The properties of whole numbers include the properties of Closure, Associative, Commutative, and Identity for the operations of addition and multiplication. These properties are not true for the operations of subtraction and division. Another common property is the Distributive property. The Closure property states that the result of the addition or multiplication of any two numbers in a set will produce another number in the same set. Since the sum of any two whole numbers results in a whole number, the property of Closure exists for the set of whole numbers for the operation of addition. The closure property also exists for the multiplication of whole numbers. Symbolically, for any whole numbers a and b: The Associative property for the operation of addition states that the order in which we add three whole numbers does not matter. The result will always be the same sum. This property also works for the operation of multiplication. Therefore, for any whole numbers a, b, and c: The Associative property is useful for computations. For example, the sum of the three whole numbers 13, 23, and 47 is easily found when performed in the order of 13 + (23 + 47). By applying the Associative property, these numbers can be re-grouped (13 + 23) + 47 giving the same result. This property exists because the sum is the same no matter how the numbers are grouped. The Commutative property provides the opportunity to change the order of terms under the operations of addition and multiplication. The order in which we add two whole numbers does not change the result. For example, 6 + 5 = 11 as does 5 + 6. 3(2) = 6 and (2)3 = 6. This seems trivial, but it is a powerful property for algebra. 2x + 3 + 5x can be written as 2x + 5x + 3 using the Commutative property.  Then, following the order of operations and combining like terms, 2x + 5x can be combined resulting in 7x. The expression 2x + 3 + 5x can then be expressed in the simplified form of 7x+3. Symbolically, for any whole numbers a and b: The Identity property of addition identifies the numbers that do not change the value of a whole number under the operations of addition and multiplication. For example, when you add zero to any whole number, the result is the original whole number: 5 + 0 = 5. The additive identity is zero. One is the multiplicative identity. Any whole number multiplied by 1 will result in the original whole number. 5 x 1 = 5. This property becomes even more useful later in algebra. Therefore, for any whole number a: The Distributive property is a useful tool for the multiplication of whole numbers. In the real world we frequently use this property for mental multiplication. For example, 7(34) is easily done by completing 7(30) + 7(4) which can be represented by 7(30+4) = 210+28=238. Symbolically, the property is represented as: