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One of the four basic arithmetic operations

Subtraction (which is signified by the minus sign ) is one of the four arithmetic operations along with addition, multiplication and division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are 5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the difference of 5 and 2 is 3; that is, 1=5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices. In a sense, subtraction is the inverse of addition. That is, 1=c = a − b if and only if 1=c + b = a. In words: the difference of two numbers is the number that gives the first one when added to the second one. Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is also not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because is the additive identity, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract algebra. In computability theory, considering subtraction is not well defined over natural numbers, operations between numbers are actually defined using "truncated subtraction" or monus.

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, a line segment labeled with the numbers , , and From position 3, it takes no steps to the left to stay at 3, so 1=3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 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 numbers, 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 1=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, ). This way, it takes 4 steps to the left from 3 to get to −1:
1=3 − 4 = −1.

Subtraction of natural numbers is not 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.

The field of real numbers can be defined specifying only two binary operations, addition and multiplication, together with unary operations yielding additive and 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, 1=3 − π = 3 + (−π). Alternatively, instead of requiring these unary operations, the binary operations of subtraction and division can be taken as basic.

Subtraction is anti commutative, meaning that if one reverses the terms in a difference left to right, the result is the negative of the original result. Symbolically, if a and b are any two numbers, then
a − b = −(b − a).

In the context of integers, subtraction of one also plays a special role: for any integer a, the integer (a − 1) is the largest integer less than a, also known as the predecessor of a.

When subtracting two numbers with units of measurement such as kilograms or pounds, they must have the same unit. In most cases, the difference will have the same unit as the original numbers.

Changes in percentages can be reported in at least two forms, percentage change and percentage point change. Percentage change represents the relative change between the two quantities as a percentage, while percentage point change is simply the number obtained by subtracting the two percentages. As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is = − = %, while the percentage point change is −10 percentage points.

Methods used to teach subtraction to elementary school vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as traditional mathematics, a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.

Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after William A. Brownell published a study—claiming that crutches were beneficial to students using this method. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country.

In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example: