 Hey friends welcome again to another session on number system and in this session. We are going to discuss conversion of decimal numbers Into rational numbers of the form p yq now There are two possibilities here one is we have either terminating decimal For example like point five and hence you can you have to convert into a form of p by q Otherwise you have a recurring non-terminating decimal representation like point three bar and that you have to convert into p by q now Non-terminating recurring decimal itself can be of two types one is pure non-terminating that is all the digits after the decimal are repeating like Point three bar on the other hand it can be mixed non-terminating recurring as well mixed So take note of these words these terms pure and mixed so pure is all the digits are repeated But mixed is only few not all are repeating basically so example point two three three three right or zero point two three Bar and this session we are going to take up only the a case a this one and in the subsequent sessions will take up case B Okay. Now. Let us do it except with two examples and also we will write the The algorithm so first step is count the number of digits after the decimal point so in this case I have taken a first example as five point six seven so the number of Digits after the decimal is two isn't it now remove the decimal So hence after you remove the decimal you will be left with five six seven and then you simply Divide that by divide the number obtained by one followed by Number of zeros obtained in one so now how many zeros we obtained two right? Sorry. How many you know? Now number of digits obtained after decimal is two. So you'll have to divide the The numbers so obtained in step number two by one followed by those many zeros So if you see I'm dividing five six seven by one Followed by two zeros why two zeros because two digits were there after the decimal point So five sixty seven by hundred and you have to reduce it to the most simplest form if it is possible Right in this case it is not reducible because there is five sixty seven is neither a multiple of two Nor a multiple of five so it cannot be reduced further Another example is let us say I have to take point seven five So how many decimals after how many digits after the decimal again two So you have to remove the decimal part from point seven five you'll get 75 and then divide by one followed by two zeros Which can see here and if you reduce it you will get three by four Ultimately, you'll have to reduce it to the most simplest form of the fraction Let us say instead of point seven five. We had point seven zero five one zero in between seven five How many digits after the decimal you can see count one two three, isn't it? So remove the decimal to get seven zero five and divide now it by one thousand and if you simplify Both are multiples of five denominator and numerator. So hence in the top will get one forty one Divide by two hundred in the denominator. This is the most simplest form And finally I've taken another example where it is two point five six double zero, right? So two point five six double zero how many zeros how many decimals sorry how many digits after decimal four So after removing the decimal the number is two five six double zero and I have divided by One followed by four zeros because four digits were there after the decimal So finally it is two fifty six by hundred which is reduced to sixty four upon twenty five Now you also know that two point five six Double zero can also be written simply as two point five six. You don't need to count the last two zeros in The number so hence it is simply two fifty six upon One followed by two zeros why two zeros because there were two digits after the decimal it is reduced to further There are sixty four upon twenty five So I hope you learned how to convert a terminating decimal form into a P by Q form in the subsequent sessions will see how to convert a non-terminating recurring decimal representation Both mixed as well as pure form into rational number of the form P by Q Thank you