Properties of Real Numbers

By the end of this section, you will be able to: use the commutative and associative properties; use the properties of identity, inverse, and zero; and simplify expressions using the distributive property.

Use the commutative and associative properties

The order we add two numbers doesn’t affect the result. If we add $8 + 9$ or $9 + 8$ , the results are the same — they both equal $17$ . So, $8 + 9 = 9 + 8$ . The order in which we add does not matter!

Similarly, when multiplying two numbers, the order does not affect the result. If we multiply $9 \cdot 8$ or $8 \cdot 9$ the results are the same — they both equal $72$ . So, $9 \cdot 8 = 8 \cdot 9$ . The order in which we multiply does not matter!

These examples illustrate the Commutative Property.

Commutative Property.

of Addition — If $a$ and $b$ are real numbers, then $a + b = b + a$ .

of Multiplication — If $a$ and $b$ are real numbers, then $a \cdot b = b \cdot a$ .

When adding or multiplying, changing the order gives the same result.

The Commutative Property has to do with order. We subtract $9 - 8$ and $8 - 9$ , and see that $9 - 8 \neq 8 - 9$ . Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.

Division is not commutative either. Since $12 \div 3 \neq 3 \div 12$ , changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!

Addition and multiplication are commutative.
Subtraction and division are not commutative.

When adding three numbers, changing the grouping of the numbers gives the same result. For example, $(7 + 8) + 2 = 7 + (8 + 2)$ , since each side of the equation equals $17$ .

This is true for multiplication, too. For example, $\left(5 \cdot \tfrac{1}{3}\right) \cdot 3 = 5 \cdot \left(\tfrac{1}{3} \cdot 3\right)$ , since each side of the equation equals $5$ .

These examples illustrate the Associative Property.

Associative Property.

of Addition — If $a$ , $b$ , and $c$ are real numbers, then $(a + b) + c = a + (b + c)$ .

of Multiplication — If $a$ , $b$ , and $c$ are real numbers, then $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ .

When adding or multiplying, changing the grouping gives the same result.

The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order — the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

\begin{array}{rcl} (10 - 3) - 2 &\neq& 10 - (3 - 2) \\[4pt] 7 - 2 &\neq& 10 - 1 \\[4pt] 5 &\neq& 9 \end{array} \qquad \begin{array}{rcl} (24 \div 4) \div 2 &\neq& 24 \div (4 \div 2) \\[4pt] 6 \div 2 &\neq& 24 \div 2 \\[4pt] 3 &\neq& 12 \end{array}

When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.

Example. Simplify: $18p + 6q + 15p + 5q$ .

\begin{array}{lrcl} \text{} && & 18p + 6q + 15p + 5q \\[4pt] \text{Reorder so like terms are together.} && & 18p + 15p + 6q + 5q \\[4pt] \text{Add like terms.} && & 33p + 11q \end{array}

Simplify: $23r + 14s + 9r + 15s$ .

Simplify: $37m + 21n + 4m - 15n$ .

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.

Example. Simplify: $\left(\tfrac{5}{13} + \tfrac{3}{4}\right) + \tfrac{1}{4}$ .

\begin{array}{lrcl} \text{} && & \left(\tfrac{5}{13} + \tfrac{3}{4}\right) + \tfrac{1}{4} \\[10pt] \text{The last 2 terms share a denominator, so regroup.} && & \tfrac{5}{13} + \left(\tfrac{3}{4} + \tfrac{1}{4}\right) \\[10pt] \text{Add in parentheses first.} && & \tfrac{5}{13} + \left(\tfrac{4}{4}\right) \\[10pt] \text{Simplify the fraction.} && & \tfrac{5}{13} + 1 \\[10pt] \text{Add.} && & 1\tfrac{5}{13} \\[10pt] \text{Convert to an improper fraction.} && & \tfrac{18}{13} \end{array}

Simplify: $\left(\tfrac{7}{15} + \tfrac{5}{8}\right) + \tfrac{3}{8}$ . Enter your answer as an improper fraction.

Simplify: $\left(\tfrac{2}{9} + \tfrac{7}{12}\right) + \tfrac{5}{12}$ . Enter your answer as an improper fraction.

Use the properties of identity, inverse, and zero

What happens when we add $0$ to any number? Adding $0$ doesn’t change the value. For this reason, we call $0$ the additive identity. The Identity Property of Addition states that for any real number $a$ , $a + 0 = a$ and $0 + a = a$ .

What happens when we multiply any number by one? Multiplying by $1$ doesn’t change the value. So we call $1$ the multiplicative identity. The Identity Property of Multiplication states that for any real number $a$ , $a \cdot 1 = a$ and $1 \cdot a = a$ .

We summarize the Identity Properties here.

Identity Property.

of Addition — For any real number $a$ : $a + 0 = a$ and $0 + a = a$ . $0$ is the additive identity.

of Multiplication — For any real number $a$ : $a \cdot 1 = a$ and $1 \cdot a = a$ . $1$ is the multiplicative identity.

What number added to $5$ gives the additive identity, $0$ ? We know

5 + (-5) = 0

The missing number was the opposite of the number!

