Real numbers

 Real numbers


All numbers on the number line or the numbers which exists..Real numbers are usually denoted by the letter R.

 Real numbers contain:

• (1, 2, 3, ...)  natural numbers or counting numbers denoted by N
• (0, 1, 2, 3, ...)  whole numbers denoted by W
•  (-3,-2, -1, 0, 1, 2,3, ...)  integers denoted by Z
•  rational numbers (p/q, where p and q are any two numbers (except that q cannot be 0))
•  irrational numbers (such as √2 or π)
• square roots,cube roots etc

Every natural numbers are whole numbers.Every whole numbers are integers .Every integers are rational numbers and every rational numbers are real numbers. So real number is the combination of rational numbers and irrational numbers 

                 Whole number =  natural number  and  zero

                 Integer = whole numbers and  negatives

                 Rational number = integer/integer(we can make any integer a rational number by putting it over 1)

               Example:   0, -9 ,6/3 ,π,24,or 0.333333 .......

Rational Numbers

The numbers that can be written as the ratio o f two integers are called rational numbers or fractions.

Rational numbers 

• can be represented as simple fraction 
• both numerator and denominator are integers and denominator cannot be zero
• the result of a fraction is a terminating or repeating decimal.
• can be put on a number line.

Examples

 2.5 is rational, because it can be written as the ratio 5/2

78 is rational, because it can be written as the ratio 78/1

 3.333... (3 repeating) is also rational, because it can be written as the ratio 10/3

sqr(25) is also rational since it can be written as 5/1

-60/3 is rational ,because it can be written as  -20/1

  An integer is a rational number since any integer can be made as a fraction by dividing it with 1.
But  a rational number  always may not be an integer

Example: 10/1 is an integer, but 58/3 is not

Irrational Numbers

These are the numbers that cannot be expressed as a simple fraction or ratio.

Irrational numbers 

• real numbers
• cannot be represented as integer/integer
• non repeating, non terminating decimal

   Examples

sqr(2)=1.414213........

π = 3.141592..........

In both the above example,they cannot be written as a fraction and also the digits after the decimal are non repeating and non terminating,  

A number can either be  rational or  irrational, but not both.

 An irrational  number is either non-terminating or non-repeating but rational number terminates or repeates. There is no overlap between these two number types.

Examples

0.25

This is a terminating decimal, so it can be written as a fraction: 25/100 = 1/4. 

3.141592653... 

Here the decimal does not repeat, so this is an irrational.  the answer is: irrational

3.1415

This is  a  terminating number. The answer is: rational, real

Find whether the numbers are rational or irrational

         

  58/9 ,7 8/3 ,–sqrt(64) ,sqr(3)


58/9 is a fraction ,so rational


7 8/3 = (3*7+8)/3 = 29/3,so rational


-sqr(64) = -8, is an integer.so rational


sqr(3)  = 1.732050............,so irrational