Subtraction

Subtraction

Figure: Subtraction of numbers 0–10. Line labels = minuend. X axis = subtrahend. Y axis = difference.Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). For example, if there are 5 apples and if 2 are taken away, we will have a total of 3 apples left. Therefore, 5 − 2 = 3. Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 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.  

Video: Subtracting Integers

Notation and terminology

Subtraction is written using the minus sign "−" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,

 (verbally, "two minus one equals one")
 (verbally, "four minus two equals two")
 (verbally, "six minus three equals three")
 (verbally, "four minus six equals negative two")

There are also situations where subtraction is "understood" even though no symbol appears:

  • 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. This is most common in accounting.

Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. The result is the difference.

Properties
Anticommutativity

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. Should the expression

......

 

 

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