1.6: Properties of Addition
Properties of Addition of Real Numbers
Objectives
The lesson objectives for The Properties Addition of Real Numbers are:
- The commutative property of addition
- The closure property of addition
- The associative property of addition
- The identity property of addition
- The inverse property of addition
Introduction
In this concept you will learn the properties of addition that apply to real numbers. These properties are the commutative property, the closure property, the associative property, the identity property and the inverse property. You will learn what is meant by each of these properties and how each one applies to the addition of real numbers. You will learn to apply the properties to addition, how to recognize the property as it is applied to addition and to name the property that justifies a given addition statement.
Guidance
On the first day of school, you are all dressed in your new clothes. When you got dressed, you put one sock on your left foot and one sock on your right foot. Would it have made a difference if you had put one sock on your right foot first and then one sock on your left foot? The order in which the process was done, did not affect the outcome – you have one sock on each foot.
In algebra, the operation of addition is commutative. The order in which you add two real numbers does not change the result. Try these problems yourself.
\begin{align*}(+7)+(+20)&= ? && (+20)+(+7)=?\\ (+7)+(+20)&={\color{blue}+27} && (+20)+(+7)={\color{blue}+27}\end{align*}
The order in which you added the numbers did not affect the answer.
Example A
Does \begin{align*}(5)+(-3)=(-3)+(5)\end{align*}?
Let’s use a number line to determine each result. You add a positive number by moving to the right on the number line and you add a negative number by moving to the left on the number line.
\begin{align*}(5)+(-3)=+2\end{align*} The red dot is placed at +5. Then the (-3) is added by moving three places to the left. The result is +2.
\begin{align*}(-3)+(5)=+2\end{align*} The blue dot is placed at -3. Then the (+5) is added by moving five places to the right. The result is +2.
The result was the same regardless of the order in which the addition was performed. This is called the commutative property of addition. In general, the commutative property of addition states that the order in which two numbers are added, does not affect the sum. If \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers, then \begin{align*}\boxed{a+b=b+a}\end{align*}.
Example B
Does \begin{align*}(-6)+(-2)=\end{align*} a real number?
\begin{align*}(-6)+(-2)=-8\end{align*} The red dot is placed at -6. Then the (-2) is added by moving two places to the left. The result is -8.
The result is -8. This is an integer. An integer is a real number. The sum of any two real numbers will result in a real number. This is known as the closure property of addition. The result will always be a real number. In general, the closure property states that the sum of any two real numbers is a unique real number. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}\boxed{a+b=c}\end{align*}.
Example C
Does \begin{align*}(-4+7)+5=-4+(7+5)\end{align*}?
\begin{align*}(-4+7)+5=\end{align*} The red dot is placed at -4. Then the (+7) is added by moving seven places to the left. Then (+5) is added by moving five places to the right. The result is +8.
\begin{align*}-4+(7+5)=\end{align*} The blue dot is placed at +7. Then the (+5) is added by moving five places to the left. Then (-4) is added by moving four places to the left. The result is +8.
\begin{align*}(-4+7)+5=-4+(7+5)\end{align*}
The numbers in the problem were the same but on the left side of the equal sign, the numbers -4 and +7 were grouped in parenthesis. The addition in the parenthesis was completed first and then +5 was added to that sum. The final result was +8.
On the right side of the equal sign, the numbers +7 and +5 were grouped in parenthesis. The addition in the parenthesis was completed first and then (-4) was added to the sum. The final result was +8.
The order in which three or more real numbers are grouped for addition, will not affect the sum. This is known as the associative property of addition. The result will always be the same real number. In general, the associative property states that order in which the numbers are grouped for addition does not change the sum. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}\boxed{(a+b)+c=a+(b+c)}\end{align*}.
Example D
Does \begin{align*}(-5)+0=-5\end{align*}?
\begin{align*}(-5)+0=-5\end{align*} The red dot is placed at -5. If zero is being added to the number, there is no movement to the right and no movement to the left. Therefore the result is -5.
