# Some Properties of Whole Numbers and their Operations

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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?

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