Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. 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 operations in mathematics.

Commutative Properties

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

$a+b=b+a$

We can better see this relationship when using real numbers.

$\left(-2\right)+7=5\text{ and }7+\left(-2\right)=5$

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

$a\cdot b=b\cdot a$

Again, consider an example with real numbers.

$\left(-11\right)\cdot\left(-4\right)=44\text{ and }\left(-4\right)\cdot\left(-11\right)=44$

It is important to note that neither subtraction nor division is commutative. For example, $17 - 5$ is not the same as $5 - 17$. Similarly, $20\div 5\ne 5\div 20$.

Associative Properties

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.

$a\left(bc\right)=\left(ab\right)c$

Consider this example.

$\left(3\cdot4\right)\cdot5=60\text{ and }3\cdot\left(4\cdot5\right)=60$

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

$a+\left(b+c\right)=\left(a+b\right)+c$

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

$[15+\left(-9\right)]+23=29\text{ and }15+[\left(-9\right)+23]=29$

Are subtraction and division associative? Review these examples.

$\begin{array}\text{ }8-\left(3-15\right) \hfill& \stackrel{?}{=}\left(8-3\right)-15 \\ 8-\left(-12\right) \hfill& =5-15 \\ 20 \hfill& \neq 20-10 \\ \text{ }\end{array}$
$\begin{array}\text{ }64\div\left(8\div4\right)\hfill&\stackrel{?}{=}\left(64\div8\right)\div4 \\ 64\div2 \hfill& \stackrel{?}{=}8\div4 \\ 32 \hfill& \neq 2\end{array}$

As we can see, neither subtraction nor division is associative.

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.

$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$

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

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, 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.

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

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

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

$a-b=a+\left(-b\right)$

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

$12+\left(-1\right)\cdot \left(5+3\right)$

Now, distribute $-1$ and simplify the result.

$\begin{array}12-\left(5+3\right) \hfill& =12+\left(-1\right)\cdot\left(5+3\right) \\ \hfill& =12+[\left(-1\right)\cdot5+\left(-1\right)\cdot3] \\ \hfill& =12+\left(-8\right) \\ \hfill& =4 \end{array}$

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. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

$\begin{array}12-\left(5+3\right) \hfill& =12+\left(-5-3\right) \\ \hfill& =12+\left(-8\right) \\ \hfill& =4\end{array}$

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.

$a+0=a$

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.

$a\cdot 1=a$

For example, we have $\left(-6\right)+0=-6$ and $23\cdot 1=23$. There are no exceptions for these properties; they work for every real number, including 0 and 1.

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, 0.

$a+\left(-a\right)=0$

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

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

$a\cdot \frac{1}{a}=1$

For example, if $a=-\frac{2}{3}$, the reciprocal, denoted $\frac{1}{a}$, is $-\frac{3}{2}$ because

$a\cdot \frac{1}{a}=\left(-\frac{2}{3}\right)\cdot \left(-\frac{3}{2}\right)=1$

A General Note: Properties of Real Numbers

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

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

$a+0=a$
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a

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

$a+\left(-a\right)=0$
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted $\frac{1}{a}$, such that

$a\cdot \left(\frac{1}{a}\right)=1$

Example 7: Using Properties of Real Numbers

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

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

Solution

1. $\begin{array}\text{ }3\cdot6+3\cdot4 \hfill& =3\cdot\left(6+4\right) \hfill& \text{Distributive property} \\ \hfill& =3\cdot10 \hfill& \text{Simplify} \\ \hfill& =30 \hfill& \text{Simplify}\end{array}$
2. $\begin{array}\text{ }\left(5+8\right)+\left(-8\right) \hfill& =5+\left[8+\left(-8\right)\right] \hfill& \text{Associative property of addition} \\ &\hfill =5+0 \hfill& \text{Inverse property of addition} \\ \hfill& =5 \hfill& \text{Identity property of addition}\end{array}$
3. $\begin{array}6-\left(15+9\right) \hfill& =6+[15\left(-15\right)+\left(-9\right)] \hfill& \text{Distributive property} \\ \hfill& =6+\left(-24\right) \hfill& \text{Simplify} \\ \hfill& =-18 \hfill& \text{Simplify}\end{array}$
4. $\begin{array}\text{ }\frac{4}{7}\cdot\left(\frac{2}{3}\cdot\frac{7}{4}\right) \hfill& =\frac{4}{7} \cdot\left(\frac{7}{4}\cdot\frac{2}{3}\right) \hfill& \text{Commutative property of multiplication} \\ \hfill& =\left(\frac{4}{7}\cdot\frac{7}{4}\right)\cdot\frac{2}{3}\hfill& \text{Associative property of multiplication} \\ \hfill& =1\cdot\frac{2}{3} \hfill& \text{Inverse property of multiplication} \\ \hfill& =\frac{2}{3} \hfill& \text{Identity property of multiplication}\end{array}$
5. $\begin{array}\text{ }100\cdot[0.75+\left(-2.38\right)] \hfill& =100\cdot0.75+100\cdot\left(-2.38\right)\hfill& \text{Distributive property} \\ \hfill& =75+\left(-238\right) \hfill& \text{Simplify} \\ \hfill& =-163 \hfill& \text{Simplify}\end{array}$

Try It 7

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

a. $\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right]$
b. $5\cdot \left(6.2+0.4\right)$
c. $18-\left(7 - 15\right)$
d. $\frac{17}{18}+\cdot \left[\frac{4}{9}+\left(-\frac{17}{18}\right)\right]$
e. $6\cdot \left(-3\right)+6\cdot 3$

Solution