We call $-a$ the additive inverse of $a$ . The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number $a$ , $a + (-a) = 0$ .

What number multiplied by $\tfrac{2}{3}$ gives the multiplicative identity, $1$ ? In other words, $\tfrac{2}{3}$ times what results in $1$ ? We know

\frac{2}{3} \cdot \frac{3}{2} = 1

The missing number was the reciprocal of the number!

We call $\tfrac{1}{a}$ the multiplicative inverse of $a$ . The reciprocal of a number is its multiplicative inverse. This leads to the Inverse Property of Multiplication that states that for any real number $a$ , $a \neq 0$ , $a \cdot \tfrac{1}{a} = 1$ .

We’ll formally state the inverse properties here.

Inverse Property.

of Addition — For any real number $a$ , $a + (-a) = 0$ . $-a$ is the additive inverse of $a$ . A number and its opposite add to zero.

of Multiplication — For any real number $a$ , $a \neq 0$ , $a \cdot \tfrac{1}{a} = 1$ . $\tfrac{1}{a}$ is the multiplicative inverse of $a$ . A number and its reciprocal multiply to one.

The Identity Property of addition says that when we add $0$ to any number, the result is that same number. What happens when we multiply a number by $0$ ? Multiplying by $0$ makes the product equal zero.

What about division involving zero? What is $0 \div 3$ ? Think about a real example: If there are no cookies in the cookie jar and $3$ people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets $0$ cookies. So, $0 \div 3 = 0$ .

We can check division with the related multiplication fact. So we know $0 \div 3 = 0$ because $0 \cdot 3 = 0$ .

Now think about dividing by zero. What is the result of dividing $4$ by $0$ ? Think about the related multiplication fact: $4 \div 0 = {?}$ means ${?} \cdot 0 = 4$ . Is there a number that multiplied by $0$ gives $4$ ? Since any real number multiplied by $0$ gives $0$ , there is no real number that can be multiplied by $0$ to obtain $4$ . We conclude that there is no answer to $4 \div 0$ and so we say that division by $0$ is undefined.

We summarize the properties of zero here.

Properties of Zero.

Multiplication by Zero: For any real number $a$ , $a \cdot 0 = 0$ and $0 \cdot a = 0$ — the product of any number and $0$ is $0$ .

Division by Zero: For any real number $a$ , $a \neq 0$ :

$\tfrac{0}{a} = 0$ — zero divided by any real number, except itself, is zero.
$\tfrac{a}{0}$ is undefined — division by zero is undefined.

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Example. Simplify: $-84n + (-73n) + 84n$ .

\begin{array}{lrcl} \text{} && & -84n + (-73n) + 84n \\[4pt] \text{The first and third terms are opposites; re-order.} && & -84n + 84n + (-73n) \\[4pt] \text{Add left to right.} && & 0 + (-73n) \\[4pt] \text{Add.} && & -73n \end{array}

Simplify: $-27a + (-48a) + 27a$ .

Simplify: $39x + (-92x) + (-39x)$ .

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals — their product is $1$ .

Example. Simplify: $\tfrac{7}{15} \cdot \tfrac{8}{23} \cdot \tfrac{15}{7}$ .

\begin{array}{lrcl} \text{} && & \tfrac{7}{15} \cdot \tfrac{8}{23} \cdot \tfrac{15}{7} \\[10pt] \text{The first and third terms are reciprocals; re-order.} && & \tfrac{7}{15} \cdot \tfrac{15}{7} \cdot \tfrac{8}{23} \\[10pt] \text{Multiply left to right.} && & 1 \cdot \tfrac{8}{23} \\[10pt] \text{Multiply.} && & \tfrac{8}{23} \end{array}

Simplify: $\tfrac{9}{16} \cdot \tfrac{5}{49} \cdot \tfrac{16}{9}$ .

Simplify: $\tfrac{6}{17} \cdot \tfrac{11}{25} \cdot \tfrac{17}{6}$ .

The next example makes us aware of the distinction between dividing $0$ by some number or some number being divided by $0$ .

Example. Simplify: (a) $\tfrac{0}{n + 5}$ , where $n \neq -5$ ; (b) $\tfrac{10 - 3p}{0}$ , where $10 - 3p \neq 0$ .

(a) Zero divided by any real number except itself is $0$ :

\frac{0}{n + 5} = 0

(b) Division by $0$ is undefined:

\frac{10 - 3p}{0} \text{ is undefined.}

Simplify: $\tfrac{0}{m + 7}$ , where $m \neq -7$ .

Simplify: $\tfrac{18 - 6c}{0}$ , where $18 - 6c \neq 0$ .

Simplify expressions using the distributive property

Suppose that three friends are going to the movies. They each need $9.25 — that’s $9$ dollars and $1$ quarter — to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need $3$ times $9 so $27 and $3$ times $1$ quarter, so $75$ cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.

Distributive Property.

If $a$ , $b$ , and $c$ are real numbers, then

\begin{array}{rcl} a(b + c) &=& ab + ac \\[4pt] (b + c)a &=& ba + ca \\[4pt] a(b - c) &=& ab - ac \\[4pt] (b - c)a &=& ba - ca \end{array}

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.