When zero was added to the number -5, the number did not change. If zero is added to any real number the answer is always the real number. Zero is known as the additive identity or the identity element of addition. The sum of a number and zero is the number. This is called the identity property of addition. If \begin{align*}a\end{align*} is a real number, then \begin{align*}\boxed{a+0=a}\end{align*}.
Example E
Does \begin{align*}(+6)+(-6)=0\end{align*}?
\begin{align*}(+6)+(-6)=0\end{align*} The red dot is placed at +6. Then the (-6) is added by moving six places to the left. The result is 0.
When any real number is added to its opposite, the result is always zero. If \begin{align*}a\end{align*} is any real number, its opposite is \begin{align*}-a\end{align*}. The opposite, \begin{align*}-a\end{align*}, is also known as the additive inverse of \begin{align*}a\end{align*}.
The sum of any real number and its additive inverse is zero. You learned in example D that zero is also called the identity element of addition. Therefore, it can be said that the sum of any real number and its additive inverse is the identity element. This is called the inverse property of addition. If \begin{align*}a\end{align*} is a real number, then \begin{align*}\boxed{a+(-a)=0}\end{align*}.
Vocabulary
- Additive Identity
- The additive identity for addition of real numbers is zero.
- Additive Inverse
- The additive inverse of addition is the opposite of the real number and the sum of the real number and its additive inverse is zero. If \begin{align*}a\end{align*} is any real number, its additive inverse is \begin{align*}-a\end{align*}.
- Associative Property
- The associative property of addition states the order in which three or more real numbers are grouped for addition, will not affect the sum. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}\boxed{(a+b)+c=a+(b+c)}\end{align*}.
- Closure Property
- The closure property of addition states that the sum of any two real numbers is a unique real number. If \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are real numbers, then \begin{align*}\boxed{a+b=c}\end{align*}.
- Commutative Property
- The commutative property of addition states that the order in which two numbers are added, does not affect the sum. If \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers, then \begin{align*}\boxed{a+b=b+a}\end{align*}.
- Identity Element of Addition
- The identity element of addition is another term for the additive identity of addition. Therefore, the identity element of addition is zero.
- Identity Property
- The identity property of addition states that the sum of a number and zero is the number. If \begin{align*}a\end{align*} is a real number, then \begin{align*}\boxed{a+0=a}\end{align*}.
- Inverse Property
- The inverse property of addition states that the sum of any real number and its additive inverse is zero. If \begin{align*}a\end{align*} is a real number, then \begin{align*}\boxed{a+(-a)=0}\end{align*}.
Guided Practice
1. Add using the properties of addition: \begin{align*}-1.6+4.2+1.6\end{align*}
2. What property justifies the statement? \begin{align*}(-21+6)+8=-21+(6+8)\end{align*}
3. Apply the commutative property of addition to the following problem. \begin{align*}17x-15y\end{align*}
Answers
1. \begin{align*}& -1.6+4.2+1.6\\ & -1.6+1.6+4.2 \rightarrow {\color{blue}\text{Communicative Property}}\\ & (-1.6+1.6)+4.2 \rightarrow {\color{blue}\text{Associative Property}}\\ & 0+4.2 \rightarrow {\color{blue}\text{Inverse Property}}\\ & =4.2 \rightarrow {\color{blue}\text{Identity Property}}\end{align*}
2. \begin{align*}(-21+6)+8=-21+(6+8)\end{align*}
The numbers on each side of the equal sign are the same but they are not grouped the same.
\begin{align*}&(-21+6)+8 && -21+(6+8)\\ &=(-15)+8 && =-21+(14)\\ &=-7 && =-7\end{align*}
The order in which the numbers were grouped did not affect the answer. The property that is being used is the associative property of addition.
3. \begin{align*}& 17x-15y\\ & 17x+(-15y) \ \mathbf{Rewrite \ the \ statement \ as \ an \ addition \ statement.}\\ & 17x+(-15y)=-15y+17x\end{align*}
The commutative property of addition states that the order in which two numbers are added, does not affect the sum.
