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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 ….
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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 xb=bx 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 (bxc) = (a xb) xc ? • 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?