 Hello folks welcome again to yet another session on number system today in this session We are going to discuss decimal representation of rational numbers now Looks intimidating isn't it? But don't worry. We will try and simplify this concept and you'll be able to use it in multiple, you know in application. So what what does it mean decimal representation of? Rational numbers so we have to just recapitulate what is a rational number So if you remember what are rational numbers rational numbers are the numbers of the form of p by q where q was not equal to 0 and P and q are p and q belongs to the integer set yeah, and p and q are integers and Gcd or hcf of p and q was Equal to 1 that is there is no common factor between p and q it that means in terms of fraction it is most simplified fraction Okay. Now. What does decimal representation of rational numbers mean? So let us take an example and then first understand So if someone says one upon two clearly, it's a rational number There's no doubt about it, but what is one upon two? So basically, you know if it was a bigger number divided by a smaller number then You usually know there will be a quotient and there will be a remainder in this case one is less than two So how do we represent it in some decimal form? So it's nothing, but you have to basically divide the one by two So how to divide that you know the process the long division method So if there is a smaller number then you have to put a decimal after zero and then the moment you put a decimal point One becomes ten one zero gets associated with it so hence now it is zero two times five you follow the normal division process and You get quotient as zero point five So one point one by two is nothing but zero point five Okay, let us take another example one upon three. So in this case what you do you this is three and You divide one so for that you put zero first and then you put a decimal so that one becomes ten Then it goes by three so three threes are nine. So the remainder is one Now again, you can put a zero and then it goes by three again So it becomes nine and if you see the process is repeating, isn't it? so Again, you take nine and then one and then if you see there is no end to this process because You know, you see every time the same three nine comes and the remainder one goes and then It keeps on going like that. So what do we say we say one by three is equal to nothing But zero point three three three three three three and there will be infinite number of three So we just put a dot like that But it's very difficult to write like that because how many threes would you recommend would you write basically? So hence there is a process to it. So you see only three is repeating So whatever portion is repeating you put a bar over it, right? So zero point three bar So hence this is read as zero point three bar. So one upon three is nothing but Zero point three bar. Let us take some other example. Let us take we have four upon let's say 25 Okay, so how do we go about it? So, you know 25 and then what you do you again go by the long method So this is for so you put you put zero and then a decimal so it becomes 40 is it now you go by 25 ones at 25 So hence you get remainder as 15 Then what again you put a zero then what 25 times six is 150 and it ends the process ends so hence four by 25 is nothing but zero point one six This is the decimal representation of four upon 25 for that matter any fraction it is already rational number will there be There will be a decimal representation of it Let's take another example. Let us say we have three upon seven Okay, now three of them seven again. Let's do the process division process seven and this is three You put zero then the symbols it becomes 30. So seven three times. No, not three times my bad So zero point four four times 28 You get what two right then again put a zero 20 then what seven two times 14, isn't it then what six? So 60 now what seven seven the forty nine seven eight the fifty six so seven eight times fifty six So what is it for right? Oh, it seems it is going to do to be now again 40. So hence what seven five So 35 right again five. Oh, it doesn't seem to be ending anywhere soon. So 50 so seven seven's a 49 correct. What is left one? So 10 so seven ones are seven and three so hence 30 so if you notice we have hit 30 again and there is no reason why the entire process will repeat Isn't it so you can try for a few. So let us see let us say 30, right? So again seven four So 28 remainder two so 20 Two and then 14 and then six and so on and so forth. So what do we see we see that three upon seven Is equal to zero point four two Eight five seven one Then again same digits will repeat four two eight five seven one Then again four two eight five seven one like that like that like that, right Which can be now in our notation can be written as zero point four two eight five seven one Bar, isn't it? This is the decimal representation of three upon seven Okay, so let us take another one. So let me take What should I take? I said take let's say 21 upon 11 Okay, this is another rational number both numerator and denominator are integers And they can be the gcd is one and denominator is not equal to zero So let us go by the long division method. So 11 and this is 21. So it is one Times 11. So 10. So you put a decimal. So not becomes hundred 11 9s are 99 Right. So what is left one again? You put what? zero so hence zero comes Is it it? Zero will anyways come so it will become hundred right now now what again nine So you will get 99 And then you see the process repeats it is one. So you put zero, but it is not sufficient So you put zero again So this is hundred and again nine and 99 and It carries on so hence 21 upon 11 can be written as 1.90 9090 like that. So hence it is 1.90 Bar understood. So this is what? 21 by 11 would look like So, uh, what do we infer if you if you see what are the observations? There are two types of decimal representations we are getting. Yeah, so I can write that so observations are observations Observations are these one one the one of the There are two types of observations are two two types of Two types of decimal decimal representation is there Are there representation, right? Two types are there one if you notice is called Terminating decimal terminating the term is terminating why terminating because The decimal representation ends after a point. So for example here four upon 25 you see Doesn't keep on going on forever, right? It is 0.16. That's it. Similarly, if you see here one by two was 0.5 Correct. So these are called terminating, right? So there will be an end to the process, but there is another one which is called non-terminating Non-terminating, right? There is no end point of the division process but It not only it is non-terminating. It is also called it is also recurring recurring means repeating right after some time The decimal representation repeats if you see best example was our three by seven So four two eight five seven four two eight five seven four two eight five seven you see a pattern, right? It repeats after the Certain times similarly here. It is one point nine zero nine zero nine zero like that Similarly one by three was point three three three three repeats keeps on repeating. So there is a Repetition, right? So what do we observe? There are two types of decimal representation one either it will be a terminating or it will be non-terminating recurring so you now know the process of expressing a fraction Into a decimal form. You just simply need to divide it. You know the process of division And that's all so there are and then you you observed this particular thing that there are two types of representation one is terminating decimal Another that is non-terminating recurring decimal Representation right now. I'll give you another good, you know Trick here if you see terminating decimal only a few types of Fractions will end in terminating decimal that we will take up in next lecture. Thank you