For some activities we perform, the order of certain processes does not matter, but the order of others do. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for addition and multiplication.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

Caution! It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex]. Similarly, [latex]20\div 5\ne 5\div 20[/latex].

Associative Properties – Grouping

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Example

Use the distributive property to show that [latex]4\cdot[12+(-7)]=20[/latex]

Show Solution

Note that [latex]4[/latex] is outside the grouping symbols, so we distribute the  [latex]4[/latex] by multiplying it by [latex]12[/latex], multiplying it by [latex]–7[/latex], and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition.

The reverse is not true as we can see in this example.

[latex]\begin{array}{ccc}\hfill 6+\left(3\cdot 5\right)& \stackrel{?}{=}& \left(6+3\right)\cdot \left(6+5\right) \\ \hfill 6+\left(15\right)& \stackrel{?}{=}& \left(9\right)\cdot \left(11\right)\hfill \\ \hfill 21& \ne & \text{ }99\hfill \end{array}[/latex]

A special case of the distributive property occurs when a sum of terms is subtracted.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted a, that, when added to the original number, results in the additive identity, [latex]0[/latex].

For example, if [latex]a=-8[/latex], the additive inverse is [latex]8[/latex], since [latex]\left(-8\right)+8=0[/latex].

The inverse property of multiplication holds for all real numbers except [latex]0[/latex] because the reciprocal of [latex]0[/latex] is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\dfrac{1}{a}[/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[/latex].