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

For example, group and add:

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

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**

**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.

## Arithmetic properties worksheets

**Arithmetic properties - Integers** (127.4 KiB, 1,853 hits)

**Arithmetic properties - Decimals** (159.3 KiB, 769 hits)

**Arithmetic properties - Fractions** (199.4 KiB, 800 hits)

**Distributive property** (311.9 KiB, 766 hits)