*On exponential growth, its alarming consequences, and Prof Al Bartlett’s useful approximations for understanding it.*

Prof Bartlett sets out clearly how quite small annual percentage growth rates accumulate rapidly over only a few years to give a doubling in the amount of whatever is growing in this “exponential” way. Over further time, spectacularly greater increases occur.

We think little on hearing for example that the price of ski tickets rises steadily at 7% a year. But over 40 years (e.g., from 1963 to 2003 in Colorado), this gives a total increase by a multiple of many times. Prof Bartlett says the 40-year increase is 16 times. This is an approximation, but the actual value, about 14.97 times [1.07^40], is still alarmingly large.

The same pattern occurs with steady annual growth in other more significant quantities, including demand for oil, electricity, and other resources. Prof Bartlett relates this to a crisis in energy supply. Such growth patterns and increases cannot be sustained forever: we live in a finite world.

In the video, Prof Bartlett gives two rules of thumb for quick calculations about exponential growth. It’s worth noting that these are approximations, quite accurate for small percentage growth values, but not correct for large rates of growth.

**Doubling time**

The first rule is for calculation of doubling time. Here’s a still from 1 minute 40 seconds:

The approximation here is that if n is a “small” annual percentage growth rate, then there is an overall doubling after about 70 / n years.

As Prof Bartlett notes, 70 is an approximation for 100 ln 2 = 69.3… (where “ln 2” is the natural logarithm of 2). However, there is another approximation in the formula itself.

You can see at once that the formula is not accurate for large n, by taking n = 70. The formula would tell you that 70% annual growth causes a doubling in 70 / 70 years, that is, in one year. But of course 70% annual growth yields 70% growth in one year, not a doubling (100% growth).

The exact doubling time in years for n% annual growth is ( ln 2 )/( ln (1+n/100) ). Here’s a table showing, for various values of n, firstly the actual doubling time; then Prof Bartlett’s estimation of 70 / n for the doubling time; and also the value 69.3147180559945 / n with a more accurate value of ln 2 than just 70.

Annual percentage growth n | Actual doubling time (years) | Prof Bartlett’s estimate 70 / n | 69.3147180559945 / n |

1 | 69.66 | 70.00 | 69.31 |

2 | 35.00 | 35.00 | 34.66 |

3 | 23.45 | 23.33 | 23.10 |

4 | 17.67 | 17.50 | 17.33 |

5 | 14.21 | 14.00 | 13.86 |

6 | 11.90 | 11.67 | 11.55 |

7 | 10.24 | 10.00 | 9.90 |

8 | 9.01 | 8.75 | 8.66 |

9 | 8.04 | 7.78 | 7.70 |

10 | 7.27 | 7.00 | 6.93 |

12 | 6.12 | 5.83 | 5.78 |

14 | 5.29 | 5.00 | 4.95 |

16 | 4.67 | 4.38 | 4.33 |

18 | 4.19 | 3.89 | 3.85 |

20 | 3.80 | 3.50 | 3.47 |

30 | 2.64 | 2.33 | 2.31 |

40 | 2.06 | 1.75 | 1.73 |

50 | 1.71 | 1.40 | 1.39 |

60 | 1.47 | 1.17 | 1.16 |

70 | 1.31 | 1.00 | 0.99 |

80 | 1.18 | 0.88 | 0.87 |

90 | 1.08 | 0.78 | 0.77 |

70 / n is a good approximation for the actual doubling time for n up to 10% or even 20% say. But for larger n, 70 / n increasingly overestimates underestimates the doubling time. Replacing 70 by a more accurate value of 100 ln 2 makes little difference.

Here’s an attempt to state Prof Bartlett’s rule with the inclusion of a warning about approximation.

**For a growth rate of n% per year, 70 / n gives a good estimate of the number of years it will take for the original value to double, provided n is less than 20.**

**Steady growth for 70 years**

Prof Bartlett’s second rule appears later in the video, e.g., around 10 minutes and 0 seconds:

He states that annual growth of n% will mount up over 70 years to total growth of multiplication by a factor of 2 to the power n, that is, n twos multiplied together. I’ll represent this nth power of 2 as 2^n.

As I’ve pointed out in a comment on an earlier Joss Winn post, this is also an approximation. It becomes less accurate more rapidly than for the first rule above.

Annual percentage growth n | Total growth over 70 years | Prof Bartlett’s estimate 2^n |

1 | 2.01 | 2.00 |

2 | 4.00 | 4.00 |

3 | 7.92 | 8.00 |

4 | 15.57 | 16.00 |

5 | 30.43 | 32.00 |

6 | 59.08 | 64.00 |

7 | 113.99 | 128.00 |

8 | 218.61 | 256.00 |

9 | 416.73 | 512.00 |

10 | 789.75 | 1024.00 |

11 | 1488.02 | 2048.00 |

12 | 2787.80 | 4096.00 |

Note that already for n = 7, total growth over 70 years is about 114 times, not the 128 times given by the value 2^7.

Hence we could say:

**For a growth rate of n% per year, a good estimate for total growth over 70 years is 2^n times (2 to the power of n, or n twos multiplied together), provided n is less than 6.**

**Conclusion**

Both of Prof Bartlett’s rules are very useful, as long as they aren’t applied for excessively large values n of annual percentage growth. They give helpful quick ways to calculate the effect of exponential growth, via simple arithmetic, avoiding the more advanced mathematics which gives the exact values.

None of this affects Prof Bartlett’s main point: exponential growth mounts up alarmingly over time, and cannot be sustained indefinitely. Resources such as oil are finite, and will run out rapidly if demand grows steadily.

This has deep implications for our way of life and the future of the Earth.