University of Colorado Professor Albert Bartlett was widely viewed as a pioneer on explaining the arithmetic involved in the world’s population, energy, and sustainability issues. He asserted that “the greatest shortcoming of the human race is our inability to understand the exponential function.” Having previously been a high school math teacher, I can confirm that statement with great remorse. It doesn’t have to be this way, though.

I would recommend watching Professor Bartlett’s lecture on exponential growth https://www.youtube.com/watch?v=TBtW51D_q2Q , but to give a disclaimer it’s a bit dated. He does a great job of explaining how to conceptualize exponential growth by calculating the doubling time. A neat trick that can be used to calculate the doubling time is by taking 70 divided by the rate of growth, but I digress. Discussing what exponential growth is unfortunately requires delving into the idea of linearity.

Linear growth or decay is achieved through constantly adding one number to the previous one. We graph these functions using the infamous “y = mx + b,” and if you haven’t run away from or decided to read a different blog post by now, then I must say you make me proud. No worries, there will be no fractions here. Fundamentally, linear functions are just addition. Unfortunately, this is the conception of growth most people are stuck looking at the world through. The vast majority of modeled growth relationships, however, utilize exponential functions and multiplication.

The percentage increase or decrease happening for exponential functions involves each iteration or incremental development multiplying the previous term by 1 + r, where “r” is the rate of growth or decay. Doing so creates a snowball effect that escalates quickly, and is why exponential functions are so different. Here’s where things get interesting.

Sound is not only measured but it also perceived logarithmically (read “exponentially”). Not everyone realizes that when you increase the volume on your music, even if your stereo or phone displays a linear increase from 10 to 11, that is an exponential scale of difference happening. Our perceptions register exponential changes better than linear ones. Increasing your volume from 10 to 12 is like doubling the amount of power being output by your speakers. This same phenomenon of linear perception and exponential reality occurs when we dim or brighten the lights in our home, and when we drive the square root of 2 times faster in our car we’re actually doubling our kinetic energy. Our world is exponential!

All of this seems counterintuitive to our basic understanding of arithmetic, which is why more people need to be engaged, and understanding the exponential nature of not only how we perceive the world but how it functions is what is underlying many of the issues with population growth, energy, and sustainability. To remedy this, I have created a manipulatable graph on Desmos to help anyone reading this fully understand the difference between linear and exponential functions if they choose. Personally, I highly recommend pressing the play button and letting the graph go wild. Most importantly, having armed your Brian with mathematics, I invite you to watch how you conceive of growth and to be diligent when looking at information.

Desmos link: https://www.desmos.com/calculator/m2tzegeln6

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