VIEWS: 2 PAGES: 13 POSTED ON: 11/12/2012 Public Domain
5.5 Real Numbers and Their Properties Name Description Examples Natural Numbers {1, 2, 3, 4, 5,…} 3, 17, 435 Whole Numbers {0, 1, 2, 3, 4, 5,…} 0, 1, 84, 267 Integers { … , -3, -2, -1, 0, 1, 2, 3, … } -649, -17, 0, 25, 453 Rational Numbers The set of numbers that can be -5, -⅔, 0, ⅝, 246 expressed as fractions. a/b where a and b are integers and b is not zero Irrational Numbers Numbers that cannot be expressed as 2.020020002… , π, a ration of integers. The decimal roots expansion does not terminate nor repeat. 4 15, 3 5 , 34, , , 1.7, 36 , 2.35 , 0, 5 Which are Natural Numbers? 7 Which are Whole Numbers? Which are Integers? Which are Rational Numbers? Which are Irrational Numbers? Name Description Examples Natural Numbers {1, 2, 3, 4, 5,…} 3, 17, 435 Whole Numbers {0, 1, 2, 3, 4, 5,…} 0, 1, 84, 267 Integers { … , -3, -2, -1, 0, 1, 2, 3, … } -649, -17, 0, 25, 453 Rational Numbers The set of numbers that can be -5, -⅔, 0, ⅝, 246 expressed as fractions. a/b where a and b are integers and b is not zero Irrational Numbers Numbers that cannot be expressed as 2.020020002… , π, a ration of integers. The decimal roots expansion does not terminate nor repeat. Natural The union of the Numbers Rational Numbers Whole numbers Irrational Numbers and the Irrational Integers Numbers is the set of Rational Numbers the Real Numbers The closure property for addition means that if any two elements of a set are added then the sum is also in the set. The Natural Numbers are closed for addition. a + b = c If a and b are natural numbers, then c will also be a Natural Number. 2+5=7 The closure property for multiplication means that if any two elements of a set are multiplied then the product is also in the set. The Natural Numbers are closed for multiplication. a x b = c If a and b are Natural Numbers, then c will also be a Natural Number. 2 x 5 = 10 The Natural Numbers are not closed for the operations of subtraction or division. 2 – 5 has no solution in the Natural Numbers. 2 ÷ 5 has no solution in the Natural Numbers The closure property for addition means that if any two elements of a set are added then the sum is also in the set. The Integers are closed for addition and subtraction. a+b=c a-b=c If a and b are Integers, then c will also be an Integer. 2 + 5 = 7 2 – 5 = −3 The closure property for multiplication means that if any two elements of a set are multiplied then the product is also in the set. The Integers are closed for multiplication. axb=c If a and b are Integers, then c will also be an Integer. −2 x 5 = −10 The Integers are not closed for the operation of division. 2 ÷ 5 has no solution in the Integers. The Rational Numbers are closed for addition, subtraction, multiplication, and division. The Irrational Numbers are not closed for addition, subtraction, multiplication, nor division. 2 2 0 3 3 3 The Real Numbers are closed for addition, subtraction, multiplication, and division. Name Meaning Examples Commutative Property Two real numbers can be added in any 3+2=2+3 of Addition order. a + b = b + a 3 + (−5) = −5 + 3 Commutative Property Two real numbers can be multiplied in 3x2=2x3 of Multiplication any order. a x b = b x a 3 x (−5) = −5 x 3 Associative Property If three real numbers are added, it (2 + 3) + 7 = 2 + (3 + 7) of Addition makes no difference which two are added first. (a + b) + c = a + (b + c) Associative Property If three real numbers are multiplied, it (2 x 3) x 7 = 2 x (3 x 7) of Multiplication makes no difference which two are multiplied first. (a x b) x c = a x (b x c) Distributive Property Multiplication distributes over addition. 2(3 + 5) = 2 x 3 + 2 x 5 of Multiplication over a(b + c) = ab + ac Addition 2 7 7 2 Commutative Property of Addition 2 3 5 23 2 5 Distributive Property of Multiplication over Addition Name Meaning Examples Commutative Property Two real numbers can be added in any 3+2=2+3 of Addition order. a + b = b + a 3 + (−5) = −5 + 3 Commutative Property Two real numbers can be multiplied in 3x2=2x3 of Multiplication any order. a x b = b x a 3 x (−5) = −5 x 3 Associative Property If three real numbers are added, it (2 + 3) + 7 = 2 + (3 + 7) of Addition makes no difference which two are added first. (a + b) + c = a + (b + c) Associative Property If three real numbers are multiplied, it (2 x 3) x 7 = 2 x (3 x 7) of Multiplication makes no difference which two are multiplied first. (a x b) x c = a x (b x c) Distributive Property Multiplication distributes over addition. 2(3 + 5) = 2 x 3 + 2 x 5 of Multiplication over a(b + c) = ab + ac Addition 2 17 5 217 5 Associative Property of Multiplication 2 3 7 2 3 2 7 Distributive Property of Multiplication over Addition Name Meaning Examples Identity Property There is an identity element 0 such 3+0=3 of Addition that a + 0 = a and 0 + a = a 0 + (−5) = −5 Identity Property There is an identity element 1 such 3x1=3 of Multiplication that a x 1 = a and 1 x a =a 1 x (−5) = −5 Inverse Property There is an inverse such that 3 + (−3) = 0 of Addition a + (−a) = 0 (the identity element) and −a + a = 0 Inverse Property There is an inverse such that 3x⅓=1 of Multiplication a x b = 1 (the identity element) and ⅓ x 3 =1 bxa=1 2 2 0 Inverse property of addition. 2 1 2 Identity property of multiplication. a b c d e Is the system closed? a a a a a a b a b c d e Is the operation commutative? c a c e b d Is there an identity element? d a d b e c Does the system have the inverse e a e d c b property? Is the operation associative? a b c d e Is the system closed? a a b c d e b b c d e a Is the operation commutative? c c d e a b Is there an identity element? d d e a b c e e a b c d Does the system have the inverse property? Is the operation associative? Is the system closed? Is the operation commutative? Is there an identity element? Does the system have the inverse property? Is the operation associative? Is the system closed? Is the operation commutative? Is there an identity element? Does the system have the inverse property? Is the operation associative?