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Narrator: In a previous video
when we were looking at a very simple case
of compounding interest, we got the expression
(1+1/n)^n and the way we got this,
we saw an example where a loan shark is charging
100% interest and that's where this 1 is,
and then if they only compound once in the year,
so it's 100% over the year, then n is 1.
So, you get 1+100%/1^1,
you're going to have to pay back
twice the amount of the original amount of money.
If n is 2, (1+1/2)^2, gets you 2.25
If you compound half the interest,
so 100%/2, but you compound it twice.
Then we kept going and going and going,
we saw interesting things happen.
I want to review that right over here
using this calculator.
I want to see what happens as we get larger
and larger and larger n's.
In that last video we went as high as n=365
and it seemed to be approaching a magical number,
but now let's go even further.
So, let's type in ... let's throw some really large numbers here.
1+1/1,000,000 so that's a million to the millionth power.
(1+1/1,000,000)^1,000,000
Did I get the right number of zero's?
Yeah, that looks right.
Before I even press enter, which is exciting,
let's just think about what's going on here.
This part that we have here is that
n gets larger and larger, it's getting closer
and closer to 1, but never quite exactly 1.
This is 1 and 1 millionth.
So, it's very close to 1, but not exactly 1.
We're going to raise that thing to the millionth power
and normally when you raise something
to the millionth power, that's just going to
be unbounded, just become some huge number,
but there's a clue that 1 to the millionth power will just be 1.
If we're getting really close to 1,
well maybe this won't just be some unbounded number.
When we calculate it, we see that that's the case.
It's 2.71828 and just keeps going.
Now, let's go even higher.
Let's take it ... let's do 1+1/ and actually
I can now use scientific notation.
Let's just say (1+1/1'10^7)^1'10^7,
so what do we get here?
So, now we went 2.718281692.
Let's go even larger.
Let's get our last entry here.
Let's go, instead of the 7th power,
let's go to the eighth power,
so now we're (1+1/100,000,000^100,000,000)
I don't even know if this calculator can handle this
and we get 2.71828181487
and you see that we are quickly approaching,
or maybe not so quickly, we have to raise this
to a very large power, to the number e.
The number e in our calculator.
You see we've already gotten 1, 2, 3, 4, 5, 6, 7 digits
to the right of the decimal point
by taking it to the 100 millionth power.
So, we are approaching this number.
We are approaching, so one way to talk about it
is we could say the limit, as n approaches infinity.
As n becomes larger and larger, it's not becoming unbounded.
It's not going to infinity.
It seems to be approaching this number
and we will call this number,
we will call this magical and mystical number e.
We'll call this number e and we see
from our calculator that this number
and these are kind of, these are almost
as famous digits as the digits for Pi,
we are getting 2.7182818 and it just
keeps going and going and going.
Never, never repeating, so it's an infinite
string of digits, never, never repeating.
Just like Pi. Pi, you remember, is the ratio
of the circumference to the diameter of the circle.
e is another one of these crazy numbers
that shows up in the universe.
And in other videos on Econ Academy
we go into depth, why this is so magical and mystical.
Already this is kind of cool.
That I can take an infinite ...
If I just add 1 over a number to 1
and take it to that number and I make that number
larger and larger and larger, it's approaching
this number, but what's even crazier about it
is we'll see that this number,
which you can view, one way of it,
it is coming out of this compound interest.
That number, Pi, the imaginary unit
which is defined as that imaginary unit squared
is a negative 1,
that they all fit together in this magical
and mystical way
and we'll see that again in future videos.
But just for the sake of e,
what you could imagine what's happening here
is going to our previous example
of borrowing $1 and trying to charge 100% over a year,
when our n was 1, that means you're just
charging over 1 period.
When n is 2, you're charging over 2 periods
and then compounding, or you're compounding over 2 periods.
When n is 3, you're compounding over 3 periods.
When n approaches infinity,
you could view it as you're continuously compounding
every zillionth of a second.
Every moment you're compounding
in a super small amount of interest,
but you're doing it, essentially you're approaching
an infinite number of times
and you get to this number.