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Lesson 1: Properties of numbers

# Properties of multiplication

Explore the commutative, associative, and identity properties of multiplication.
In this article, we'll learn the three main properties of multiplication. Here's a quick summary of these properties:
Commutative property of multiplication: Changing the order of factors does not change the product. For example, 4, times, 3, equals, 3, times, 4.
Associative property of multiplication: Changing the grouping of factors does not change the product. For example, left parenthesis, 2, times, 3, right parenthesis, times, 4, equals, 2, times, left parenthesis, 3, times, 4, right parenthesis.
Identity property of multiplication: The product of 1 and any number is that number. For example, 7, times, 1, equals, 7.

## Commutative property of multiplication

The commutative property of multiplication says that changing the order of factors does not change the product. Here's an example:
4, times, 3, equals, 3, times, 4
Notice how both products are 12 even though the ordering is reversed.
Here's another example with more factors:
1, times, 2, times, 3, times, 4, equals, 4, times, 3, times, 2, times, 1
Notice that both products are 24.
Which of these is an example of the commutative property of multiplication?

## Associative property of multiplication

The associative property of multiplication says that changing the grouping of the factors does not change the product. Here's an example:
start color #11accd, left parenthesis, 2, times, 3, right parenthesis, times, 4, end color #11accd, equals, start color #e07d10, 2, times, left parenthesis, 3, times, 4, right parenthesis, end color #e07d10
Remember that parentheses tell us to do something first. So here's how we evaluate the left-hand side:
empty space, start color #11accd, left parenthesis, 2, times, 3, right parenthesis, times, 4, end color #11accd
equals, 6, times, 4
equals, 24
And here's how we evaluate the right-hand side:
empty space, start color #e07d10, 2, times, left parenthesis, 3, times, 4, right parenthesis, end color #e07d10
equals, 2, times, 12
equals, 24
Notice that both sides equal 24 even though we multiplied the 2 and the 3 first on the left-hand side, and we multiplied the 3 and the 4 first on the right-hand side.
Which of these is an example of the associative property of multiplication?

## Identity property of multiplication

The identity property of multiplication says that the product of 1 and any number is that number. Here's an example:
7, times, 1, equals, 7
The commutative property of multiplication tells us that it doesn't matter if the 1 comes before or after the number. Here's an example of the identity property of multiplication with the 1 before the number:
1, times, 6, equals, 6
Which of these is an example of the identity property of multiplication?

## Want to join the conversation?

• Does these rules apply for subtraction and division?
• Nice question!

Subtraction and division are neither commutative nor associative. Rather than just memorizing this, let's look at some examples.

Example 1: 5-3 = 2, but 3-5 = -2.
Example 2: 7-(3-1) = 7-2 = 5, but (7-3)-1 = 4-1 = 3.
Example 3: 6/2 = 3, but 2/6 = 1/3.
Example 4: 8/(4/2) = 8/2 = 4, but (8/4)/2 = 2/2 = 1.

Subtraction has a partial identity of 0, and division has a partial identity of 1, but this only works if the identity is on the right.

x - 0 is always x, and x/1 is always x.

However, 0 - x is usually not x, and 1/x is usually not x.
Example 5: 0 - 2 = -2, not 2.
Example 6: 1/2 is one-half, not 2.

In math, it is very important to learn to distinguish properties from non-properties. Otherwise, you could end up using an invalid shortcut and getting a wrong answer, or end up failing to recognize an opportunity to use a valid shortcut.
-2(5 x 7) = (-2 x 5) x (-2 x 7) True or False? I know this is false but does the distributive property only apply to addition and subtraction and not multiplication?
• The distributive property distributes multiplication across addition or subtraction. In your example, someone is trying to use it to distribute multiplication across multiplication. The distributive property does no do this. And, if you do the math on each side of the example, you can see that the steps are invalid because they don't create the same result.
• What is this property? 1/8 * 8 = 1 or 1/3 * 3 = 1.
• That is the inverse property of multiplication.
Any number * its reciprocal = 1.

Hope this helps.
• What is the property of 4*(5+8)= (4*8)?
• It isn't a property. The 2 sides are not equal.
There is a distributive property that tells you:
4*(5+8) = 4*5+4*8

Hope this helps.
• 2+2+2=3+3 Above said this was incorrect selection. Both sides equal 6. The one below has two different answers. Why wouldn’t it be this one.
• There is also the associative property of multiplication over addition.
• The inverse property that I've heard of states that multiplication and division are inverse operations, so you multiply a number by another number and then divide it by that same other number, you'd get the first number back again. This is the same as when you add and then subtract the same number, and get what you started with. Is that what you're talking about?
• how can you mutiply each and every number in order to get the correct answer
• Split each number into pairs and multiply. For best results use a calculator.
• For the question (a+c)+b=(a+b)+c what property is it