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Current time:0:00Total duration:5:39

in a previous video when we were looking at a very simple case of compounding interest we got the expression 1 plus 1 over n 1 plus 1 over N to the nth power and the way we got this we saw an example where a loan shark is charging 100 percent interest and that's where this one is and then if they only if they only compound once in the year so it's 100 percent over the year that n is 1 so you get 1 plus 100 percent over 1 to the first power you're gonna have to pay back twice the amount the original amount of money if n is 2 1 plus 1/2 is 1 and 1/2 to the second power gets you 2.25 is if you compound half if you compound half the interest so 100 percent over - but you compound it twice and then we kept going and going going we saw interesting things happen and I want to review that right over here using this calculator I want to see what happens is we get larger and larger and larger ends in that last video we went as high as n is equal to 365 and it seemed to be approaching a magical number but now let's let's go even further so let's get let's let's type in and let's throw some really large numbers here 1 plus 1 over let's to a million 1 2 3 1 2 3 so that's a million to the to the millionth power 1 2 3 1 1 2 3 2 I get the right number zeroes yeah that looks right and 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 n gets larger and larger is 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 and they're going to raise that thing to the millionth power and normally when you raise something to the millionth powers that's just going to go be unbounded just become some huge number but there's a clue that well 1 to the millionth power would just be 1 if we're getting really close to 1 well maybe this this won't just be some unbounded number and when we calculate it we see that that's the case where it's 2.71828 and just keeps going now let's go even higher let's let's take it let's let's do one plus one over over and actually I can now use I can you now use scientific notation let's say 1 times 10 to the seventh power so that's literally this right over here is 10 million to the 10 millionth power so what do we get here so now we went to point seven one eight two eight one six nine two let's go even larger so let's get our last entry here so let's go instead of the seventh power let's go to the eighth power so now we're now we're one plus one over one hundred million to the hundred millionth power I don't even let's this calculator can handle this and we get two point seven one eight two eight one eight one eight one four eight seven 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 II the number II in our calculator you see we've already gotten to we've already gotten one two three four five six seven digits to the right of the decimal point by taking it to the hundredth millionth power so we are approaching this number we are approaching so one way to talk about it is we could say the limit 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 get we're getting 2.71828 one eight 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 he 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 of cool that I can take a I can take an infinite if I just make I just add one over a number to one and take it to that number and I make that number larger and larger and larger it's approaching this this this number but what's even crazier about it is we'll see that this number which you can view one way of it is coming out of this continuous compound interest that number PI the imaginary unit which is defined as that imaginary unit squared is a negative one that they all fit together in this magical 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 a dollar and trying to charge a hundred percent over year that when our n was one that means you're just charging over one period when n is two you're charging over two periods and then compounding or you're compounding over two periods what n is three compounding over three periods when n approaches infinity you could view it as you're continuously compounding every zillion of a second every every just every moment you're compounding just a super small amount of interest but you're doing it a essentially you're approaching an infinite number of times and you get to this number