Let's say that you are desperate for a dollar.
So you come to me the local loan shark,
and you say hey I need to borrow a dollar for
a year. I tell you I'm in a good mood, I willing
to lend you that dollar that you need for a
year. I will lend it to you for the low interest
of 100% per year. 100% per year.
How much would you have to pay me in a year?
You're going to have to pay the original
principal what I lent you plus 100% of that.
Plus one other dollar. Which is clearly going
to be equal to $2. You say oh gee, that's a lot
to have to pay to pay back twice what I borrowed.
There's a possibility that I might have the money
in 6 months. What kind of a deal could you get
me for that Mr. Loanshark?
I say oh gee, if your willing to pay back in
6 months, then I'll just charge you half the
interest for half the time.
You borrow one dollar, so in 6 months, I will
charge you 50% interest. 50% interest over 6 months.
This, of course, was 1 year. How much would you
have to pay? Well, you would have to pay the
original principal what you borrowed.
The one dollar plus 50% of that one dollar.
Plus 0.50, and that of course is equal to 1.5.
That is equal to $1.50. I'll just write it like this.
$1.50. Now you say well gee that's I guess better.
What happens if I don't have the money then?
If I still actually need a year.
We actually have a system for that.
What I'll do is just say that okay, you don't
have the money for me yet. I'll essentially ...
we could think about it. I will just lend that
amount that you need for you for another 6 months.
We'll lend that out.
We'll lend that out for another 6 months at the
same interest rate at 50% for the next 6 months.
Then you'll owe me the principal a $1.50 plus
50% of the principal, plus 75 cents. Plus 75 cents, and that
gets us to $2.25. That equals $2.25.
Another way of thinking about it is to go from
$1 over the first period, you just multiply
that times 1.5. If your going to grow something
by 50%, you just multiply it times 1.5.
If your going to grow it by another 50%, you can
multiply by 1.5 again.
One way of thinking about it that 50% interest is
the same thing as multiplying by 1.5.
Multiplying by 1.5.
If you start with 1 and multiply by 1.5 twice,
this is going to be the same thing.
$2.25 is going to be 1 multiplied by 1.5 twice.
1.5 multiplied twice is the same thing as 1.5 squared.
You can see the same thing right over here.
This is the same thing. 100% is the same thing
as multiplying by 2. As we be multiplying 1 plus 1.
This is multiplying by 2, so you could do this
right over here. You could do this as 1 times 1.
1 times 1 to the first ... I'm sorry 1 times 2 to the first power , because
your only doing it over one period over that year.
You say once again where's that 2? Well, if
someone is asking for 100%, that means over the
period you're going have to pay twice.
You're going have to pay the principal plus 100%.
You're going have to pay twice what you originally
borrowed. If someone is charging you 50% over
every period, you're going have to pay whatever
you borrowed. That's kind of the one part plus
50% of it. So 1.5 times what you borrowed. You multiply
times 1.5 every time. If you wanted to see how
this actually related to the interest, you could
view this as ... this right over here is equal
to 1 times, the interest part is 1 plus 100%
divided by 1 period to the first power.
I know this seems like a crazy way of rewriting
what we just wrote over here. Writing 1 plus 1,
but you'll see that we can keep writing this
as we compound over different periods.
This one right over here, we can rewrite.
We can write as 1 times 1 plus 100%.
Here we took our 100% for the year, and we
divided into 2 periods. Two 6 month periods.
Each of them at 50%. 1 plus 100% over 2 is the
same thing as 1.5, and we compounded it over
2 periods. Let me do that 2 periods into a different
color. The periods, let me do in this orange
color right over here. You might start to see a
pattern forming. Let's say, well gee, I might
have the money back in ... and you don't really
like this. This is $2.25. That was more than
the original $2, so you say, well what if we do
this over every 12 months. I say, "Sure. We
got a program for that."
After every 12 months ... or after every month
I should say, I'm just going to charge you 100%
divided by 12 interest. This is equal to 8 1/3%.
Having to pay back the principal plus 8 1/3%,
that's the same thing as multiplying times 1.083 repeating.
