commutative

(adjective)

Referring to a binary operation in which changing the order of the operands does not change the result (e.g., addition and multiplication).

Related Terms

(adjective)

A binary operation is commutative if changing the order of the operands does not change the result, for example addition and multiplication.

Related Terms

Examples of commutative in the following topics:

  • Basic Operations

    • The commutative property describes equations in which the order of the numbers involved does not affect the result.
    • Addition and multiplication are commutative operations:
    • As with the commutative property, addition and multiplication are associative operations:

  • The Identity Matrix

    • Note that the definition of [I][I] stipulates that the multiplication must commute, that is, it must yield the same answer no matter in which order multiplication is done.
    • This stipulation is important because, for most matrices, multiplication does not commute.
    • There is no identity for a non-square matrix because of the requirement of matrices being commutative.

  • Addition, Subtraction, and Multiplication

    • = $ac + bidi + bci + adi$ (by the commutative law of addition)
    • = $ac + bdi^2 + (bc + ad)i$ (by the commutative law of multiplication)

  • Inverses of Composite Functions

    • The functions $g$ and $f$ are said to commute with each other if $g ∘ f = f ∘ g$.
    • In general, the composition of functions will not be commutative.

  • Introduction to Variables

    • For example, a basic property of addition is commutativity, which states that the order of numbers being added together does not matter.
    • Commutativity is stated algebraically as $\displaystyle (a+b)=(b+a)$.

  • Adding and Subtracting Polynomials

    • When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together.

  • Negative Numbers

    • The basic properties of addition (commutative, associative, and distributive) also apply to negative numbers.

  • Multiplying Polynomials

    • For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing $M_1(x)$ and end with the terms containing $M_n(x)$.

  • Addition and Subtraction; Scalar Multiplication

    • Matrix addition is commutative and is also associative, so the following is true:

  • Adding and Subtracting Algebraic Expressions

    • The Commutative Property of Addition says that we can change the order of the terms without changing the sum.