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𝑒 as a limit

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

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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.