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).