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# Compound interest and e (part 3)

Video transcript

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.