Subtraction

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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 (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage; Linderholm charges, "This terminology is of no use whatsoever."^[1] However, "difference" is very common.

Subtraction is used to model several closely related processes:

1. From a given collection, take away (subtract) a given number of objects.
2. Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal.
3. 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.

Contents 1 Basic subtraction: integers 2 Algorithms for subtraction 3 See also 4 Notes and references 5 External links

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.

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. In traditional mathematics, these were taught to children in elementary school for use with multi-digit numbers, starting in the 2nd or last 1st year, and the fourth or fifth grade for decimals.

A method based on borrowing from the minuend popular in the USA.

100
 11
---

0 is smaller than 1, so we borrow 10 from the 100

 10
 90
 11
---
  9

10 minus 1 gives 9. In the tens, we now have 8 minus 1, giving 7

 90
 11
---
 79

A different method giving the correct result of 89 and based on carrying to the subtrahend is popular in Europe.

The most common hand method is used when making change and involves no actual subtraction, but rather the change-maker counts forward.

The method of complements is used to subtract in digital computers.

Fractions may be subtracted traditionally by converting to a least common denominator.

Such standard methods, including borrowing, are often omitted from standards-based mathematics curricula in the belief that manual computation fosters failure and is irrelevant in the age of calculators, and time is better spent studying statistics, cutting, pasting, and learning how to write about mathematics. In texts such as TERC, students are encouraged to invent their own methods of computation. For example, one study showed that a student introduced errors when using the traditional borrowing method, but another 2nd grade student used his understanding of the properties of negative numbers to achieve the correct result. The use of reform mathematics, however is currently an issue of debate, as education bureacrats, and education theorists have tended to support reform, while parents and mathematical professsionals have often opposed such adoptions in the math wars.

See also

• Elementary arithmetic
• Decrement
• Negative and non-negative numbers

Algorithms

• Method of complements
• Subtraction without borrowing

Notes and references

• Linderholm, Carl (1971). Mathematics made difficult. Wolfe. ISBN 0-7234-0415-1. 

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