Crash Course Chapter 3: Exponential Growth

In the Crash Course, we will learn a few foundational Key Concepts. None are more important than exponential growth. Understanding this will greatly enhance our chances to form a better future.

Here’s a classic chart displaying exponential growth – a chart pattern that is often called a “hockey stick.” We are charting an amount of something over time. The only requirement for a graph to end up looking like this is that the thing being measured grows by some percentage over each increment of time.

The slower the percentage rate of growth, the greater the length of time we’d need to chart in order to visually see this hockey stick shape.

Another thing I want you to take away from this chart is that once an exponential function “turns the corner,” even though the percentage rate of growth might remain constant and possibly quite low, the amounts do not. They pile up faster and faster.

In this particular case, you are looking at a chart of something that historically grew at less than 1% per year. It is world population, and because it is only growing at roughly 1% per year, we need to look at several thousands of years to detect this hockey stick shape. The green is history and the red is the most recent UN projection of population growth for just the next 42 years.

Certainly by now, math-minded folks might be starting to get a little uncomfortable here, because they might feel that I am not presenting this information in a classical or even accurate way.

Where mathematicians have been trained to define exponential growth in terms of the rate of change, we are going to focus on the amount of change. Both are valid; it’s just one way is easier to express as a formula and the other is easier for most people to intuitively grasp.

Unlike the rate of change, the amount of change is not constant; it grows larger and larger with every passing unit of time, and that’s why it is more important for us to appreciate than the rate. This is such an important concept that I will dedicate the next chapter to illustrating it.

Also, mathematicians would say that there is no “turn the corner” stage of an exponential chart, because this is just an artifact of where we draw the left hand scale. That is, an exponential chart always looks like a hockey stick at every moment in time, as long as we adjust the left axis properly.

But if you know the limits, or boundaries, of what you are measuring, then you can fix the left axis, and the “turn the corner” stage is absolutely real and vitally important.

This is a crucial distinction, and our future depends on more of us appreciating this.

For example, the total carrying capacity of the earth for humans is thought to be somewhere in this zone, give or take a few billion. Because of this, the “turn the corner” stage is very real, of immense importance to us, and not an artifact of graphical trickery.

The critical take-away for exponential functions, the one thing I want you to recall, relates to the concept of “speeding up.”

You can think of the key feature of exponential growth either as the AMOUNT that is added growing larger over each additional unit of time, oryou can think of it as the TIME shrinking between each additional unit of amount added. Either way, the theme is “speeding up.”

To illustrate this using population: If we started with 1 million people and set the growth rate to a measly 1% per year, we’d find that it would take 694 years before we achieved a billion people. But we’d be at 2 billion people after only 100 more years, while the third billion would require just 41 more years. Then 29 years, then 22, and then finally only 18 years to add another, to bring us to 6 billion people. That is, each additional billion people took a shorter and shorter amount of time to achieve. Here we can see the theme of speeding up.

This next chart is of oil consumption, perhaps the most important resource of them all, which has been growing at the much faster rate of nearly 3% per year. So we can detect the ‘hockey-stick’ shape over the course of just one hundred and fifty years. And here, too, we can fix the left axis, because we know with reasonable accuracy how much oil the world can maximally produce. So, again, having “turned the corner” is extremely relevant and important to us.

And here’s the US money supply, which has been compounding at incredible rates, ranging between 5% and 18% per year. So this chart only needs to be a few decades long to see the hockey stick effect.

And here’s world-wide water use, species extinction, fisheries exploited, and forest cover lost. Each one of these is a finite resource, as are many other critical resources, and quite a few are approaching their limits.

And here is the world you live in. If it seems like the pace of change is speeding up, well, that’s because it is. You happen to live at a time when humans will finally have to confront the fact that our exponential money system and resource use will encounter hard, physical limits.

And behind all of this, driving every bit of every graph is the number of people on the surface of the planet.

Taken one at a time, any one of these charts could command the full attention of every earnest person on the face of the planet, but we need to understand that they are, in fact, all related and connected. They are all compound graphs, and they are being driven by compounding forces.

To try and solve one, you’d need to understand how it relates to the other ones that you see, as well as others not displayed here, because they all intersect and overlap.

The fact that you live here, in the presence of multiple exponential graphs relating to everything from money to population to species extinction, has powerful implications for your life and the lives of those who will follow you.

It deserves your very highest attention.

Let’s move onto an example that will help you understand these graphs a little bit better. Please join me for Chapter 4: Compounding Is The Problem.

Thank you for listening.


  • Sun, Jun 17, 2012 - 5:29pm



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    Joined: Jun 17 2012

    Posts: 2


    question about oil

    How do you know what is the maximum amount of fuel the earth holds? How do you know that the reason fuel is being extracted at a lower rate than before is not financial? maybe oil companies want to control the price by producing less oil ???

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  • Tue, Aug 14, 2012 - 9:42am



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    Joined: Sep 02 2011

    Posts: 1


    question about oil

    Hi Jacob,The important point is rather that these resources are NOT finite.
    Whatever amount of oil the earth holds or what rate we extract oil from the earth is not the point.
    The point is, we simple cannot keep over using "any" of our precious resources without causing permanent changes on how we develop as humans.

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  • Sat, Aug 31, 2013 - 1:14pm

    yummy jellybaby

    yummy jellybaby

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    Joined: Aug 31 2013

    Posts: 1


    Please revise :D

    P { margin-bottom: 0.21cm; }A:link { }

    exponential: The logarithm of a number is the power [ exponent ] by which the base must be raised to give the original number. Reverse the process and you get a silver star.
    Exercise: ( for a gold star ) Get a pad of community college ruled paper and plot it out. [ e exp(kt) ] where:
    e is the base of natural logarithms =2.71828182845904523536028747135266249775724709369995
    k is an arbitrary constant,
    t is time
    Hockey sticks and positioning of axes have noting to do with it.
    Rate is a mathematical term and must not be abused. The rate is the first differential of the displacement [Yes, we all know, but I don't propose to bring Werner Heisenberg into this – I'm not a pedant ] . It is the slope of the line if you will.
    Having said all that, the thrust of your thesis is essentially correct albeit just a small part of the story. You have my thanks and appreciation for your good work.
    Peace, love and power to your elbow

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