Summary
The properties of addition of real numbers do not apply to subtraction or division. All of the properties could be represented on a number line and the results paralleled with the property being presented. You learned that the order in which two numbers were added did not change the sum. This was the commutative property of addition. An extension to this property was the associative property which stated that the order in which three or more numbers were grouped had not affect on the result. The closure property simply stated that the sum of any two real numbers is a unique real number. The term unique means that the sum is a different real number than those being added. The identity property stated the sum of any real number and zero is the real number. You also learned that zero is the additive identity or the identity element of addition. The final property that you learned was the inverse property. This property stated that the sum of any real number and its opposite is always zero.
The guided examples showed you how to apply the properties to a given addition statement. You were also shown how to use the properties to justify an addition statement.
Problem Set
Match the following addition statements with the correct property of addition.
A. \begin{align*}(-5)+5=0\end{align*}
B. \begin{align*}(-16+4)+5=-16+(4+5)\end{align*}
C. \begin{align*}-9+(-7)=-16\end{align*}
D. \begin{align*}45+0=45\end{align*}
E. \begin{align*}9+(-6)=(-6)+9\end{align*}
a) Commutative Property
b) Closure Property
c) Inverse Property
d) Identity Property
e) Associative Property
Add the following using the properties of addition:
- \begin{align*}24+(-18)+12\end{align*}
- \begin{align*}-21+34+21\end{align*}
- \begin{align*}5+\left(-\frac{2}{5}\right)+\left(-\frac{3}{5}\right)\end{align*}
- \begin{align*}19+(-7)+9\end{align*}
- \begin{align*}8+\frac{3}{7}+\left(-\frac{3}{7}\right)\end{align*}
Name the property of addition that is being shown in each of the following addition statements:
- \begin{align*}(-12+7)+10=-12+(7+10)\end{align*}
- \begin{align*}-18+0=-18\end{align*}
- \begin{align*}16.5+18.4=18.4+16.5\end{align*}
- \begin{align*}52+(-75)=-23\end{align*}
- \begin{align*}(-26)+(26)=0\end{align*}
Answers
Match the following...
A. \begin{align*}(-5)+5=0\end{align*} \begin{align*}\rightarrow\end{align*} c) Inverse Property
B. \begin{align*}(-16+4)+5=-16+(4+5)\end{align*} \begin{align*}\rightarrow\end{align*} e) Associative Property
C. \begin{align*}-9+(-7)=-16\end{align*} \begin{align*}\rightarrow\end{align*} b) Closure Property
D. \begin{align*}45+0=45\end{align*} \begin{align*}\rightarrow\end{align*} d) Identity Property
E. \begin{align*}9+(-6)=(-6)+9\end{align*} \begin{align*}\rightarrow\end{align*} a) Commutative Property
Add the following...
- \begin{align*}24+(-18)+12\end{align*} Use the Associative Property \begin{align*}(24+12)-18\end{align*} \begin{align*}36-18\end{align*} Write the statement as an addition statement \begin{align*}36+(-18)\end{align*} Use the closure property \begin{align*}=+18\end{align*}
- \begin{align*}5+\left(-\frac{2}{5}\right)+\left(-\frac{3}{5}\right)\end{align*} Use the Associative Property \begin{align*}5+\left(-\frac{2}{5}-\frac{3}{5}\right)\end{align*} Add the fractions in the parenthesis \begin{align*}5+\left(-\frac{5}{5}\right)\end{align*} \begin{align*}5+(-1)\end{align*} Use the Closure Property \begin{align*}=+4\end{align*}
- \begin{align*}8+\frac{3}{7}+\left(-\frac{3}{7}\right)\end{align*} Use the Associative Property \begin{align*}8+\left(\frac{3}{7}+\left(-\frac{3}{7}\right)\right)\end{align*} Use the Inverse Property \begin{align*}8+0\end{align*} Use the Identity Property \begin{align*}=+8\end{align*}
Name the property...
- \begin{align*}(-12+7)+10=-12+(7+10)\end{align*} Associative Property
- \begin{align*}16.5+18.4=18.4+16.5\end{align*} Commutative Property
- \begin{align*}(-26)+(26)=0\end{align*} Inverse Property