The Crash Course - Chris Martenson [16]
In Figure 5.1, we’re graphing an amount of something over time. It could be the number of yeast grown in a flask of freshly squeezed grape juice every 10 minutes, or it could be the number of McDonald’s hamburgers sold each year. It doesn’t really matter what it is or what’s driving the growth; all that is required to create a line on a graph that looks like the curve seen in Figure 5.1 is that whatever is being measured must grow by some percentage over each increment of time. That’s it. Any percentage will do: 50 percent, 25 percent, 10 percent, or even 1 percent. It doesn’t matter: 10 percent more yeast per hour, 5 percent more hamburgers per year, and 0.25 percent interest on your savings account will all result in a line on a chart that looks like a hockey stick.
Figure 5.1 Linear Growth Compared to Exponential Growth
Linear growth is the dotted line; exponential growth is the solid line. The units on both axes are arbitrary; amount is on the vertical (or Y) axis and time is on the horizontal (or X) axis.
Looking at the figure a bit more closely, we observe that the curved line on the chart begins on the left with a flat part, seems to turn a corner (at what we might call the elbow), and then has a steep part.
A more subtle interpretation of Figure 5.1 reveals that once an exponential function turns the corner, even though the percentage rate of growth might remain constant (and low!), the amounts do not. They pile up faster and faster. For example, imagine that a long-ago ancestor of yours put a single penny into an interest-bearing bank account for you some 2,000 years ago and it earned just 2 percent interest the whole time. The difference in your account balance between years 0 and 1 would be just two one-hundredths of a cent. Two thousand years later, your account balance would have grown to more than $1.5 quadrillion dollars (more than 20 times all the money in the world in 2010) and the difference in your account between the years 1999 and 2000 alone would have been more than $31 trillion dollars. Where the amount added was two one-hundredths of a cent at the beginning, it was roughly equivalent to half of all the money in the entire world at the end. That’s a rather dramatic demonstration of how the amounts vary over time, but it gets the point across.
Now let’s look at an exponential chart of something with which you are intimately familiar that has historically grown at roughly 1 percent per year. It is a chart of world population; the solid part is historical data and the dotted line is the most recent UN projection of population growth for just the next 42 years.1
Again I want to draw your attention to the fact that the chart has a flat part, then a corner that gets turned, and then a steep part. By now, it is quite possible that any mathematicians reading this are hopping up and down because of what they might view to be an enormous error on my part.
A first point of departure is that where mathematicians have been trained to define exponential growth in terms of the rate of change, we’re going to concentrate here on the amount of change. Both are valid, it’s just that rates are easier to express as a formula and amounts are easier for most people to intuitively grasp. So we’re going to focus on amounts, even though this is not where classical mathematicians would train their logical eyes.
Unlike the rate of change, the amount of change is not constant in exponential growth; it grows larger and larger with every passing unit of time. For our purposes, it is more important that we appreciate what exponential growth demands in terms of physical amounts than whatever intellectual gems are contained within the rate of growth.
A second point of contention that I expect most mathematicians would vigorously dispute is the idea that there’s a turn-the-corner stage