e and compound interest발음듣기
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e and compound interest
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출처 : 칸아카데미
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So you come to me the local loan shark, and you say hey I need to borrow a dollar for a year. I tell you I'm in a good mood, I willing to lend you that dollar that you need for a year. I will lend it to you for the low interest of 100% per year. 100% per year.발음듣기
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You're going to have to pay the original principal what I lent you plus 100% of that. Plus one other dollar.발음듣기
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Which is clearly going to be equal to $2. You say oh gee, that's a lot to have to pay to pay back twice what I borrowed.발음듣기
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I say oh gee, if your willing to pay back in 6 months, then I'll just charge you half the interest for half the time.발음듣기
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How much would you have to pay? Well, you would have to pay the original principal what you borrowed.발음듣기
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What I'll do is just say that okay, you don't have the money for me yet. I'll essentially ... we could think about it.발음듣기
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I will just lend that amount that you need for you for another 6 months. We'll lend that out.발음듣기
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We'll lend that out for another 6 months at the same interest rate at 50% for the next 6 months.발음듣기
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Another way of thinking about it is to go from $1 over the first period, you just multiply that times 1.5. If your going to grow something by 50%, you just multiply it times 1.5.발음듣기
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Multiplying by 1.5. If you start with 1 and multiply by 1.5 twice, this is going to be the same thing.발음듣기
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This is the same thing. 100% is the same thing as multiplying by 2. As we be multiplying 1 plus 1.발음듣기
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1 times 1 to the first ... I'm sorry 1 times 2 to the first power , because your only doing it over one period over that year.발음듣기
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You say once again where's that 2? Well, if someone is asking for 100%, that means over the period you're going have to pay twice.발음듣기
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If someone is charging you 50% over every period, you're going have to pay whatever you borrowed.발음듣기
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If you wanted to see how this actually related to the interest, you could view this as ... this right over here is equal to 1 times, the interest part is 1 plus 100% divided by 1 period to the first power.발음듣기
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Writing 1 plus 1, but you'll see that we can keep writing this as we compound over different periods.발음듣기
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Each of them at 50%. 1 plus 100% over 2 is the same thing as 1.5, and we compounded it over 2 periods.발음듣기
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This is $2.25. That was more than the original $2, so you say, well what if we do this over every 12 months. I say, "Sure. We got a program for that."발음듣기
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After every 12 months ... or after every month I should say, I'm just going to charge you 100% divided by 12 interest.발음듣기
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Having to pay back the principal plus 8 1/3%, that's the same thing as multiplying times 1.083 repeating.발음듣기
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And this isn't the scale that actually looks more than 2 months, but it's not completely at scale.발음듣기
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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?발음듣기
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If you wanted to write it in this form right over here, this would be the same thing as the original principal.발음듣기
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Our original principal times 1 plus 100% divided by 12. Now we've divided our 100% into 12 periods, and we're going to compound that 12 times.발음듣기
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I just did that there to kind of hopefully you'd see the kind of structure in this expression.발음듣기
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So approximately 2.613. You say well this is an interesting game you all most forgot about your financial troubles, and you're just intrigued by what happens if we keep going this.발음듣기
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If I borrowed a one dollar, and I'd say well gee I'm just going to ... each day I'm going to charge you charge you one three hundred sixty-fifth of a 100%.발음듣기
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So, 100% divided 365, and I'm going to compound that 365 times. You're curious mathematically.발음듣기
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You're going to have your original principal times 1 plus 100% divided by not 12. Now we've divided the 100% into 365 periods.발음듣기
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Every time we have to multiply by 1 plus 100% over 365 everyday that the loan is not paid. 365th power.발음듣기
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Then you say well maybe not so bad, because 100% divided by 365 is going to be a small number.발음듣기
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This is the same thing as 1 plus. 100% is the same thing as 1 divided by 365 to the 365th power.발음듣기
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Then we get ... This is approximately equal to ... this approximate is a very precise approximation, but 2.7 ...발음듣기
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It looks as if we take larger and larger numbers here, it just doesn't just balloon into some crazy ginormous number.발음듣기
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If you were to take your 100% 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.발음듣기
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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.발음듣기
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You see all ready raising some ... doing 1 plus 1 over 365 to the 365th power, we got pretty ... we're starting to get really really close to E. I encourage you try this with larger and larger numbers, and your going to get closer and closer to this magical mystery.발음듣기
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