After 1 month you would have to pay 1.083 repeating.
After 2 months ... and this isn't the scale
that actually looks more than 2 months, but
it's not completely at scale. After 2 months
your going have to multiply by this again.
Times 1.083 repeating, and so that would get you
1.083 repeating squared. If you went all the way
down 12 months ... let me get myself some space
here. If you went all the way down 12 months ... let me just.
I should way from the beginning 12 months, so
another 10 months. What's the total interest
you would have to pay over a year if you weren't
able to keep coming up with the money?
If you had to keep re-borrowing it.
I kept compounding that interest.
Well, you're going have to pay 1.083 to the ...
this is for 1 month. You could view this as to
the first power. This is for 2 months, so you're
going have to pay this to the 12th power.
We have compounded over 12 periods, 8 1/3% over
12 periods. If you wanted to write it in this
form right over here, this would be the same
thing as the original principal.
Our original principal times 1 plus 100% divided
by 12. Now we've divided our 100% into 12 periods,
and we're going to compound that 12 times.
We're going to take that to the 12th power.
What is this going to equal to? This buisness
over here. We can get a calculator out for that.
I'll get my TI-85 out. What is this going to be
equal to? We could do it a couple of ways.
This is 1.083 repeating. Let's get our calculator
out. We could do it a couple of ways. Let me write it this way.
Your going to get the same value. I don't have to rewrite this one.
I just did that there to kind of hopefully you'd
see the kind of structure in this expression.
1 plus ... 100% is the same thing as 1.
1 divided by 12 to the 12th power. 2.613, I'll just round.
So approximately 2.613. You say well this is an
interesting game you all most forgot about
your financial troubles, and you're just intrigued
by what happens if we keep going this.
Here we compounded just ... we have 100% over
here. Here we do 50% every 6 months. Here we
do a 12th of 100%, 8 1/3% every 12 months
until we get to this number.
What happens if we did every day?
Every day.
If I borrowed a one dollar, and I'd say well
gee I'm just going to ... each day I'm going to charge you
charge you one three hundred sixty-fifth of a 100%.
So, 100% divided 365, and I'm going to compound
that 365 times. You're curious mathematically.
You say well, what do we get then?
What do we get after a year?
You have your original principal. Let me scroll
over a little bit more to the right, so we have more space.
You're going to have your original principal
times 1 plus 100% divided by not 12. Now we've
divided the 100% into 365 periods.
365 periods. We're going to compound it.
Every time we have to multiply by 1 plus 100%
over 365 everyday that the loan is not paid.
365th power.
You say oh gee taking somebody to 365th power
that's going to give me some huge number.
Then you say well maybe not so bad, because
100% divided by 365 is going to be a small number.
This thing is going to be reasonable close to 1.
Obviously, we can raise 1 to whatever power we
want, and we don't get anything crazy.
Let's see where this one goes.
This is the same thing as 1 plus. 100% is the
same thing as 1 divided by 365 to the 365th power.
We get 2.71456. Let me put it over here. Then we get ...
This is approximately equal to ... this approximate
is a very precise approximation, but 2.7 ...
but my calculators precision only goes so far.
2.7145675 and it keeps going on and on.
This is really really interesting. It looks
as if we take larger and larger numbers here,
it just doesn't just balloon into some crazy
ginormous number. It seems to be approaching some
magical and mystical number. It is, in fact,
the case. That if you would just take larger and
larger ... if you were to take your 100% and
divide by larger and larger numbers, but take
it to that power, you're going to approach
perhaps the most magical and mystical number
of all. The number E. You can see it right over
here in your calculator. They have this E to the X.
I can do that, so E to the ... I'll raise it to
the first power so you can look at the
calculators internal representation of it.
You see all ready raising some ... doing 1 plus 1
over 365 to the 365th power, we got
pretty ... we're starting to get really really
close to E. I encourage you try this with larger
and larger numbers, and your going to get
closer and closer to this magical mystery.
You almost wouldn't mind paying the loan shark
E dollars, because it's such a beautiful number.