Use Properties of Real Numbers

Learning Outcomes

  • Define and use the commutative property of addition and multiplication
  • Define and use the associative property of addition and multiplication
  • Define and use the distributive property
  • Define and use the identity property of addition and multiplication
  • Define and use the inverse property of addition and multiplication

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.

[latex]a+b=b+a[/latex]

We can better see this relationship when using real numbers.

Example

Show that numbers may be added in any order without affecting the sum. [latex]\left(-2\right)+7=5[/latex]

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

[latex]a\cdot b=b\cdot a[/latex]

Again, consider an example with real numbers.

Example

Show that numbers may be multiplied in any order without affecting the product.[latex]\left(-11\right)\cdot\left(-4\right)=44[/latex]

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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.

[latex]a\left(bc\right)=\left(ab\right)c[/latex]

Consider this example.

Example

Show that you can regroup numbers that are multiplied together and not affect the product.[latex]\left(3\cdot4\right)\cdot5=60[/latex]

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
[latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]

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

Example

Show that regrouping addition does not affect the sum. [latex][15+\left(-9\right)]+23=29[/latex]

Are subtraction and division associative? Review these examples.

Example

Use the associative property to explore whether subtraction and division are associative.

1) [latex]8-\left(3-15\right)\stackrel{?}{=}\left(8-3\right)-15[/latex]

2) [latex]64\div\left(8\div4\right)\stackrel{?}{=}\left(64\div8\right)\div4[/latex]

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.

[latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]

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]

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.

[latex]a-b=a+\left(-b\right)[/latex]

For example, consider the difference [latex]12-\left(5+3\right)[/latex]. We can rewrite the difference of the two terms [latex]12[/latex] and [latex]\left(5+3\right)[/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\left(5+3\right)[/latex], we add the opposite.

[latex]12+\left(-1\right)\cdot \left(5+3\right)[/latex]

Now, distribute [latex]-1[/latex] and simplify the result.

[latex]\begin{array}{l}12-\left(5+3\right)=12+\left(-1\right)\cdot\left(5+3\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=12+[\left(-1\right)\cdot5+\left(-1\right)\cdot3]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=12+\left(-8\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=4\end{array}[/latex]

Example

Rewrite the last example by changing the sign of each term and adding the results.

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.

[latex]a+0=a[/latex]

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.

[latex]a\cdot 1=a[/latex]

Example

Show that the identity property of addition and multiplication are true for [latex]-6 \text{ and }23[/latex].

Inverse Properties

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].

[latex]a+\left(-a\right)=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].

[latex]a\cdot\dfrac{1}{a}\normalsize =1[/latex]

Example

1) Define the additive inverse of [latex]a=-8[/latex], and use it to illustrate the inverse property of addition.

2) Write the reciprocal of [latex]a=-\dfrac{2}{3}[/latex], and use it to illustrate the inverse property of multiplication.

A General Note: Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

Addition Multiplication
Commutative Property [latex]a+b=b+a[/latex] [latex]a\cdot b=b\cdot a[/latex]
Associative Property [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex] [latex]a\left(bc\right)=\left(ab\right)c[/latex]
Distributive Property [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]
Identity Property There exists a unique real number called the additive identity, 0, such that, for any real number a

[latex]a+0=a[/latex]
 There exists a unique real number called the multiplicative identity, 1, such that, for any real number a

[latex]a\cdot 1=a[/latex]
Inverse Property Every real number a has an additive inverse, or opposite, denoted [latex]–a[/latex], such that

[latex]a+\left(-a\right)=0[/latex]
 Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted [latex]\dfrac{1}{a}[/latex], such that

[latex]a\cdot \left(\dfrac{1}{a}\normalsize\right)=1[/latex]

Example

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

  1. [latex]3\left(6+4\right)[/latex]
  2. [latex]\left(5+8\right)+\left(-8\right)[/latex]
  3. [latex]6-\left(15+9\right)[/latex]
  4. [latex]\dfrac{4}{7}\normalsize\cdot \left(\dfrac{2}{3}\normalsize\cdot\dfrac{7}{4}\normalsize\right)[/latex]
  5. [latex]100\cdot \left[0.75+\left(-2.38\right)\right][/latex]