Commutative Property

(noun)

States that changing the order of numbers being added does not change the result.

Related Terms

Examples of Commutative Property 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:
    • The associative property describes equations in which the grouping of the numbers involved does not affect the result.
    • As with the commutative property, addition and multiplication are associative operations:
    • The distributive property can be used when the sum of two quantities is then multiplied by a third quantity.

  • The Dot Product

    • The dot product is a commutative property, which means that the order of the terms does not change the outcome: $\vec a \cdot \vec b = \vec b \cdot \vec a$
    • The dot product is a distributive property: $\vec a \cdot ( \vec b+ \vec c ) = \vec a \cdot \vec b + \vec a \cdot \vec c$
    • Formulate properties of the dot product, including the algebraic and geometric methods used to calculate it

  • Adding and Subtracting Polynomials

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

  • Writing Chemical Equations

    • Also, please note that, as in the mathematical commutative property of addition, chemical equations are commutative.

  • Multiplying Polynomials

    • Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties.
    • Recall that the distributive property says that
    • for all real numbers $a,b$ and $c.$ The associative property says that
    • 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 $M_n(x)$.
    • Explain how to multiply polynomials using the distributive property and describe the results of doing so

  • The Identity Matrix

    • The matrix that has this property is referred to as the identity matrix.
    • The identity matrix, designated as $[I]$, is defined by the property:
    • This stipulation is important because, for most matrices, multiplication does not commute.
    • What matrix has this property?
    • There is no identity for a non-square matrix because of the requirement of matrices being commutative.

  • Introduction to Variables

    • Variables may describe some mathematical properties.
    • 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 $\displaystyle (a+b)=(b+a)$.

  • Addition, Subtraction, and Multiplication

    • The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit.
    • = $ac + bidi + bci + adi$ (by the commutative law of addition)
    • = $ac + bdi^2 + (bc + ad)i$ (by the commutative law of multiplication)
    • = $(ac - bd) + (bc + ad)i$ (by the fundamental property of the imaginary unit)

  • The Growth of Suburbs

    • Suburbs first emerged on a large scale in the 19th and 20th centuries as a result of improved rail and road transport, which led to an increase in commuting. 
    • The streetcar lines in Boston and the rail lines into Manhattan made daily commutes possible.
    • No metropolitan area in the world was as well served by railroad commuter lines at the turn of the twentieth century as New York, and it was the rail lines to Westchester from the Grand Central Terminal commuter hub that enabled its development.
    • Because these properties were summarily declared to be "in decline," families were given pittances for their properties, and were forced into federal housing called "the projects."

  • Optional Collaborative Classrom Exercise

    • Nineteen people were asked how many miles, to the nearest mile they commute to work each day.The data are as follows:
    • True or False: Three percent of the people surveyed commute 3 miles.If the statement is not correct, what should it be?
    • What fraction of the people surveyed commute 5 or 7 miles?
    • What fraction of the people surveyed commute 12 miles or more?