Compound Interest and e (part 2) Compounding 100% annual interest continuously over a year converges to e (2.71...)
Compound Interest and e (part 2)
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- 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.
- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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