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SAI International School
                                                      CLASS - X

               Mathematics
               CHAPTER-1: Real Numbers-3

               Lesson Notes-3

               SUBTOPIC :Revisiting Irrational Numbers


               Irrational Numbers



               Any number that cannot be expressed in the form of   (where p and q are integers

               and q≠0.) is an irrational number. Examples: √2, √3, √5, π    etc.

                       Theorem 1.3 :

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                       Let p be a prime number. If p divides a , then p divides a, where a is a
                       positive integer.


                       Proof to be done.

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                       Example: 3 divides 6  i.e 36, which implies that 3 divides 6 also.
                       Properties of Irrational numbers


                     The sum or difference of a rational and an irrational number is irrational.
                     The product and quotient of a non-zero rational and irrational number is
                       irrational.
                     √p is irrational where ‘p’ is a prime.


                        For example, 7 is a prime number, so √7 is irrational.

               Proof by Contradiction


               In the method of contradiction, to check whether a statement is TRUE
               (i)  We assume that the given statement is not TRUE.
               (ii) We arrive at some result which contradicts our assumption, thereby proving the
               contrary.


               Theorem 1.4 :  √2 is irrational.

               Proof:
               Let us assume that √2  is rational.

               Since it is rational √2 can be expressed as √2 =  , where a and b are co-

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