The Essential Exponential! For the Future of Our Planet

By Albert A. Bartlett with Robert G. Fuller, Vicki L. Plano Clark, and John A. Rogers
Reviewed by Manish Gupta

Center for Science, Mathematics, and Computer Education,University of Nebraska, Lincoln, NE, 2004

Professor Emeritus Albert Bartlett began his career in the physics department at the University of Colorado, Boulder in 1950. In 1969, he began delivering lectures on exponential growth and its often unintuitive results. The focus of his talks involved educating people about the consequences and arithmetic of steady growth, a centerpiece of domestic and global economies, and was motivated by the belief that “the greatest shortcoming of the human race is our inability to understand the exponential function.” The talks evolved into a series of papers on the exponential function and its relation to the energy crisis, population growth, and nonrenewable resources. This book is a collection of these papers and also includes some additions by M. King Hubbert and L. David Roper regarding depletion theory.

Many of the papers in the book discuss continuous growth in energy usage. In such cases, the time required for the growing quantity to double in size is T2 = (ln 2)/k, where k is the fractional growth per year. If the size of the resource is R (e.g. tons of coal or barrels of oil), the exponential expiration time (time at which the total consumption is equal to the total resource size), is EET = (1/k)ln(kR/r0 + 1), where r0 is the current rate of consumption. Professor Bartlett then applies these equations to oil and coal usage to estimate the life expectancy for these energy reserves under several circumstances. For example, if the world crude oil production rate continues to rise by 7%/year, the world’s known oil supply (crude oil and oil shale) will expire in 35 years. The simple model is also used to derive scenarios in which a finite resource can be extended indefinitely (e.g. “Sustained Availability”) by decreasing the rate of extraction such that the total of all future extractions equals the size of the remaining resource at present. Many of these papers rely on a very simple exponential model in which the rate of extraction continues to grow at a fixed rate despite declining resources. Later in the book, this assumption is refined to include more sophisticated depletion theories by Hubbert and Roper, and more recent data (e.g. up to 2003) is used to show depletion of oil reserves. Additionally, Bartlett notes that several political projections for energy reserves rely on the “current rate of usage” and can thus be very misleading.

The book also includes a series of articles discussing the exponential function and its impact on population growth. Here, there are several interesting points on methods to achieve zero-growth populations (both voluntary and involuntary), the impact of overpopulation on the decline of democracy, and the marginalization of Thomas Malthus’s original population growth analysis. Finally, the last chapter of the book contains 14 papers written for The Physics Teacher that detail the mathematics of exponential growth and additional simple applications to compound interest, inflation, and miles of highway.

The bulk of the work is presented at a very simple level that is easily comprehensible to undergraduates and advanced high school students. Since the book is collection of Professor Bartlett’s papers on exponential growth, there is substantial redundancy in both the mathematics and the message. Moreover, many of the papers rely on an overly simplistic continuous growth model that may, without the proper caveats, lead to erroneous conclusions; however, the inclusion of Hubbert’s and Roper’s papers allow the interested reader to make more sophisticated calculations. Despite these criticisms, I would recommend the book for anyone interested in the current energy crisis or overpopulation as it provides a good background to simple models associated with these subjects.

Manish Gupta
Los Gatos Research

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