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									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  23  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  217  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?

								
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