Properties of Real Numbers & Distributive Property

Properties of Real Numbers & Distributive Property
Some Properties of Real Numbers

Commutative Property of Addition

The problem 5 + 7 = 7 + 5 demonstrates the commutative property of addition.

The commutative property of addition states that changing the order of the addends does not change the sum.

In the example above, we can easily observe that:

5 + 7 = 7 + 5
12 = 12

Since 12 equals 12, 5 + 7 must be equal to 7 + 5.

In general, the commutative property of addition can be written as a + b = b + a.

Commutative Property of Multiplication

The problem 6 × 10 = 10 × 6 illustrates the commutative property of multiplication.

The commutative property of multiplication states that changing the order of the factors does not change the product.

In the example above, we can easily observe that:

6 × 10 = 10 × 6
60 = 60

Since 60 equals 60, 6 × 10 must be equal to 10 × 6.

In general, the commutative property of multiplication can be written as:
a × b = b × a

Associative Property of Addition

The problem (3 + 6) + 8 = 3 + (6 + 8) demonstrates the associative property of addition.

Observe that the addends are the same on either side of the equal sign: 3 plus 6 plus 8

The associative property of addition says that when we add more than two numbers the grouping of the addends does not change the sum.

In the example above, we can easily observe that:

(3 + 6) + 8 = 3 + (6 + 8)
9 + 8 = 3 + 14
17 = 17

Notice that the SUM is the same no matter what way you group the addends.

In general, the associative property of addition can be written as:
(a + b) + c = a + (b + c)

Associative Property of Multiplication

The problem (2 × 4) × 3 = 2 × (4 × 3) demonstrates the associative property of multiplication.

Observe that the factors are the same on either side of the equal sign: 2 times 4 times 3

The associative property of multiplication says that when we multiply more than two numbers the grouping of the factors does not change the product.

In the example above, we can easily observe that:

(2 × 4) × 3 = 2 × (4 × 3)
8 × 3 = 2 × 12
24 = 24

Notice that the PRODUCT is the same no matter what way you group the factors.

In general, the associative property of multiplication can be written as:
(a × b) × c = a × (b × c)

Distributive Property

Simplify the problem 4(5 + 6) using distributive property.

Distributive property tells us that if we have a sum or a difference of two numbers, we can multiply both of those by another number by distributing it.

In the example above, we have a sum of two numbers (5 + 6) and we want to multiply this sum by 4.

Let’s distribute the 4 that is outside the parenthesis to every number inside the parenthesis.

4(5 + 6) = 4 times 5 + 4 times 6
= 4 × 5 + 4 × 6
= 20 + 24
= 44

The following is one way of checking your answer:

4(5 + 6) = 4(11)
= 44

Since 44 equals 44, 4(5 + 6) must be equal to 4 × 5 + 4 × 6.

In general, the distributive property can be written as:
a (b + c) = a × b + a × c

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I’m Chandrajeet, an in-house writer for iCoachMath, the providing of All Free Solved Exampled for Math from K – 12. In All USA State Curriculum and also cover all Mathematics Curriculum topics & lessons. I am a regular reader and writer of Education articles.
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