3Speak - YouTube - Telegram - Notes - Playlist - Sequences and Series - MES Links In this video, I go over an example of converting the number 2.3171717... (with infinite repeating 17) into a fract...
Transcript
For example, 4, and states, write the number, 2.317, 17, etc. It just goes on and on with the 1-7 as the ratio of integers. So let's write this out. It's right as a ratio of integers. In other words, 2.317, and put a line above, indicating that this equals to 2 .317. 1-7, 1-7, and it keeps going on and on with the 1-7. This equals 2. What we can do is separate this 2 .3, like that. Then we could write this as a ratio of integers, in other words, the decimal place is put it as a fraction. And then from there, we can write this whole thing as integer, or ratio of integers. So we'll get this in fraction, this equals 2, plus what we have here is, well, 17, this part right here is 1-7. That's the same thing as writing 1-7, divide this by, well, 1,000. The decimal place here, 1, 2, 3. So we have 1,000 like that. And this 1,000 is just 10 to power of 3. So 10 times 10 is 100 times again, 10 is 1,000. And then the next decimal place over here, that just goes to more zeros. So in other words, instead of 10,000, we have 1,000. Actually, I mean 100,000, yeah, so 10 ,000. This is 1,000, it goes to 10 ,000, 100,000. So in other words, we have 10, the power of 5. So just add another 2 zeros, or 10 to 3. And so on. So the next one here, this is going to be a distance that far away to get in fractions, 17 over another 2 zeros, 10 to the 7. So in other words, we have a geometric series right here. This is a geometric series from here to here. I'll just write this down. Geometrics. We have a common ratio, if you can see it. So the geometric series from this side, where our A is this value, so this is our A. That's our A is 17 over 1,000, or 10 over 3. And then the common ratio is just a multiple between these two, and that's just, well, R is equal to 1 over 10th power of 2. So everyone here, so this 10 over 10 over 3 times it by 1 over 10 of the power of 2, add those up, it becomes this 1 over 10 to the power of 5, then 1 over 10 at power of 7. That's what's separated, so that's our R. And of course, we have the 17 on top there. because of our A starts off like that. So, yes, so after the first one, we have a geometric series with A equals 17 over 10 to power of 3, and R is equal to 1 over 10 to power of 2. So, therefore, what we end up having is this number is equal to, so 2.317 like that, and then 1-7 to the line above, indicating it's repeating, equals to 2 .3, times it by, or 2.3 plus, and then we have here, yeah, this is a geometric series, so the summation of all of these, I was put this dot, dot, dot, dot, keeps going on and on for infinity, this just equals to, oh, we know the sum, this is going to be A over R. This is the sum of the geometric series, like that, so sum of gs. Yeah, and so that's where the geometric series comes in handy. So now this equals 2.2.3 plus A, which is a 17 over 10 to 3, 17 over 10 to 3, which is 1,000. So, like this, over 1,000, divide this by, then 1 minus R. 1 minus, and then, well, 1,000, this is 100, so 1 over 10 squared, that's just, 10 times 10 is 100, so 1 over 100, like that. Yeah, so all we have to do now is just write this as just one full fraction, so let's just see what we can do, do a bunch of algebra. So what we can do here is, you know, multiply this one here by, well, 100 over 100, and also this 2.3, I'm going to write this as 23 over 10, like that, so everything's in a fraction, plus... So we have this 17 over 1,000 times it by. Now we have at the bottom here, I'll just write this like this, 1 over 100 minus 1. Over 100. So yes, you could just write it all like that. As you could see here, this becomes 99, and then this right here, 100 cancels with two of these zeros. Like that, so two of the zeros are gone. So what we end up having is plus 23, right here, 23 over 10, plus now, we have 17 over, there's going to be a 10 times 9. Yeah, it's over 10 times by 99. In other words, we're going to have a 99 with 0 or 990. So let's go further and see what we could do. So we want to get the same common denominator, so we're going to have to multiply this by 99 over 99, like that. So this equals to, I'll just put a, like this, underneath 9-9-9-0, so 10 times 9-99, and now I'll just put a note and then write this down separately. Just do this all by hand. So note, Yeah, note here we'll have 23 times 99, so this equals 2, let's put it times, so 9 times 3 is 27, put a 2 there, 9 times 2 is 18, add the 2 is 20, just do a multiplication by hand, and then we have a 0 like that, and then 9 times 3 is again, 7, we're going to get the exact same thing, 9 times 23 is 2 of 7, but put the extra 0, now we can just add these up. Yeah, so adding these up, what we end up having is a 7, plus a 2, plus a 2, like that. So what we have here is 2277 plus 17, so there's another 17 there, let's just add it there, as well, let's do addition by hand. 7 plus 7 is 14, and then we have to add 1 there, so 1 plus 7 is 8, and then plus another 1 is 9. Yes, we just carry that one above, so now 2294, like that. So let's write this down here, let's put this in a bubble, like that, and we have here, yes, this whole thing, equals 2, 227 plus 17, that's 2294, divide this by 990. So we could stop here, but my calculus book goes even further, because you could notice that are both even numbers, so we can divide those by 2. So we just factor out of 2, factor out of 2 like this. So 9-90, divide that by 2, that's just going to be, well, if it's 500, that's going to be, 500 times 2 is 1,000. So it's a minus 5 times, yeah, it's minus 10 divided by 2. So it's going to be instead of 500, 4, 4, 5. So that 400 times, actually not 4-4 -495, so yes. So 500, yeah, 500 times 2 is 1 ,000, so we need to subtract a 10 from it. So in other words, minus 5 here times by 2. So there it is. And then here, let's see if we could do this one in my head too. This one's an easier one here. We can just take it out. Yeah, this number is similar actually. So, 23, if you think of 2300 divided by 2, all right, just quickly just write this down to get my brain going. So, 2, 300 divided by 2. Think of this one, you divide by 2 would become 1-1-5-0, so 150 times 2 is 300 there. We just have to subtract another 10 from it. I mean, not 10, we need to subtract another 6 from it to get this 9 -4, so minus 6. This one's going to be minus 3, so this is going to be 1-1-4 -7. Anyways, I'll put that for brainstorm. Storming. So, yes, we have here 1-1-4 -7, like that. Instead of using a calculator, so this cancels, and we're left with 11-47, and the series is convergent because R is less than 1. It's... or absolute value of r is less than 1, and it is equal to 1 over 10 squared or 1 over 100, and this is our ratio. There it is 1147 over 495.