditional names for the parts of the formulaare
minuend −
subtrahend =
difference . The words "minuend" and "subtrahend" are virtually absent from modern usage; Linderholm charges "This terminology is of no use whatsoever." However, "difference" is very common.
Subtraction is used to model several closely related processes:
#From a given collection, take away a given number of objects.
Encyclopedia
Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of
addition. Subtraction is denoted by a
minus sign in infix notation.
The traditional names for the parts of the formula
- c − b = a
are
minuend −
subtrahend =
difference . The words "minuend" and "subtrahend" are virtually absent from modern usage; Linderholm charges "This terminology is of no use whatsoever." However, "difference" is very common.
Subtraction is used to model several closely related processes:
- From a given collection, take away 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 .
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 . From 3, it takes 4 steps to the left to get to −1, so
- 3 − 4 = −1.
Algorithms for Subtraction
There are various algorithms for subtraction, and they differ in their suitability for various applications. A number of methods are adapted to hand calculation and these are taught to children in elementary school. In particular, there is a method based on borrowing from the minuend popular in the USA, and a different method based on carrying to the subtrahend popular in Europe.
The most common hand method is used when making change and involves no actual subtraction, rather the change-maker counts forward.
The method of complements is used to subtract in digital computers.
See also
- Elementary arithmetic
- Decrement
- Negative and non-negative numbers
;Algorithms
- Method of complements
- Subtraction without borrowing
Notes and references
External links
Printable Worksheets: , , and
- at cut-the-knot
- selected from
