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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?