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Arithmetic properties

arithmetic properties

There are few arithmetic properties of multiplication and addition that help us define those operations. They are the commutative, associative, distributive, identity and inverse properties. So let’s begin…

First is the commutative property.

The commutative property of addition says the order in which we are adding numbers does not matter. In other words, we can change the placement of addends to any order we like, and the result will not change.

Likewise, the commutative property of multiplication says that we can change places of factors in any way we like without affecting the result.

Commutative property

commutative property

Where do we use it? Well, if we are working with expressions that are a little bit more complicated and we’d like to simplify them somehow, the commutative property makes it easier to do so.


Associative property

associative property

The second property is the associative property.

The associative property of addition tells us that we can group any addends we like together and then add the rest of them.

The associative property of multiplication tells us that we can group any factors we like together and then multiply with the rest of them.

Where do we use it? Again, this property enables us to group together numbers that are bounded by the same operation. This will come in handy especially in cases with many numbers in it and you’d like to calculate it swiftly and accurately.

For example, group and add:

\ 1 + 5 + 9 + 5 = ? To simplify this, we’ll use the commutative property to twist them around, and then we’ll use associative property to group 1 and 9, and 5 and 5, because these pairs both add up to 10, so the final result is 20.

Distributive property

The following property is the property that combines addition and multiplication – the distributive property.

The distributive property is a property which tells us that, if we have a number which multiplies a sum in a parenthesis, we can remove the parenthesis if we multiply every term in that parenthesis with that same number.

distributive property

No matter how many terms you have inside the brackets, this will always be valid.
This property is usually applied when you have an unknown as a part of your addition, and it enables you to single them out. Just remember to always multiply all of the terms, or the result will be incorrect.

There are two more basic arithmetic properties:

Identity element

identity element




Inverse element

inverse element




If you’re having trouble understanding these properties, try to play with small numbers. Test everything. You’ll get a hang of it soon enough.

Arithmetic properties worksheets

  Arithmetic properties - Integers (127.4 KiB, 343 hits)

  Arithmetic properties - Decimals (159.3 KiB, 280 hits)

  Arithmetic properties - Fractions (199.4 KiB, 246 hits)

  Distributive property (311.9 KiB, 268 hits)