Properties of Real Numbers

Properties: Addition Commutative Property For all real a, b a + b = b + a Associative Property For all real a, b, c a + (b + c) = (a + b) + c Identity Property There exists a real number 0 such that for every real a a + 0 = a Additive Inverse (Opposite) For every real number a there exist a real number, denoted (-a), such that a + (-a) = 0 Multiplication Commutative Property For all real a, b ab = ba Associative Property For all real a, b, c (ab)c = a(bc) Identity Property There exists a real number 1 such that for every real a a (1) = a Multiplicative Inverse (Reciprocal) For every real number a except 0 there exist a real number, denoted , such that

More Properties: Closure Property A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set.  If an element outside the set is produced, then the operation is not closed.  Distributive Property states that the product of a number and a sum is equal to the sum of the individual products of the addends and the number. Note: Distribution is more than this simplified definition. Just remember that you need both multiplication or division along with addition or subtraction. Multiplicative property of zero The product of any number an zero is zero.

Which property does this represent? Example 1 Which property does this represent? -41 + 0 = -41 This is the Additive Identity because adding zero does not change the value.

Which property is this an example of? 14 + 16 = 16 + 14 This is the Commutative Property because the order of the addends have been changed.

Which property does this represent? Example 3 Which property does this represent? ab + cd + (-ab) = cd This is an example of the Additive Inverse because ab and –ab are opposites. When added together you get the Additive Identity or 0.

Are odd numbers closed under addition? Example 4 Are odd numbers closed under addition? No, because if you add 3 and 5 you do not get an odd number.

Which property does this represent? Example 5 Which property does this represent?   a (b + c) = ab + ac This is the Distributive Property because you multiply a over the sum of b and c. Remember you need two opposing operations.

Which property does this represent? Example 6 Which property does this represent?  (4m)(n) = 4(mn) This is an example of the Associative Property. I know this by looking at the order of the factors. The factors are written in the same order, but we multiply in a different order because of the parentheses.