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, 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, c = a − b if and only if 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. Subtraction is anti-commutative, 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 0 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.
Notation and terminology Subtraction is usually written using the minus sign "−" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example, 2 − 1 = 1 {\displaystyle 2-1=1} (pronounced as "two minus one equals one") and 4 − 6 = − 2 {\displaystyle 4-6=-2} (pronounced as "four minus six equals negative two"). Nonetheless, there are some situations where subtraction is understood, even though no symbol appears; in accounting, a column of two numbers, with the lower number in red, usually indicates that the lower number in the column is to be subtracted, with the difference written below, under a line. The number being subtracted is the subtrahend, while the number it is subtracted from is the minuend. The result is the difference, that is:
m i n u e n d − s u b t r a h e n d = d i f f e r e n c e . {\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}.}
All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, which in turn is a compound of sub "from under" and trahere "to pull". Thus, to subtract is to draw from below, or to take away. Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted". Likewise, from minuere "to reduce or diminish", one gets "minuend", which means "thing to be diminished".
Of integers and real numbers
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, a line segment labeled 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, 5, 6, ...). 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, ...). This way, it takes 4 steps to the left from 3 to get to −1:
3 − 4 = −1.
Natural numbers 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.
Real numbers 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, 3 − π = 3 + (−π). Alternatively, instead of requiring these unary operations, the binary operations of subtraction and division can be taken as basic.
Properties
Anti-commutativity 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).
Non-associativity Subtraction is non-associative, which comes up when one tries to define repeated subtraction. In general, the expression
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