Main content

### Course: Algebra 2 > Unit 8

Lesson 2: The constant e and the natural logarithm# 𝑒 as a limit

Sal continues the discussion on e, this time digging deeper into the mathematical definition of 𝑒. Created by Sal Khan.

## Want to join the conversation?

- so if the interest was lets say 7% instead of 100% would you put .07/n or instead of 1/n?(58 votes)
- Exactly! The general compound interest formula is (1 + r/n)^n, where r is the rate. Obviously 100% = 1 and 7% = 0.07, so you did a good job.(83 votes)

- What other applications are there for e?(57 votes)
- also if you ever see ln or ln() that is a reference to e as the base for log. Log base e of a number is called the natural log(21 votes)

- I don't understand what limits are when Sal talks about it at3:30. Can someone explain?(17 votes)
- Let's consider the function:

𝑓(𝑥) = [1 + (1/𝑥)]ˣ

If we plug in larger and larger values of 𝑥, we see that the value of 𝑓(𝑥) seems to be getting closer and closer to a certain value around 2.71.... It turns out, when we use an infinitely large value for 𝑥, we get the exact value of 𝑒. More succinctly, we can say that the limit of 𝑓(𝑥) as 𝑥 tends to ∞ is 𝑒. Essentially, the limit helps us find the value of a function 𝑓(𝑥) as 𝑥 gets closer and closer to some value. You will learn more about limits and a more rigorous definition later in Precalculus and Calculus.(33 votes)

- Will this work if n is equal to a very large number i.e 999,999,999,999,999 because i tested it and the results were more than 3 so is there like a limit for n?(8 votes)
- Interestingly Google's "hidden" calculator does the same thing. I'm pretty sure this is a result of the limited precision with which floating point numbers are stored.

In other words, since computer chips can't store an infinite number of digits (that would take an infinite amount of of memory) numbers for intermediate steps get approximated and this can lead to inaccuracies like the one you noticed.

So, yes this works better for larger numbers, but**only**if you keep**all**of the information!

For more information:

http://stackoverflow.com/questions/2100490/floating-point-inaccuracy-examples(23 votes)

- Sal references another video or videos at4:20. What are those videos?(12 votes)
- It might be covered in some video later in this sequence of logarithm-related videos, though I doubt it, but it is mentioned in Vi Hart's pi videos under Recreational Math -> Doodling in math class.(6 votes)

- Instead of using 100% in (1+100%/n)^n can we take any other interest rates? Will that reach e as well?(8 votes)
- If you use a number other than 100% for the infinite series you won't get
`e`

, no.(9 votes)

- are there any other magical numbers other than pi e and i(3 votes)
- There are hundreds of magical numbers in math, like φ (phi) also known as the golden ratio. It has numerous properties, but vi explains them better here: https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/spirals-fibonacci/v/doodling-in-math-spirals-fibonacci-and-being-a-plant-1-of-3.

Besides Phi, there is Graham's number, Tau, √2 and many more.(7 votes)

- does this mean that (1+1/∞)^∞=e?(3 votes)
- 1. ∞ is merely a concept, rather than something you can substitute into an equation.

2. lim x->∞ means x converges to ∞, but never reach ∞. x can reach 100, can reach 10^10, can also reach 10^10000000, but never reach ∞.(8 votes)

- I can't make an intuitive sense of why it happens. As x approaches infinity, 1/x should approach 0(since, 1/infinity = 0). (1+0)^infinity should be 1 because 1 raised to any power is 1. So, shouldn't (1+1/x)^x be 1 as x approaches infinity? Why 2.7182818...?(3 votes)
- You could just as well say that (1+1/x) is greater than 1, and (something greater than 1)^x goes to infinity as x goes to infinity, so the limit should go to infinity.

The answer is that you don't get to pick parts of the limit to evaluate first. The exponent is going to infinity and 1/x is going to 0*at the same time*. When you evaluate the limit your way, you're assuming the exponent is constant while 1/x goes to 0, which it isn't.(6 votes)

- So what actually IS e? He says what pi represents but never explains what e represents. All he said was that is was some cool number. plz help(2 votes)
- Most of the more fundamental properties of e are rooted in calculus.

The main one is this: consider the set of functions a^x, where a is some positive constant. So 2^x, 5^x, π^x, etc.

If you take a point on one of these functions (x, a^x) and draw the tangent line to the function there (that is, the line that touches the curve at the chosen point and nowhere else), the slope of the line will be ln(a)·a^x.

If we take the function e^x, then, the slope of the tangent line is ln(e)·e^x, or just e^x. The value of the function is equal to the slope of its tangent line. And e is the only choice of number that will cause this.(6 votes)

## Video transcript

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 Khan 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.