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.
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.
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:
To simplify this, use the commutative property to switch the order and then use the associative property to group and , and and , since these pairs both add up to , so the final result is .
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 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.
Multiplicative inverse or reciprocal for a number , denoted by , is a number which when multiplied by yields the multiplicative identity, 1. The multiplicative inverse of a fraction is
Additive inverse of a number is the number that, when added to , 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 , and the oppostie number of 5 is -5.
Arithmetic properties worksheets
Arithmetic properties - Integers (127.4 KiB, 737 hits)
Arithmetic properties - Decimals (159.3 KiB, 444 hits)
Arithmetic properties - Fractions (199.4 KiB, 421 hits)
Distributive property (311.9 KiB, 391 hits)