 An important concept in higher mathematics is the change of a quantity. We define the change of a quantity to be the difference between the ending amount and the starting amount. For example, suppose that at the start of a vacation you had $1,000, and later you had $20, by how much did your money change? So definitions are the whole of mathematics, all else is commentary. The change is the difference between the ending amount and the starting amount. And so the amount of change was the ending amount minus the starting amount. Now the ending amount was $20. The starting amount was $1,000, and so our difference $20 minus $1,000. And very important, wherever you have them always include units. So here the ending amount is $20, the starting amount is $1,000, and so the difference is minus $980. One important idea to keep in mind, when speaking about amounts of change, choose one. Either include the sign or omit the sign and include a direction. So in this case we might say your money changed by minus $980, where we've included the sign. Or in this case, since the amount of money you had decreased, you might say your money decreased by $980, where we've omitted the sign. And this leads to a very important rule, never include a signed number with a direction. Now the absolute change is useful, but the average rate of change is often more important. And so we define the average rate of change to be the change of a quantity divided by how long that change took. So at the start of the vacation, you had $1,000, five days later you had $20. What was the average rate of change? So definitions are the whole of mathematics. All else's commentary, our rate of change, is the change in the quantity divided by how long the change took. Now we already determined the change was minus $980. And since it took five days, our rate of change is the change in quantity divided by how long that change took. That's minus $980 divided by five days. We'll divide our numbers minus $980 divided by five. That's minus $196. Our units will also be divided, giving us dollars per day.