If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:11:38

Video transcript

let's say that you are desperate for a dollar and so you come to me the local loan shark and you say hey I need to borrow a dollar for a year and I tell you well you know I'm in a good mood I'm willing to lend you that dollar that you need for a year and I will lend it to you for the low interest of 100 percent per year so 100 percent 100 percent per year so how much would you have to pay me in a year where you're going to have to pay the original principal what I lent you plus 100 percent of that so plus one other dollar which is clearly going to be equal to clearly going to be equal to two dollars now you say well gee that's a lot to have to pay to have to have to pay back twice what I borrowed and there's a possibility that I might have the money in six months what kind of a deal can you get me for that mr. loan shark and I say oh gee if you're willing to pay back in six months then I'll just tap charge you half the interest for half the time so you borrow $1 and so in six months in six months I will charge you 50 percent interest 50 percent interest over six over six months this of course was one one year and so how much would you have to pay well you would have to pay the original principal what you borrowed the $1 plus fifty percent of that $1 so plus 0.5 oh and that of course is equal to one point five so that is equal to or a dollar fifty I'll just write it like this a dollar fifty now you say well gee that's I guess better but what happens if I don't have the money then if I still actually need a year and I say well 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 can think about it that I'll just lend that amount that you need for you for another six months so we'll lend that out we'll lend that out for another six months at the same interest rate at fifty percent for the next six months and so then you'll owe me you only the principal a dollar fifty a dollar fifty plus fifty percent of the principal plus seventy-five cents plus seventy-five cents and that gets us to two dollars and twenty-five cents so that equals two dollars and twenty-five cents or another way of thinking about it is to go from one dollar over the first period you just multiply that times 1.5 if you're going to grow something by 50% you just multiply it times 1.5 and if you're going to grow it by another 50% you can multiply by one point five again so one way of thinking about is that 50% interest is the same thing as multiplying by one point five multiplying by one point five so if you start with one and multiply one point 5 multiplied by one point five twice this is going to be the same thing two dollars twenty-five is going to be 1 multiplied by 1 point five twice 1 point 5 multiplied twice is the same thing as 1 point 5 squared and you can see the same thing right over here this is the same thing a hundred percent is the same thing as multiplying as multiplying by two as literally multiplying by 1 plus 1 so this is multiplying by 2 so you could view this right over here you could view this as 1 times 1 1 times 1 to the first one I sorry 1 times 2 I should say 1 times 2 to the first power because you're only doing it over one period over that year and you say once again where is that - well if someone's asking for 100% that means over the period you're going to have to pay twice you have to pay the principal plus 100% you're going to pay twice what you originally borrowed if someone's charging you 50% over every period you're going to have to pay whatever you borrowed so that's kind of the 1 part plus 50% of it so 1.5 times what you borrowed so you multiply times 1.5 every time and if you wanted to see how this actually related to the interest you could view this as so this right over here is equal to 1 times times the interest part is 1 plus 100% divided by 1 period to the first power and this seems like a crazy way of just try rewriting what we just wrote over here writing 1+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 hundred percent for the year we divided it into two periods two six-month periods each of them at 50% 1 plus 100% over 2 is the same thing as 1 point 5 and then we compounded it we compounded it over 2 periods and actually let me do that two periods into a different color so this the periods let me do in this orange color right over here and you might start to see a pattern forming so let's say well gee maybe I have the money back in and you don't really like this because this is $2 25 that was more than the original $2 so you say well what if I if we do this over every 12 months I say sure we got a program for that so after every 12 months after every 12 or after every month I should say I'm just going to charge you I'm going to charge you 100% divided by 12 a interest so this is equal to 8 and 1/3 percent and taking an or or get taking the having to pay back the principal plus 8 and 1/3 percent that's the same thing as multiplying times one point zero eight three repeating so after one month you would have to pay one point zero eight three repeating after two months after two months and this isn't to scale that actually looks more like more than two months it's not completely at scale after two months you're gonna have to multiply by this again so times one point zero eight three repeating and so that would get you to one point zero eight three repeating squared and if you went all the way down 12 months so let me get myself some space here so if you went all the way down twelve months so let me just so twelve months or I should say from the beginning twelve months oh another ten months so 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 have to keep re borrowing and I kept compounding that interest well you're going to have to pay one point zero eight three to the well this is for one month so you could view this as to the first power this is for two months so you're going have to pay this to the 12th power we have compounded over twelve periods eight and 1/3 percent over twelve periods or if you wanted to write it in this form right over here this would be the same thing as our original principal our original principal x times 1 1 plus 100% divided by 12 so now we've divided our 100% into twelve periods and we're going to compound that 12 times so we're going to take that to the take that to the twelfth power so what is this going to equal to this business right over here so we can get a calculator out for that so get my ti-85 out and so what is this going to be equal to well we could do a couple of way and this is one point zero eight three let me write three this is repeating right over here let's get our calculator out so we can do it a couple of ways so let me just write it this way you're gonna get the same value and I don't have to rewrite this one here I just did that there to kind of hopefully you see this the kind of the structure in this expression so one plus 100% is the same thing as 1 so 1 divided by 12 to the 12th power to the 12th power 2.61 3 I'll just round so approximately approximately two point six one three now you say well this is an interesting game you've almost forgot about your financial troubles and you're just intrigued by what happens if we keep going this if we you know here we compounded just we have one hundred percent over a year here we do 50 percent every six months here we do a twelfth of a hundred percent eight and 1/3 percent every 12 months we get to this number what happens if we did it every day every day so 5 bar to one dollar I said well gee I'm just going to each day I'm just going to charge you I'm just going to charge you one 365th of a hundred percent so one hundred percent divided by 365 and I'm going to compound that 365 times and so you're you're curious mathematically and you say well what do we get then what do we get what do we get after a year well you have your original principle let me scroll over a little bit more to the right so we have more space you're going to have your original principle times 1 plus 100% divided by not 12 now we've divided 100% into 365 periods 365 periods and but we're going to compound it remember every time we have to multiply by 1 plus 100% over 3 and 65 every day that the loan is not paid so the 360 fifth 360 fifth power and you say oh gee you know taking some of the 365th power that's going to give me some huge number but then you say well actually maybe not so bad because 100 percent divided by 365 is going to be a small number this thing is going to be reasonably close to 1 and obviously we could raise 1 to whatever power we want and we don't get anything crazy so let's see where this one goes let's see where this one goes so this is the same thing as 1 plus 100% is the same thing as 1 divided by 365 to the 360 fifth power and we get to point seven one four five six let me let me put it over here so then we get so this is approximately equal to approximately equal to two and you know this approximate is a very it's it's a very precise approximation but two point seven but my calculators precision is it only goes so far two point seven one four five six seven five and it keeps going on and on and this is really really interesting it looks like we as we take larger and larger numbers here it doesn't just balloon into just some crazy ginormous number it seems to be approaching some magical and mystical number and it is in fact the case that if you were to just take larger and larger if you were to take your hundred percent 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 burr II and you could see it right over here in your calculator they have this this e to the X and 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 and you see just already raising some doing one plus one over three hundred sixty-five to three hundred five power we got pretty we got we're starting to get really really really really close to e and I encourage you to try this with larger and larger numbers and you're going to get closer and closer to this magical mister you almost wouldn't mind paying the loan shark $1 because it's such a beautiful number