 Hello and welcome to the session. In this session we will understand how a rational number terminates or eventually repeats. Now first of all let us see what does terminate mean. Now when we have ended our conversation on phone, we terminate the phone call. So to terminate means to end. Now we can write a rational fraction in decimal and that decimal number either terminates or repeats. Now consider a fraction 7 by 8. Now we can change this fraction in a decimal by dividing the numerator with the denominator. So now we will divide the numerator with the denominator. Now as 7 is never than 8, so we put a decimal in the quotient and we will annex 0 to 7. And then we will divide as we do with the whole numbers. Now 8 into 8 is 64 and 79 is 64 is 6. Now we get the remainder as 6 which is again smaller than 8. So we annex another 0 with 7d because we have already put decimal in the quotient. Now 60 is greater than 8 and 8 into 7 and 60 minus 56 is 4. Now we have got the remainder as 4 which is again smaller than 8. So we annex another 0 with 700. So the remainder will become 50. Now 8 into 5 and now the remainder is 0. So the division terminates here as we have got the remainder as 0. Therefore 7 by 8 is equal to 0.875. Here we got the remainder as 0. This means 0.875 is a terminating decimal. So we can say that 7 by 8 is a rational number that terminates. Now let us see the definition of a terminating decimal and ends or terminates. That is when the remainder is 0, when the decimal is a terminating decimal and we say it in rational number. Let us see one more example for this. We are considering the fraction 9 by 4. Now here also for converting this fraction into decimal that is divide the numerator by the denominator. Now 4 into 2 is 8 and the remainder is 1. Now here we have got the remainder as 1 which is smaller than 4. So now we will put a decimal in the quotient and we will annex a 0 with 9. So the remainder will become 10. Now 10 is greater than 4 and 4 into 2 is 8. So 10 minus 8 is 2 and the remainder as 2 which is smaller than 4. So again we will annex a 0 with 90. So the remainder will become 20. Now 4 into 5 is 20 and we get the remainder as 0. That means the division terminates. We have got 9 by 4 is equal to 2.25 which is a terminating decimal. Now not all fractions can be written as terminating decimals. Sometimes a digit or book of digits repeats without end in the quotient. That is called non-terminating decimals. Now let us see one example for this. Here we consider the fraction 1 by 6. Now to convert it in decimal form let us divide the numerator with the denominator. Now 1 is smaller than 6. So we will put a decimal in the quotient and we will annex a 0 with 1. Now 10 is greater than 6 and 6 into 1 is 6. And we get the remainder as 4 against smaller than 6. So we will annex one more 0 with 10. So the remainder will become 40. Now 6 into 6 is 36 and the remainder is 4 which is smaller than 6. So we will annex one more 0 with 100. Now again the remainder becomes 40. Now 6 into 6 is 36 and again we get the remainder as... Now we can put as many 0s with 1 till we get the remainder of 0 or numbers in the quotient start repeating the division. Now here we can see that the number 6 is repeated in the quotient because every time we are getting the same remainder 4. It means it is endless division. So we can write 1 by 6 is equal to 0.166 and so on which can be written as 0.16 bar. Now here we will put a bar on the digit which is repeating. So here we can say that 1 by 6 is a non-terminating fraction and 0.16 bar is a non-terminating decimal. The division never ends and digit or group of digits repeats that is a non-terminating decimal. And we say the non-terminating rational number. Now let us see one example where group of digits repeat. So here let us take the correction 6 by 11. So let us start the division. Now here 6 is smaller than 11. So we will put a decimal in the quotient and we will annex a 0 with 6. So this becomes 60. Now 11 into 5 is 55. So we get the remainder as 5 which is smaller than 11. So again we will annex a 0 with 60. So the remainder will become 50. Now 11 into 4 is 30 further. So we get the remainder as 6. Again we will annex a 0 with 600. And the remainder will become 60. Now 11 into 5 is 55. So the remainder is 5. Now again we will repeat the same process. Now 11 into 4 is 44. So the remainder is again 6. Now here you can see we get the remainder as 6 alternatively. So division does not end. It is repeated in the quotient by 11 is equal to 0.5454 and so on. Which is equal to 0.54 power. Now here we have put the power on the group of digits which is repeating. Now let us see some remarks. And the first one is if the fraction is negative the decimal is also negative. And the division remains same. Now here you can see we have got 6 by 11 is equal to 0.54. Now according to this remark minus 6 by 11 will be equal to minus 0.5454 and so on which is equal to minus 0.54 power. When denominator is 100,000 and so on then the fraction 3 by 100. Now let us divide the numerator with the denominator. Now 3 is smaller than 100 a decimal in the quotient and we will annex a 0 with 3. Now 30 is again smaller than 100. So we have to attach one more 0. Now for putting two 0's together we put a 0 after the decimal in the quotient and then we will start division. Now here 100 into 3 is 300 so we will get the remainder as 0. So 3 by 100 is equal to 0.03. Similarly 9 by 100 is equal to 0.09 5 by 100 is equal to 0.55 9 by 33 by 10 is equal to 3.3 and so on. So in this session we have learnt how a rational number or eventually repeats and this completes our session. Hope you all have enjoyed the session.