Example. Simplify: $3(x + 4)$ .

\begin{array}{lrcl} \text{} && & 3(x + 4) \\[4pt] \text{Distribute.} && & 3 \cdot x + 3 \cdot 4 \\[4pt] \text{Multiply.} && & 3x + 12 \end{array}

Simplify: $4(x + 2)$ .

Simplify: $6(x + 7)$ .

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the example above would look like this — an arrow from the $3$ to the $x$ and an arrow from the $3$ to the $4$ in $3(x + 4)$ .

Example. Simplify: $8\left(\tfrac{3}{8}x + \tfrac{1}{4}\right)$ .

\begin{array}{lrcl} \text{} && & 8\left(\tfrac{3}{8}x + \tfrac{1}{4}\right) \\[10pt] \text{Distribute.} && & 8 \cdot \tfrac{3}{8}x + 8 \cdot \tfrac{1}{4} \\[10pt] \text{Multiply.} && & 3x + 2 \end{array}

Simplify: $6\left(\tfrac{5}{6}y + \tfrac{1}{2}\right)$ .

Simplify: $12\left(\tfrac{1}{3}n + \tfrac{3}{4}\right)$ .

Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.

Example. Simplify: $100(0.3 + 0.25q)$ .

\begin{array}{lrcl} \text{} && & 100(0.3 + 0.25q) \\[4pt] \text{Distribute.} && & 100(0.3) + 100(0.25q) \\[4pt] \text{Multiply.} && & 30 + 25q \end{array}

Simplify: $100(0.7 + 0.15p)$ .

Simplify: $100(0.04 + 0.35d)$ .

When we distribute a negative number, we need to be extra careful to get the signs correct!

Example. Simplify: $-11(4 - 3a)$ .

\begin{array}{lrcl} \text{} && & -11(4 - 3a) \\[4pt] \text{Distribute.} && & -11 \cdot 4 - (-11) \cdot 3a \\[4pt] \text{Multiply.} && & -44 - (-33a) \\[4pt] \text{Simplify.} && & -44 + 33a \end{array}

Notice that you could also write the result as $33a - 44$ . Do you know why?

Simplify: $-5(2 - 3a)$ .

Simplify: $-7(8 - 15y)$ .

In the next example, we will show how to use the Distributive Property to find the opposite of an expression.

Example. Simplify: $-(y + 5)$ .

\begin{array}{lrcl} \text{} && & -(y + 5) \\[4pt] \text{Multiplying by } {-1} \text{ results in the opposite.} && & -1(y + 5) \\[4pt] \text{Distribute.} && & -1 \cdot y + (-1) \cdot 5 \\[4pt] \text{Simplify.} && & -y + (-5) \\[4pt] \text{Simplify.} && & -y - 5 \end{array}

Simplify: $-(z - 11)$ .

Simplify: $-(x - 4)$ .

There will be times when we’ll need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be to multiply using the Distributive Property, which removes the parentheses. The next two examples will illustrate this.

Example. Simplify: $8 - 2(x + 3)$ .

We follow the order of operations. Multiplication comes before subtraction, so we will distribute the $2$ first and then subtract.

\begin{array}{lrcl} \text{} && & 8 - 2(x + 3) \\[4pt] \text{Distribute.} && & 8 - 2 \cdot x - 2 \cdot 3 \\[4pt] \text{Multiply.} && & 8 - 2x - 6 \\[4pt] \text{Combine like terms.} && & -2x + 2 \end{array}

Simplify: $9 - 3(x + 2)$ .

Simplify: $7x - 5(x + 4)$ .

Example. Simplify: $4(x - 8) - (x + 3)$ .

\begin{array}{lrcl} \text{} && & 4(x - 8) - (x + 3) \\[4pt] \text{Distribute.} && & 4x - 32 - x - 3 \\[4pt] \text{Combine like terms.} && & 3x - 35 \end{array}

Simplify: $6(x - 9) - (x + 12)$ .

Simplify: $8(x - 1) - (x + 5)$ .

Key terms

commutative property — changing the order of two numbers when adding or multiplying gives the same result ( $a + b = b + a$ , $a \cdot b = b \cdot a$ ). associative property — changing the grouping of three numbers when adding or multiplying gives the same result ( $(a + b) + c = a + (b + c)$ , $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ ). additive identity — the number $0$ , which added to any number leaves it unchanged. multiplicative identity — the number $1$ , which multiplied by any number leaves it unchanged. additive inverse — the opposite $-a$ of a number $a$ ; a number and its opposite add to $0$ . multiplicative inverse — the reciprocal $\tfrac{1}{a}$ of a nonzero number $a$ ; a number and its reciprocal multiply to $1$ . distributive property — a rule for multiplying a sum or difference by a number: $a(b + c) = ab + ac$ . undefined — having no value; division by zero is undefined.

This section is adapted from Intermediate Algebra 2e, Section 1.5: Properties of Real Numbers by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: typeset the property boxes as callouts and the worked-example steps as aligned math; omitted the Be Prepared quiz, the summary table of all properties, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Decimals Solving Linear Equations