 Now, time is a continuous quantity, but it's convenient to treat it as a discrete quantity. So for example, your paycheck comes once every two weeks, or a bill is paid once a month, or the bank pays interest on alternate leap years. And by treating time as a discrete quantity, we can talk about the nth time period. For example, the first by-weekly pay period, or the sixth month a bill is paid, or the last time the bank paid interest. Now, to avoid having a zeroth time period, the interval from n minus one to n is considered the nth time period. For example, suppose we want to find the interval corresponding to the fifth year where t is measured in years. So the fifth year corresponds to the interval between t equals five minus one, that's four, and t equal to five. And we can write this as bracket for five using our mathematical interval notation. Strictly speaking, we should not have brackets at both the four and the five, because that would include both endpoints, though in practice, including or excluding the endpoints, does not normally affect our computations. So let's consider, if an amount k is invested, the amount function ak t gives the amount that the investment is worth after time t. Since ak t depends on both t and k, it can be difficult to analyze, so it's convenient to consider the accumulation function a of t. We define a of t to be a one of t, that's the amount returned at time t if one dollar is borrowed or invested. Usually ak of t is k times a of t, in other words, every dollar gets multiplied by whatever that principal amount k is, but there are common situations where it isn't. For example, the most common might be where you get higher interest rate if you deposit more money. To maintain focus on the mathematics, we'll make the following simplification, unless otherwise specified, we'll assume that our principal k times the accumulation function a of t does actually give us the amount function ak t. So let's consider simple interest. Remember that if an amount k is invested at simple interest with a rate of s per time unit, then ak t is k times one plus st. So if k equals one, we get the accumulation function one plus st. Let's begin by trying to find a of zero. We note that a of zero is the amount when one dollar is invested for t equals zero time. Since no time has elapsed, this should be one dollar, so a of zero is equal to one. Strictly speaking, we don't have to have that. You could borrow money for zero time and still have to pay interest on it, but we'll always assume that a of zero is equal to one as it actually makes our computations easier. So let's think about this. The amount of interest between two time periods is just going to be the difference between the amount functions at those two time periods, and this is just the amount by which the balance changed during the interval from t1 to t2. The effective interest rate for this period is that difference divided by this amount at the start. And it's worth noting that this is based on the accumulation function and not on the actual amount. However, if we are assuming, as we generally are, that akt equals ka of t, then the usual definition of the interest rate would give us the difference between the amounts at t1 and t2 divided by the actual amount at t1. And if we simplify, we get... And finally, the effective interest rate during the nth time interval would be where we remember that the nth time interval is the time between t equals n minus 1 and t equals n, and we'll use the notation. Now before we continue, it's worth emphasizing one important fact. There's a lot of formulas to remember. A good mathematician or a financier doesn't rely on them. The concept of effective rate over a time interval is the ratio between the change and the starting amount. Everything else is interpretation of the question. And the two things to remember, computers can calculate, but they can't interpret, and no computer ever lost its job because of a missed interpretation. So suppose we borrow 1,000 at simple interest with a rate of 5% per year, find the effective interest rate during the first year and the third year. So remember the first year is the period between t equals 1 minus 1, 0, and t equals 1. We note that our amount function is given by where t is measured in years, and so we find the amount at time 0 and at time 1, and so the amount of interest will be the difference between these two values. And since this was the interest on the amount at time 0, $1,000, the effective rate is, which is 5%, which we should have expected. All right, so maybe sometimes we'll use a formula. So during the third year that runs from t equals 2 to t equals 3, we compute our effective interest rate according to the formula. That's the accumulation function at 3 and at 2 divided by the value at 2. Now since our accumulation function for simple interest is 1 plus 0.05 t, we find a of 3 and a of 2, and we compute our effective interest rate, which is somewhat surprising because it's only 4.5%, even though our interest rate is 5%. And it's worth noting that with simple interest, the effective rate drops. And this is a problem for several reasons. Borrowers have less incentive to repay before the due date, and investors have less incentive to leave their money. So to incentivize both, we want the effective rate to be constant. So let's take a look at that next.