## Algebra## Algebraic Operations |

## How do numbers associate with each other? |

Generally in mathematics, there are certain properties of operations that determine how numbers associate with each other. *Closure* is a property of an operation that reveals how numbers associate with each other; in particular, when two whole numbers are added, their sum will be a whole number. Closure as a property of multiplication occurs when two whole numbers are multiplied and their resulting product is a whole number.

An *associative* property means that for a given operation that combines three quantities (two at a time), the initial pairing of the quantities is arbitrary. For example, when doing an addition operation, the numbers can be combined in two ways: (*a* + *b*) + *c* = *a* + (*b* + *c*). Thus, when adding the numbers 3, 4, and 5, this means that they may be combined as (3 + 4) + 5 = 12 or 3 + (4 + 5) = 12. Following the same logic for multiplication, the associative law states that (*a* × *b*) × *c* = *a* × (*b* × *c*). In fact, in an associative operation, the parentheses that indicate what quantities are to be first combined can be omitted; an example of the associative law for addition is 3 + 4 + 5 = 12, and for multiplication, 2 × 3 × 4 = 24. But not all operations are associative. One good example is division: You can’t divide in the same way as you added or multiplied above. For example, the result of dividing three numbers differs. The operation (96÷12) ÷ 4 = 2 is not the same as 96 ÷ 4 (12 ÷ 4) = 32.

Like the associative property, the *commutative property* is another way of looking at how numbers associate with each other in operations. In particular, this law holds that for a given operation that combines two quantities, the order of the quantities is arbitrary. For example, in addition, adding 4 + 5 can be written either as 4 + 5 = 9 or 5 + 4 = 9, or expressed as *a* + *b* = *b* + *a*. When working on a multiplication operation, the same rule applies, as in *a* × *b* = *b* × *a*. Again, not all operations are commutative. For example, subtraction is not, as 6 - 3 = 3 is not the same as 3 - 6 = -3. Division also is not commutative, as 6 ÷ 3 = 2 is not the same as 3 ÷ 6 = ½.

The final property of an operation is the *distributive property.* In this rule, for any two operations, the first is distributive over the second. For example, multiplication is distributive over addition; for any numbers a, b, and c, *a* × (*b* + *c*) = (*a* × *b*) + (*a* × *c*). For the numbers 2, 3, and 4, you would have 2 × (3 + 4) = 14 or (2 × 3 ) + (2 × 4) = 14. Formally, there is a right and left distribution—left is listed above; right is (*a* + *b*) × *c* = (*a* × *c*) + (*b* × *c*). In most cases, both are commonly referred to as distributivity. Again, not all operations are distributive. For example, addition is not distributive over multiplication, as in *a* + (*b* × *c*) (*a* + *b*) × (*a* + *c*).