Question

properties of real numbers please illustrate. i need this for my math project.

Answer 1

A Real number is a value of a number, representing a quantity of something. It can be represented by a point on an infinitely long and continuous line (real number line like X axis).

Real numbers include rational numbers and irrational numbers. Rational numbers include all integers, and fractions of integers. Irrational numbers include square roots and nth roots of integers like:

A real number can be represented in decimal system also as:  132.92822223333322222.......  infinitely long. It can be expressed as a ratio or a proportion or a percentage. They are uncountable. There are infinitely many real numbers. In between two given real numbers, you can find infinitely many real numbers. 

There are more real numbers than number of integers. We have mathematical operators like +,-,/ and * to perform algebraic operations on real numbers.

Properties:

1.  Commutative property: Order of the operands does not matter.  
              A + B = B + A          
      But, the operations -, / do not obey this property. A * B = B * A            

2.  Associative property A + ( B + C )  =  (A +B ) + C 
     The order of evaluation of which + is done first does not matter.
       A * (B * C) = (A *B) *C               A / (B / C) = (A /B) / C  

3.  Distributive property   
     A * ( B + C ) =  A * B + A * C               A * ( B - C ) =  A * B - A * C 
   But  / is not distributive over + and -.  
     * is distributive over + or - .

4.  Identity property

For operators +, -, / and *, there exist identity numbers.  A number added to the  identity is equal to the number itself.
  6 + 0 = 6         6 * 1 = 6         9 / 1 = 9    11 – 0 = 11  

5.  Density property

We can always find a real number lying between given two real numbers.  Given 222.221 and 222.222, we can find  222.2215 that lies between them. There are infinite number of real numbers between given two numbers.  


6.       Closure property Sum or product of any two real numbers is a real number. Same with division or subtraction. A * B  = AB is a real number     A/ B is a real number  

7.       Inverse property
  
        For a real number a, there is a unique real number b, such that  a + b = 0 = identity. 
             b = - a and is called inverse of a for operation +. 
          a⁻¹ or 1/a  is the inverse of a for the operation /.    a +  -a  = 0          a  *  1/a  =  1

8.        Multiplication with zero For a real number a, multiplication with 0 gives 0.   a * 0 = 0  

9.       Division by 0 Division of a real number by 0 is not defined. 
            For mathematical purposes, it is denoted by and is called infinity.
                  1 / infinity     =   0