Last time, I started talking about Facebook. My goal is to make you say "whoa!" when you see how simple mathematical models can describe really complicated things in the real world. This post is a necessary digression on the way to understanding the Facebook example from last time. So you will have to postpone "whoa!" for one more day.

Since we are in the midst of an economic depression, let's talk about money. Suppose you have a savings account that yields interest on whatever you current balance is. And say that you deposit 100.41. We can write this as

[100 (1+0.004074)^1 = 100.82.]

This is called compounding. After the second month, you didn't just earn interest on your original 105.00. ]

We say that the annualized percentage rate is 5%. In other words, in terms of your bottom line, earning 0.4074% per month 12 times a year on an untouched account is that same as earning 5% one time at the end of the year.

This example so far has ben set in what we call discrete time. Compounding happens at distinct moments (for instance, the end of each month). But what if instead of happening at distinct moments, things happen at every moment in time? In case that made your brain explode, suppose that instead of earning 0.4074% once per month, you earned half of that, namely 0.2037%, twice a month. In this case, instead of ending up with 100.05 and an additional 1/2 cent. This is close to, but not the same as, our original example. And then imagine that you earned a quarter of the original interest rate, but compounding happened four times a month. Then you'd end up with 105.01. In short, you'd earn a whopping extra cent due to the continuous, rather than discrete, compounding.

A common way to describe a problem where something grows in continuous time is to use a differential equation. I'm about to write down a simple differential equation which I call "the most important differential equation in the world." My students know it by this name. If you ask them "what's the most important differential equation in the world," they will write down this equation. If you write down the equation and ask them the name, they'll tell it to you. The equation is

[frac{dx}{dt} = rx. ]

This equation can describe our continuously compounded interest problem. In that case, is the amount of money in your savings account, and it depends on time measured in months. The left side of the equation is just formal mathematical notation for change, known as a derivative. I'll post more on derivatives another time. For now, when you see , you should replace it in your head with "how quickly my bank balance is changing over time." Then there's an equals sign, which means that on the right hand side of the equation, we are going to put in the proper rule describing a savings account. The parameter is the rate of continuous compounding. So the equation says that at every moment in time, the way your bank account changes is that it accrues times the amount of money currently in the account. If we want to have an annualized rate of 5%, it turns out that we need to choose to be about 0.0040658, which is close to, but not the same as, the value of 0.004074 needed to earn 5% annual when compounding only takes place 12 times a year, rather than continuously. To restate: if your 105.

One advantage of having a differential equation description of something is that it is sometimes possible to solve the differential equation, that is, to write down a formula for the unknown function which in our case describes the amount of money in the bank account over time. If you start out with r=0.0040658ttt = 12x(12) = .

Believe it or not, we are now ready to move back to Facebook... tomorrow. But I swear, I am going to make you say "whoa!" and it will have been worth the wait.