 Suppose I have $20 and you need $20. Since I'm a decent human being, I lend it to you in the expectation that you'll return it. The principle is the amount I've lent you. Since you're a decent human being, at some point you'll return it to me. You might also give me some additional money, and this is known as the interest on the principle. Now, there's an important moral and ethical question. Should you have to pay interest? The answer is complicated, and it belongs in a philosophy or ethics class. More importantly, the basic financial transaction here is I give you some amount of money k, you return at a later point, some amount of money k, plus an additional amount m. While this is often interpreted as borrowing k and returning k plus m, it also describes investing, and it's important to realize there is no mathematical difference between borrowing and investing. So let's introduce some of the mathematics. Suppose the amount k is borrowed. The amount function akt is the amount that would be returned, principle and interest, at time t. We assume ak0 equals k, the amount borrowed. For example, if you borrow $20 today with the plan to pay me back $25 in one week, then a27 is equal to 25. But if you don't pay me back $25 in one week, you'll need to pay me back 50 in two weeks. So a2014 is equal to 50. So suppose you agreed to a27 equals 25, and a2014 equals 50. What about a21? This is the amount you'd owe if you paid me back one day after you borrowed the money. This would be, well, here's the problem. If it's all written down, it didn't happen. And in the financial and business world, this really means if it's all written down, it becomes a lawsuit. We want to avoid lawsuits. So we might phrase our borrowing terms a little bit more precisely. Suppose you borrowed $20 and agreed to return $25 if you paid me back within seven days. Find a23 and a200.001. A23 is the amount you'd pay me back if you returned the money three days after you borrowed it. Since that's within seven days, a320 is equal to 25. A200.001 is the amount you'd pay back if you returned the money 0.001 days, about 86 seconds after you borrowed it. Since this is within seven days, a200.001 is 25. It may seem unfair you'd have to pay back the full amount plus interest, even if you paid the money back almost immediately. But that's what was written down. Now, historically, this unfairness has been addressed by religious leaders telling people it's immoral, legislators passing laws that limit the power of the wealthy, or robbing banks. Instead of these work very well, what does work? Mathematics. And it's important to remember to fight injustice, quantify, and compute. Generally, we can compute the amount using a mathematical formula. And a common method is simple interest. In simple interest, let amount k be invested at simple interest rate s per unit of time. Then akt is k times 1 plus st. The interest is simple because it's based on only two things. The interest rate s, the principal k, and the time t. Now suppose $1,000 is invested at simple interest with a rate of 5% per year. How much will this become after two years, or after 30 months? We have the amount invested is 1,000. The interest rate is 5%, which we need to convert to a decimal. And so our amount function, a1000t, is 1,000 times 1 plus 0.05t. And we can compute after two years, a1000 of 2 substituting that in gives us 1,100. And after 30 months, a1000t substituting that in gives us 2,500. And at this point, it is important to keep in mind no computer was ever fired for making a mistake. So the computer makes it very easy to calculate things like a1000t, but if we did calculate this, we'd be wrong. So let's take this apart. The interest is 5% per year, so we must measure time in years. And so 30 months must be converted to an amount in years. And so to convert, there's three ideas to keep in mind. You can multiply anything by 1. 1 is a quantity divided by its equal, and units act like algebraic variables. And so we convert. 30 months is the same as 30 months. Now we want to remove this factor of months, so we'll divide by it. We'll put month in the denominator. We want to express our final answer in terms of years, so we'll put years in the numerator. And if I multiply, it's got to be multiplied by 1. Well, 1 year is the same as 12 months. So we'll put those numbers there. And completing our multiplication gives us 30 months is the same as 2.5 years. And so we need to calculate a1000 of 2.5, which works out to be 1,125.