Some Properties of Whole Numbers and their Operations

Commutative Property

• Does order matter when you add, subtract, multiply, or divide two whole numbers?

• Is a+b=b+a? • Is a-b=b-a?• Is a x b=b x a?• Is a÷b=b÷a?

Whole Numbers are Commutative under the operations of …

• Addition• Multiplication

• Note that the order of the numbers changes as they move to opposite sides of the + or x sign.

Associativity- order of numbers stays the same, grouping changes

• Is a + (b + c) = (a + b) + c ?

• Example: 3 + (4 + 5) = (3 + 4) + 5

• Is a x (b x c) = (a x b) x c ?

• Example: 2 x (5 x 3) = (2 x 5) x 3

Distributive Property of Multiplication over Addition

• a(b + c) = ab + ac

• Example: 2(5 + 3) = 2(5)+2(3)

Identity Element for Addition

• Start with any number. What number do you add to it to keep it the same?

• Zero is called the identity element for addition.

• Is there an identity element for subtraction?• Yes, it is also zero. Why?

Identity Element for Multiplication

• Start with any number. What do you multiply by to keep it the same? This is the identity element for multiplication.

• Example 5 x ___ = 5• 5 x 1 = 5• Does division have an identity element?• Yes, also 1.

Inverse Elements

• What do you add to a number to get the identity (zero)?

• Example: 6 + ____ = 0• Whole numbers do not include additive

inverses.

Multiplicative inverse in the whole numbers?

• What do you multiply by to get the identity element for multiplication?

• Example: 8 x ____ = 1

• There is no multiplicative inverse in the set of whole numbers.

Closure Property

• If you perform an operation on two elements of a set and you get a result that is also an element of the set, we say the set is closed under that operation.

Example/Non-example

• Is the set of whole numbers closed under the operation of addition?

• (Yes, since the sum of any two whole numbers is a whole number.)

• Is the set of odd whole numbers closed under the operation of addition?

• (No, since the sum any two odd whole numbers is an even, not an odd.)

• Is the set of whole numbers closed under the operation of subtraction?

• Is the set of whole numbers closed under the operation of division?

Density Property

A set of numbers has the density property if there is another member of the set between any two other members of the set.

Is the set of whole numbers a dense set? In other words, is there a whole number between any two whole numbers?