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These videos were supposed to be about e, but compound interest was the natural way to introduce it. And since I've already gone down this compound interest track, let's just keep going so that I can put these videos in both my mathematics and my finance playlists. And actually, before I continue, I just took the Excel and I just wanted to show you how it converges to e. So this is how many periods I'm compounding. So in this formula right here, this is n. This column a that I have, right here. And then this is essentially what happens when I evaluate 1 plus 1 over n to the n power. You can actually look at the formula. My Excel formula. It's 1 plus 1 over this blue cell, to that blue power. That's just that formula. And I did it for a bunch of numbers, and I just double the numbers every time. So I go to a very high number very quickly. And you see that very quickly it converges to this number: 2.7183. And this number just keeps going on and going on. And that number is e. And what's interesting is, in fact, you go to Google, and you type in search on e. They give you the number. Because Google is actually a calculator. And you could look up e other places. And I think there are sites that calculate e to arbitrary decimals. There's actually some people who, for whatever reason, they see numbers like pi and e. They can see them. And they can recite the digits to arbitrary decimal places. And I think the more you realize-- the more you see where e pops up in the world, and pi, and imaginary numbers-- I think you'll realize that these numbers, I think they're somehow scratching the surface of something very deep. I mean we're just touching on them because they pop up everywhere in completely different places in the universe. And they're all almost magically related. And I'll show you that over the course of the next videos. I think this will really give you motivation if you're in the mood for starting a new cult, perhaps. Well, anyway, let's continue with the compound interest before, because we need to be able to finance our cult. Or maybe our cult is financed by giving other people financing. By being loan sharks. Well, anyway, the example I just gave was the situation in which I'm charging 100% interest. So let's generalize it a little bit, to the situation where I'm charging some other percentage of interest. So let's say I'm charging r percent. Oh, my rate-- right. My rate is r percent. That's essentially what I'm going to charge. The rate will be r, as a decimal. So it'll be 10r%, if I were to write it. But as a decimal-- so, for example, if I'm charging 25%, my rate would be .25, and I would write that as 25%. Just to clarify. So what would you owe me at the end of the year depending on how often I compounded? Well, just going back to what we said before, you have your initial principal, which in every example we've done so far was $1.00, but I'll just write p so we can get general. And then the amount that you owe me after one compounding period is 1 times the principal, plus my annual interest rate-- so in this case it's r-- divided by the number of times I'm compounding. So that's n again. And I'm raising that to the nth power. And just so this makes sense to you in the terms we thought about, when n is-- let's say, r is equal to 10% and n is equal to 2. And this is what you owe me at the end of a year-- so if n is equal to 2, that means we're compounding twice over a year. Or that we're charging essentially half of this rate every six months. So if you were to borrow, let's say, p is equal to, I don't know, $50.00. That's how much you initially borrow from me. So all this formula says is, after every period you will owe-- so, after one period, how much will you owe? This is how much you borrowed. Then after the next period, after six months, you'll owe me this p, $50.00, plus the interest rate divided by the number of periods in the year. So this is essentially kind of an annual interest rate, but if I'm charging you every six months, I'm going to divide it by 2. So it's 10% over 2 times $50.00. And this is the same thing as what? This is the same thing as 50 times 1 plus our rate divided by the number of times we compound, right? And this is after 6 months, as I highlight right here. And then after another 6 months, I'm going to take this number, and then I-- you know, let's call this x-- and I'm going to charge you x plus 10% over 2 times x. Or I'm going to charge you x times 1 plus 10% over 2. And this was x. So after a full year, I'm charging you $50.00 times 1 plus 10% over 2, times 1 plus 10% over 2. And that's the same thing as-- well, going in the opposite direction-- as $50.00 times 1 plus-- we could write this is a decimal-- .1 over 2 to the second power. Right? I'm just multiplying this times itself. So, in general, when I compound-- and now I think you'll see the relationship between what I just wrote out there-- and experiment with some numbers on your own if you're getting a little bit confused, or if I'm going a little bit too fast. Hopefully you see that this is the same thing as this. So let's see what happens as I try to compound continuously, or as n approaches infinity. Let me get some space. So the amount that you owe me-- so we can call that final payment after a year-- payment is equal to the amount you're borrowing-- I don't like this color-- times 1 plus the interest rate over n to the nth power. Well, let's just make a substitution. Let's say that-- and I think you'll understand why I'm doing the substitution-- let's say that r over n-- and let's say I want to find the limit as n approaches infinity, as I compound continuously. So the limit as n approaches infinity of 1 plus r over n to the nth power. Let's make a substitution. Let's say that 1 over x is equal to r over n. If 1 over x is equal to r over n, what is this? Let's see, that means that n is equal to xr. Right? I just crossed multiplied. And then, if n is equal to xr, what's-- n approaching infinity is the same thing, assuming that r is constant, that's the same thing as x approaching infinity. Or we could view it the same way the other way around. x approaching infinity is the same thing as n approaching infinity. And so we can make the substitution here, and we get-- this is the same thing as the limit as x approaches infinity of what? 1 plus-- we said r over n is the same thing as 1 over x, we just defined it that way. To the nth power. But we said n-- this substitution comes into this. So n is just equal to xr. Remember, r is just a constant, right? And this is the same thing as the limit, as x approaches infinity, of 1 plus 1 over x to the x. And then when you multiply exponents like that, that's the same thing as that whole expression to the r power, right? And this r is a constant, right? We're not taking the limit on r, or anything like that. So this is the same thing as the limit, as x approaches infinity, of 1 plus 1 over x to the x, and all of that to the r power. And then what did we figure out that this was in the previous two or three videos? Well, this is equal to e. So this is equal to e to the r power. So if I charge an interest rate of 10%, and I want to compound it continuously over one year, at the end of one year, you're going to owe me e to the 10% power times the original principal. So we said that this is equal to e to the r, so p times this is equal to p. There's a p the whole time. I'll do it in the blue so you remember. I dropped the p somewhere along the way. There should have been a p here. But it's just a scaling factor. Should be a p here. I could have taken the p out, because it's a constant, put the p here, and it would have stayed there. But anyway, I'm about to run out of time and I will see you in the next video where we know how much we're going to pay if we continuously compound for a year. Let's see what we'll pay if we continuously compound for multiple years, at a rate of r percent per year.