 Hello friends welcome to the session let's discuss real numbers and given question is the following real numbers have Decimal expansion as given below in each case decide whether they are rational or not If they are rational and of the form p by q. What can you say about the prime factors of q? numbers are 43 decimal one two three four five six seven eight nine second number Zero decimal one two zero one two zero zero one two zero zero zero zero one two zero zero zero zero zero and so on and Third number is 43 decimal one two three four five six seven eight nine which is repeating so Before starting with the solution I would like to tell you what are rational numbers and what are irrational numbers Rational number when any number is of the form p by q where p and q are integers and Q is not equal to zero are known as rational numbers and a number which can either be expressed as a Terminating decimal not as a repeating decimal is called an irrational number a point to be noted is that every repeating decimal is a rational number Now let's start with the solution We are given 43 decimal one two three four five six seven eight nine This number can be written as 43 then we'll remove the decimal we get one two three four five six seven eight nine upon one one two three four five six seven eight nine zeros So here we see that this number is in the form of p by q this implies that the number is rational number Now we are given in the version that if the number is rational then what are the factors of q? therefore prime factors of Q It can be written as four three one two three four five six seven eight nine Upon five vectors of q are two to the power five and five to the power four This implies that Prime factors of q will be either two or five or both Therefore the given number is rational and prime factors of q will be either two or five or both only So hopefully understood the solution of the first part now we see the second part given number is zero decimal one two zero one two zero zero one two zero zero one two zero zero zero And so on here we see that the given number can neither be expressed as a Terminating decimal nor as a repeating decimal therefore it is an irrational number rational Number as we have already seen in the definition of a rational number That a number which can neither be expressed as a terminating decimal nor as a repeating decimal is an irrational number Hope you understood the solution of the second part and enjoyed it now. Let's see the third part our third part is 43 decimal one two three four five six seven eight nine Which is repeating Now since we know from our key idea that every repeating decimal is a rational number Therefore we can see from that the given number is a rational number now equal to 43 decimal one two three four five six seven eight nine Repeating so this we are first equation now multiply both sides Equation first by Ten to the power nine We get ten to the power nine X equal to 43 one two three four five six seven eight nine upon one one two three four five six seven eight nine zeros Decimal one two three four five six seven eight nine Which is repeating into ten to the power nine here. We see that These nine zeros will cancel out with ten to the power nine so let this be our second equation now Subtract equation first second We get ten to the power nine X minus X equal to 43 one two three four five six seven eight nine Decimal one two three four five six seven eight nine, which is repeating minus 43 decimal one two three four five six seven eight nine repeating This implies Ten to the power nine X minus X is nine nine nine nine Nine nine nine nine nine nine X equal to Equal to four three one two three four five six seven four six This implies X equal to four three one two three four five six seven four six upon nine nine nine nine nine nine nine nine nine nine nine Therefore the given number is rational time factors Q will Also other than hope you understood the solution and enjoyed the session. Goodbye and take care