Complete Detail of Irrational Number

IRRATIONAL NUMBERS

A number which can] neither be expressed as a terminating decimal nor as a repeating decimal is called an irrational number.

Thus, nonterminating, non repeating decimals are irrational numbers.

All real numbers which are not rational, are called irrational numbers. \sqrt { 2 } ,\sqrt [ 3 ]{ 2 } ,-\sqrt { 7 } are some examples of irrational numbers. Perhaps the student is used to taking \pi is an irrational number (while \frac { 22 }{ 7 } is a rational number)

There are decimals which are non-terminating and non-recurring decimal, that is, it can not be written in the form \frac { p }{ q } , where p and q are both integers and q \neq 0. Consider a decimal of the form 0.303003000300003 …

An Irrational number is a non-terminating and non-recurring decimal and cannot be put in the form \frac { p }{ q }

Examples of Irrational Numbers

Type1- (i) Clearly, 0.01001000100001 … is a nonterminating and nonrepeating decimal and therefore, it is irrational.

Similarly,

(ii) 0.02002000200002 … is irrational

     0.03003000300003 … is irrational and so on

(iii) 0.12112111211112 … is irrational,

       0.13113111311113 … is irrational, and so on.

(iv) 0.54554555455554 … is irrational,

       0.64664666466664 … is irrational, and so on

TYPE 2 – If m is a positive integer which is not a perfect square, then \sqrt { m }  is irrational.

Thus \sqrt { 2 } ,\sqrt { 3 } ,\sqrt { 5 } ,\sqrt { 6 } ,\sqrt { 7 } ,\sqrt { 8 } ,\sqrt { 10 } ,\sqrt { 11 } , etc., are all irrational numbers.

TYPE 3 – If m is a positive integer which is not a perfect cube, then \sqrt [ 3 ]{ m } is irrational.

Thus, \sqrt [ 3 ]{ 2 } ,\sqrt [ 3 ]{ 3 } ,\sqrt [ 3 ]{ 4 } ,\sqrt [ 3 ]{ 5 } ,\sqrt [ 3 ]{ 6 } ,\sqrt [ 3 ]{ 7 } ,\sqrt [ 3 ]{ 8 } ,\sqrt [ 3 ]{ 9 } , etc., are all irrational numbers.

The numbers \pi: \pi  is a number whose exact value is not \frac { 22 }{ 7 } .

In fact \pi has a value which is nonterminating and nonrepeating.

So, \pi is irrational, while \frac { 22 }{ 7 } is rational.

PROPERTIES OF IRRATIONAL NUMBERS

  1. Irrational numbers satisfy the commutative, associative and distributive laws for addition and multiplication.
  2.  (i) Sum of two irrationals need not be an irrational.

           Example: Each one of (2+\sqrt { 3 }) and (4-\sqrt { 3 } ) is irrational.

                            But, (2+\sqrt { 3 }) + (4-\sqrt { 3 } ) = 6, which is rational.

          (ii) Difference of two irrationals need not be an irrational.

          Example: Each one of (5+\sqrt { 2 }) and (3+\sqrt { 2 }) is irrational.

                           But,(5+\sqrt { 2 }) - (3+\sqrt { 2 }) = 2, which is rational.

         (iii) Product of two irrationals need not be an irrational.

          Example: \sqrt { 3 } is irrational

                           But, \sqrt { 3 } x \sqrt { 3 }  = 3, which is rational.

       (iv) Quotient of two irrationals need not be an irrational.

          Example: Each one of 2\sqrt { 3 }  and \sqrt { 3 }  is irrational.

                           But, \frac { 2\sqrt { 3 }  }{ \sqrt { 3 }  } =2, which is rational.

  1. (i) Sum of a rational and an irrational is irrational.

(ii) Difference of a rational and an irrational is irrational.

(iii) Product of a rational and an irrational is irrational.

(iv) Quotient of a rational and an irrational is irrational.

Remark-: If a is rational and \sqrt { b }  is irrational, then each one of (a+\sqrt { b }), (a-\sqrt { b }), a\sqrt { b }  and \frac { a }{ \sqrt { b }  }  is irrational.

Examples: Each one of (4+\sqrt { 3 }), (8-\sqrt { 5 }), 5\sqrt { 3 }  and \frac { 3 }{ \sqrt { 2 }  } is irrational.

Final Wordings:

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Source-: Ts aggarwal & R.L Arora References


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