 So friends, continuing from the previous session, we are going to deal with conversion of decimal numbers non-terminating recurring into rational numbers of the form P by Q. So in the last session, we saw how to convert or terminating decimal into a fraction, that is a rational number. And we also discussed that there are two cases, one case is where the fraction or sorry decimal representation is terminating, then how to convert it into a fraction or a rational number. So we saw how 0.5 became 5 upon 10 and it was 1 by 2. Today or in this session, what we are going to target is we are going to target pure non-terminating recurring decimal representation. Pure means all the digits after the decimal repeats, you know, particular trend. For example, pure non-terminating recurring are decimal are like 0.3 bar or 0.21 bar or 0.316 bar. These are all pure non-terminating recurring decimals. But numbers like 0.21 bar or 0.3251 bar are mixed. These are called mixed non-terminating non-terminating mixed non-terminating recurring. And these are called pure non-terminating recurring decimal. So in this session, we are going to learn how to convert pure non-terminating recurring decimal representation into rational numbers. Basically, we will be jotting down all the steps here and we'll simultaneously take an example. So first is express the given decimal as x, given decimal as x. What does it mean? Let us say we have to convert 0.3 bar into p by q form. So first step I'll do x is equal to 0.3333. I'll express like this. Okay, point number two. Count the number of digits repeating after a decimal. So if you see, only one digit is repeating, isn't it? And that is three in this case. So hence, and call it call it n. Okay, so n is equal to one. Okay. Now, third is multiply, multiply the equation, equation so obtained, equation so obtained, which equation? So if you see, I have an equation here, x is equal to 0.3333. Let us call it one and multiply the equation so obtained by one followed by n zeros. So in this case, what is n? n is one. So how many zero will follow one? One zero. So hence, basically that number here is, I am writing that number here is one and followed by how many zeros? One zero. So that is 10. So let us do that so if you multiply this, it becomes 10x. And if you multiply this 3.333 becomes 3.3333. Okay, call it equation number two. Now, step number four, subtract, subtract the previous subtract the previous equation that is one here from, from the, the new equation, new equation that is two. Okay, so what does it mean? So first is this, so you subtract, subtracting means LHS minus LHS should be equal to RHS minus RHS. That means 10x minus x is equal to 3.3333. minus 0.3333. If you see now, both the fraction or the decimal part is equal, right? So hence, after subtraction, what will you get? 9x is equal to, is equal to 333. So hence, x is equal to 3 by 9, right? So what is x? Finally, if you reduce it into the basic form, most simplest form, 1 upon 3. So what do I learn? So x was what? x was 0.3 bar. And by calculation, x is 1 upon 3. So hence, I can write 0.3 bar is equal to 1 upon 3. Okay, let us take another example to make it more clear. Let us say I had a number, a number, let us say 0.21, 0.21 bar. It's a pure non-terminating recurring decimal. So let us say x is equal to 0.212121 and so on and so forth. right what is the value of n n is equal to 2 why two digits are repeating after the decimal so hence what should I multiply with multiplication factor will be hundred one followed by two zeros so let us multiply let us call it equation number one multiply by hundred you will get hundred x is equal to twenty one point two one two one two one like that call it to and now what do you do you need to subtract these two equations so if you subtract you will get 99x is equal to 21 so what will be x x is 21 by 99 which in reduced form will be 7 upon 33 so hence I could convert I could convert 0.21 bar into 7 upon 33 in the form of p by q if you see right so I hope you understood the problem if there were three digits repeating then you could have multiplied by thousand and so on and so forth so I hope you understood the process and you can solve all such problems now in the next session we'll take up mixed non-terminating repeating decimal representation thank you