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
Commutative Properties
The commutative property of addition states that numbers may be added in any order without affecting the sum.Example
Show that numbers may be added in any order without affecting the sum. [latex]\left(-2\right)+7=5[/latex]Answer: [latex-display]7+\left(-2\right)=5[/latex-display]
Example
Show that numbers may be multiplied in any order without affecting the product.[latex]\left(-11\right)\cdot\left(-4\right)=44[/latex]Answer: [latex]\left(-4\right)\cdot\left(-11\right)=44[/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.Example
Show that you can regroup numbers that are multiplied together and not affect the product.[latex]\left(3\cdot4\right)\cdot5=60[/latex]Answer: [latex-display]3\cdot\left(4\cdot5\right)=60[/latex-display]
Example
Show that regrouping addition does not affect the sum. [latex][15+\left(-9\right)]+23=29[/latex]Answer: [latex]15+[\left(-9\right)+23]=29[/latex]
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]Answer: 1) [latex]8-\left(3-15\right)\stackrel{?}{=}\left(8-3\right)-15[/latex] [latex-display]\,\,\,\,\,\,\,\,\,\,\,8-\left(-12\right)=5-15[/latex-display] [latex-display]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,20\neq-10[/latex-display] 2) [latex]64\div\left(8\div4\right)\stackrel{?}{=}\left(64\div8\right)\div4[/latex] [latex-display]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,64\div2\stackrel{?}{=}8\div4[/latex-display] [latex-display]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,32\neq 2[/latex-display] 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.Example
Use the distributive property to show that [latex]4\cdot[12+(-7)]=20[/latex]Answer: 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.
[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.Example
Rewrite the last example by changing the sign of each term and adding the results.Answer: [latex]\begin{array}{l}12-\left(5+3\right)=12+\left(-5-3\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=12+\left(-8\right) \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=4\end{array}[/latex]
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.Example
Show that the identity property of addition and multiplication are true for [latex]-6 \text{ and }23[/latex].
Answer:
[latex]\left(-6\right)+0=-6[/latex]
[latex]23+0=23[/latex]
[latex]-6\cdot1=-6[/latex]
[latex]23\cdot 1=23[/latex]
There are no exceptions for these properties; they work for every real number, including [latex]0[/latex] and [latex]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.
Answer:
1) The additive inverse is [latex]8[/latex], and [latex]\left(-8\right)+8=0[/latex]
2) The reciprocal is [latex]-\dfrac{3}{2}[/latex] and [latex]\left(-\dfrac{2}{3}\normalsize\right)\cdot \left(-\dfrac{3}{2}\normalsize\right)=1[/latex]
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.- [latex]3\left(6+4\right)[/latex]
- [latex]\left(5+8\right)+\left(-8\right)[/latex]
- [latex]6-\left(15+9\right)[/latex]
- [latex]\dfrac{4}{7}\normalsize\cdot \left(\dfrac{2}{3}\normalsize\cdot\dfrac{7}{4}\normalsize\right)[/latex]
- [latex]100\cdot \left[0.75+\left(-2.38\right)\right][/latex]
Answer:
- [latex]\begin{array}{l}\\\\3\cdot\left(6+4\right)=3\cdot6+3\cdot4\,\,\,\,\,\,\,\,\,\,\,\text{Distributive property} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=18+12\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=30\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify}\end{array}[/latex]
- [latex]\begin{array}{l}\\\\\left(5+8\right)+\left(-8\right)=5+\left[8+\left(-8\right)\right]\,\,\,\,\,\,\,\,\,\,\,\text{Associative property of addition} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5+0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Inverse property of addition} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Identity property of addition}\end{array}[/latex]
- [latex]\begin{array}{l}\\\\6-\left(15+9\right) \hfill& =6+[\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}[/latex]
- [latex]\begin{array}{l}\\\\\\\\\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}[/latex]
- [latex]\begin{array}{l}\\\\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}[/latex]
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Properties of Real Numbers. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Specific attribution
- College Algebra: Using Properties of Real Numbers. License: CC BY: Attribution.