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# Properties of Addition

Addition has several properties that always hold for all real numbers. Some textbooks only list a few of them, while others note them all. Furthermore, some textbooks might name the properties of addition differently, even if the underlying principles are the same. Here is a closer look at each:

## The properties of addition: identity property

The identity property states that there is a unique real number called the additive identity, such that if you add it to a number, you get that same number back.

$a+0=a$ and $0+a=a$

For example, $5+0$ is still 5. The number zero is considered the identity element of addition for this reason.

## The properties of addition: commutative property

The commutative property states that for all real numbers the order in which you add them does not affect the outcome:

$a+b=b+a$

For instance, $3+5=8$ and $5+3=8$ . It doesn't matter which order the numbers are in.

## The properties of addition: associative property

The Associative Property of Addition states that for all real numbers a, b, and c:

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

For example, $\left(2+3\right)+4=9$ . If we instead approach the problem as. $\left(2+\left(3+4\right)\right)$ , we still get the same answer of 9. When you add real numbers, the grouping (or association) of the numbers doesn't change the result.

## The properties of addition: property of opposites

The Property of Opposites states that for all real numbers, there is a unique real number such that:

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

For instance, $6+\left(-6\right)=0$ and $\left(-6\right)+6=0$ . A number and its opposite are called additive inverses of each other because their sum is zero.

## The properties of addition: property of the opposite of a sum

The Property of the Opposite of a Sum states that for all real numbers a and b:

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

For example, $-\left(2+3\right)$ is -5. If we instead added -2 and -3, the answer would still be -5. The opposite of a sum of real numbers is equal to the sum of the opposites.

## Properties of addition practice problems

a. Which property of addition tells you that $6+3=3+6$ ?

Commutative property

b. Which property of addition tells you that $100+0=100$ ?

Identity property

c. Does it matter if you add $\left(8+4\right)+6$ or $8+\left(4+6\right)$ ? Why or why not?

No, because of the Associative Property

d. Which property of addition tells you that $7+\left(-7\right)=0$ ?

Property of Opposites

e. Which property of addition states that the opposite of a sum of real numbers is equivalent to the sum of the opposites?

Property of the Opposite of a Sum

## Properties of addition help is available with Varsity Tutors

Many of the properties of addition have analogs in other operations and are revisited when students begin studying algebra, so students need to grasp them early on. If the student in your life is struggling with rote memorization of which property means what or applying them in context, a private math tutor can provide additional practice problems until they feel more comfortable. Contact the friendly Educational Directors at Varsity Tutors today to connect with a qualified tutor.

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