Forgotten Fundamentals of the Energy Crisis - Part 3
by Prof. Al Bartlett
III. The power of two
Legend has it that the game of chess was invented by a mathematician who worked for an ancient king. As a reward for the invention the mathematician asked for the amount of wheat that would be determined by the following process: He asked the king to place 1 grain of wheat on the first square of the chess board, double this and put 2 grains on the second square, and continue this way, putting on each square twice the number of grains that were on the preceding square. The filling of the chessboard is shown in Table I. We see that on the last square one will place 2^{63} grains and the total number of grains on the board will then be one grain less than 2^{64}.
How much wheat is 2^{64} grains? Simple arithmetic shows that it is approximately 500 times the 1976 annual worldwide harvest of wheat? This amount is probably larger than all the wheat that has been harvested by humans in the history of the earth! How did we get to this enormous number? It is simple; we started with 1 grain of wheat and we doubled it a mere 63 times!
Exponential growth is characterized by doubling, and a few doublings can lead quickly to enormous numbers.
The example of the chessboard (Table I) shows us another important aspect of exponential growth; the increase in any doubling is approximately equal to the sum of all the preceding growth! Note that when 8 grains are placed on the 4th square, the 8 is greater than the total of 7 grains that were already on the board. The 32 grains placed on the 6th square are more than the total of 31 grains that were already on the board. Covering any square requires one grain more than the total number of grains that are already on the board.
Table I. |
||
Square Numbers |
Grains on the Square |
Total Grains Thus Far |
1 |
1 |
1 |
2 |
2 |
3 |
3 |
4 |
7 |
4 |
8 |
15 |
5 |
16 |
31 |
6 |
32 |
63 |
7 |
64 |
127 |
64 |
2^{63} |
2^{64} - 1 |
On April 18, 1977 President Carter told the American people, "And in each of these decades (the 1950s and 1960s), more oil was consumed than in all of man's previous history combined."
We can now see that this astounding observation is a simple consequence of a growth rate whose doubling time is T_{2} = 10 yr (one decade). The growth rate which has this doubling time is P = 70 / 10 = 7% / yr.
When we read that the demand for electrical power in the U.S. is expected to double in the next 10-12 yr we should recognize that this means that the quantity of electrical energy that will be used in these 10-12 yr will be approximately equal to the total of all of the electrical energy that has been used in the entire history of the electrical industry in this country! Many people find it hard to believe that when the rate of consumption is growing a mere 7% / yr, the consumption in one decade exceeds the total of all of the previous consumption.
Populations tend to grow exponentially. The world population in 1975 was estimated to be 4 billion people and it was growing at the rate of 1.9% / yr. It is easy to calculate that at this low rate of growth the world population would double in 36 yr, the population would grow to a density of 1 person / m2 on the dry land surface of the earth (excluding Antarctica) in 550 yr, and the mass of people would equal the mass of the earth in a mere 1,620 yr! Tiny growth rates can yield incredible numbers in modest periods of time! Since it is obvious that people could never live at the density of 1 person / m^{2 }over the land area of the earth, it is obvious that the earth will experience zero population growth. The present high birth rate and / or the present low death rate will change until they have the same numerical value, and this will probably happen in a time much shorter than 550 years.
A recent report suggested that the rate of growth of world population had dropped from 1.9% / yr to 1.64% / yr.^{2} Such a drop would certainly qualify as the best news the human race has ever had! The report seemed to suggest that the drop in this growth rate was evidence that the population crisis had passed, but it is easy to see that this is not the case. The arithmetic shows that an annual growth rate of 1.64% will do anything that an annual rate of 1.9% will do; it just takes a little longer. For example, the world population would increase by one billion people in 13.6 yr instead of in 11.7 years.
Compound interest on an account in the savings bank causes the account balance to grow exponentially. One dollar at an interest rate of 5% / yr compounded continuously will grow in 500 yr to 72 billion dollars and the interest at the end of the 500th year would be coming in at the magnificent rate of $114 / s. If left untouched for another doubling time of 14 yr, the account balance would be 144 billion dollars and the interest would be accumulating at the rate of $228 / s.
It is very useful to remember that steady exponential growth of n% / yr for a period of 70 yr (100 ln2) will produce growth by an overall factor of 2^{n}. Thus where the city of Boulder, Colorado, today has one overloaded sewer treatment plant, a steady population growth at the rate of 5% / yr would make it necessary in 70 yr (one human lifetime) to have 2^{5} = 32 overloaded sewer treatment plants!
Steady inflation causes prices to rise exponentially. An inflation rate of 6% / yr will, in 70 yr, cause prices to increase by a factor of 64! If the inflation continues at this rate, the $0.40 loaf of bread we feed our toddlers today will cost $25.60 when the toddlers are retired and living on their pensions!
It has even been proven that the number of miles of highway in the country tends to grow exponentially.^{1(e),3}
The reader can suspect that the world's most important arithmetic is the arithmetic of the exponential function. One can see that our long national history of population growth and of growth in our per-capita consumption of resources lie at the heart of our energy problem.
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