# Arithmetic properties

Multiplication and addition have specific arithmetic properties which characterize those operations. In no specific order, they are the commutative, associative, distributive, identity and inverse properties.

## Commutative property

An operation is commutative if changing the order of the operands does not change the result.

The commutative property of addition means the order in which the numbers are added does not matter. This means if you add 2 + 1 to get 3, you can also add 1 + 2 to get 3.

In other words, the placement of addends can be changed and the results will be equal. Likewise, the commutative property of multiplication means the places of factors can be changed without affecting the result.

## Associative property

Within an expression containing two or more occurrences of only addition or of only multiplication, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. This is called the associative property.

That is, rearranging the parentheses in such an expression will not change its value.

$\ 1 + 5 + 9 + 5 = ?$

To simplify this, use the commutative property to switch the order and then use the associative property to group $1$ and $9$, and $5$ and $5$, since these pairs both add up to $10$, so the final result is $20$.

## Distributive property

Distributive property combines addition and multiplication. If a number multiplies a sum in parenthesis, the parenthesis can be removed if we multiply every term in the parenthesis with the same number.

The number of terms inside the brackets doesn’t matter, this will always be valid.
This property is usually applied when an unknown is a part of addition, and it enables us to single the unknowns out.

## Identity element

The Identity element or neutral element is an element which leaves other elements unchanged when combined with them. Identity element for addition is 0 and for multiplication is 1.

## Inverse element

Multiplicative inverse or reciprocal for a number $x$, denoted by $\frac{1}{x}$, is a number which when multiplied by $x$ yields the multiplicative identity, 1. The multiplicative inverse of a fraction $\frac{x}{y}$ is  $\frac{y}{x}$

Additive inverse of a number $x$ is the number that, when added to $x$, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

For example, the reciprocal of 5 is $\frac{1}{5}$, and the oppostie number of 5 is -5.

### What is the difference between commutative and associative property?

When you are considering addition or multiplication, it’s important to know some properties or laws. In math, these are things that remain the same.

###### Commutative property vs Associative property

The commutative property or commutative law means you can change the order you add or multiply the numbers and get the same result.

For example, in the commutative property of addition, if you have 2 + 4, you can change it to 4 + 2, and you will have the same answer (6).

This is the same with the commutative property for multiplication. If you have 2 x 4, you can change it to 4 x 2 and get the same result (8).

The difference with the associative property or associative law is it involves more than two numbers. It doesn’t matter how you group the numbers or what you add or multiply first. The important thing is that it’s only addition or only multiplication.

You can change the order whether you add or multiply the numbers and get the same result.

The associative property of addition means you can add the numbers in any order. Example: 2 + 3 + 1 + 5 + 6 = 17. This is true whether you add 2 to 3 to 1 to 5 to 6 or if you add 2 and 3 together to get 5 and then add the 1, 5 and 6 together to get 12, and the 5 and 12 together to get 17.

The associative property for multiplication is the same. If you have three or more numbers, you can multiply them in any order to get the same result.

For example, in the problem: 2 x 3 x 5 x 6, you can multiply 2 x 3 to get 6 and then 5 x 6 to get 30, and then multiply 6 x 30 to get 180. You can multiply the numbers in any order and will get 180.

## Arithmetic properties worksheets

Arithmetic properties - Integers (127.4 KiB, 2,445 hits)

Arithmetic properties - Decimals (159.3 KiB, 971 hits)

Arithmetic properties - Fractions (199.4 KiB, 1,034 hits)

Distributive property (311.9 KiB, 1,085 hits)