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Compound interest and e (part 2)

Video transcript
Welcome back. In the last video, I was confusing you with compound interest, and now I will continue to do so. But the general notion is I threw out this 100% interest rate, and that was if we just say you pay me 100% of what you borrow after a year. And then we talked about what happens if instead you want half the rate for six months, but then I will compound the interest over the next six months. Then we said what happens if it's for every month or every day, and this was the case for every day. And if I charged you 2.7% every day, but I compound it for 365 days, it becomes-- I'm just using Excel here. Let me clear all of this. It becomes 1.027 to the 365th power. Oh, I'm using my Excel incorrectly. Oh, that's not right. Let's see, plus 1.027 to the 365th-- no, I'm making a mistake here. Let me make sure I got this 1.027 right. So if I takes 100% and I divide it by 365, so 100% is the same thing as 1 divided by 365, that is-- oh, it's 0.00. Sorry. So I'm not going to charge you 2% a day. Yeah, that seems high. I'm going to charge you 1.-- that is 0.00274. I'm charging you 0.2%, right? This would be-- this is the kind of one percentage place, so this is 0.2%. So I'm charging you 0.274% per day. I don't know why I put brackets around that. So if I were to compound that over 365 days, you take it to the 365th power, and what does that get me? Let's see, if I do plus 1.00274 to the 365th power, I get 2.7148. It equals 2-- that's how much you're going to owe me. If you just keep the money, you keep kind of reborrowing it every day, you'll owe me $2.7148 after the end of one year. Now, let's say that's not enough for you because this interest rate is so high, you want the option to pay it by the hour. You want this thing to compound every hour of the day. So let's see. Let's first of all figure out how many hours there are in a year. So let's see. In a year, there's 365 days times 24 hours per day, so there's 8,760 hours in a year. And then if we want to divide 100%, which is just 1, divided by that number, I could charge you 0.01-- what is that number? Yeah, it's 0.0114. So hourly compounding-- I should've been doing this in green since it's money. Hourly compounding, I will charge you 100% divided by the number of hours in a year, which equals 0.0114% per hour. So over a year, I would take it to this power, right? So let's say after one hour, you'll owe me 1.01-- sorry, it's 01% so it's 1.000114. That's how much you're going to owe me after one hour. So $1 plus a very small fraction of a cent. But then after another hour, you're going to owe that times that again. Because this'll be the new principal after an hour, so then you're going to owe that same fraction times it again, right? Then after three hours, you'll multiply it again. So after the total number of hours in the year, which is 1.000114, and there's 8,760 hours in a year, let's see what you'll owe me. So if I do plus 1.000114 to the 8,760th power: 2.71443. So now, at the end of the year after compounding roughly 8,700 hours, you'll owe me $2.71 and then some fractions of a penny. And I know you thought that these videos were about e and you were just learning about how to take advantage of someone in need, but there should be something interesting here that maybe you observed. When we started compounding, at first you owed me $2, just when I did one period, which is the whole year. Then it got to $2.25, and then it kept getting higher as we compounded shorter and shorter periods, but it seems to be approaching some number. It seems to be approaching, right? When I compounded every day, at the end of the year, you owed me $2.71 and some change. And then if I compounded every hour, which is 24 times as many compounds, you still owe me a very similar number. So it seems like it's gravitating towards this mystical number here, and that mystical number is e. So let's kind of formalize what I've been meandering around for a video and a half now. And I'll switch colors. So, in general, the amount of money you owed me at the end of the year was the amount you borrowed. Let me do that in blue. Let's call that the principal. That's not bright enough. The principal times 1 plus-- and what was the interest rate? It was 100%. 100% divided by the number of times you wanted to compound in the year. We'll call that n, right? And we raise that to the n power. So in the case of when there's only one compounding period, where you just borrowed it for the year, and the principal in our example was 1, right? So this is just 1 times-- the principal is just what you borrowed-- 1 plus-- 100% is the same thing as 1, right, or 1.00, and when there was one compounding period, we just did that, and you owed me $2 at the end of the year. And this is exactly what I've done in the last video and a half. I'm just formalizing it with a couple of variables. When we compounded it every month, it turned into this: the principal you borrowed was 1 times 1 plus 100% over 12 to the 12th power, which equaled-- let's see. I erased the numbers so I will redo it. So it's 1 plus 1 divided by 12 equals 1 plus-- I'm using Excel for those of you who have never seen it before-- 12 to the 12th power. That equaled 2.613. And when I compounded every day, I got-- the principal you borrowed was that times 1 plus 1 over 365 to the 365th power, and that equaled 2.71 and then some, some, something. So as you see, as I make n larger and larger in this original equation, I approach this magical number. I approach this magical number 2.71 something, something, something, and that magical number is e. And it amazes me that this-- and it never repeats. It's one of these transcendental numbers like pi, and later on in future videos, we'll see that it shows up all over the place. It shows up in random combinatorics, it shows up in complex analysis, and as we see here, and maybe most importantly, it shows up in compound interest. So, in general, what we could say is the limit. And the limit is just what happens as you approach something. The limit as n approaches infinity. And in our example, that's as we compound over smaller and smaller periods of time of 1 plus 1 over n to the nth power is equal to this magical number, and I'll do it in a bold color-- well, that's not that bold, but I'll highlight with another bold color-- is e. And that is equal to 2.71. I forget all the digits. It keeps going on. And it's really fun to experiment. Put in a crazy, huge-- put in like a million there. And if you put it there, you have to put a million there, too. And you'll see that the larger numbers you get, you just get closer and closer and closer to this number e. And a fun project would be to see how many digits of e you can get. But the fact that as you compound something over smaller and smaller periods it converges to this number, to me is pretty interesting. So with that out of the way, in the next video I'll show you how to figure-- and so, in the limit, as n approaches infinity, what are you doing? You're actually compounding continuously. You're compounding every zillionth of a second. And the fact that you can actually calculate an interest rate compounding every zillionth of a second to me is a fairly amazing result. But anyway, I'll see you in the next video because I've run out of time.