**This special essay is written by Prof Evar Nering – a long time friend and mentor!!**

There is a fable about a young man in ancient China who invented the game of chess.

The Emperor was fascinated with the game and wanted to reward its inventor. He

asked the young man what he would like as a gift. The young man replied that he

would like one grain of rice for the first square on a chess board, two grains of rice for

the second square, four for the third. This was to continue, doubling for each successive

square on the board. Without thinking, a common practice for emperors, the Emperor

shouted, “Granted!”

What did this award amount to? The first row of squares would contain 1, 2, 4, 8, 16, 32,

64, and 128 grains of rice. Hardly a handful. The second row would contain 256, 512,

1024, 2048, 4096, 8192, 16384 and 32768 grains. The prize still looks manageable. But

things start to get interesting in the following row. The 24th square at the end of the

third row would contain 8,388,608 grains. That is still not going to bankrupt the

Emperor. It’s just a few sacks of rice. But the total number of grains on the full

chessboard would be 2 to the 64th . Since one grain of rice is of no significance, let’s round it

off to 2 the 64th

Just how large is that number? If you have a hand held scientific or business calculator,

or an iPhone, you can do the calculations yourself. Make the following keystrokes.

Enter 2, then press the key with Yx on it (the exponential key), then 64 and the equal

key. The result, 1.844674407371e+19. The really interesting part of this number is the

“e+19” at the end. The number in decimal form has 20 digits. The national debt is a few

trillion dollars, but this is almost a billion trillion. Without going into detail, it is easy to

estimate that if the Emperor devoted all his lands to growing this crop of rice it would

take about a million years to produce it.

Almost everyone is familiar with an exponential function in the form of

a savings account in a bank. At an interest rate of 6% per year compounded monthly $1

deposited in such an account would accrue monthly but it would take almost 14 years for the account to double.

But if your great great grandfather had deposited that $1 in your account 140 years ago

it would have doubled 10 times and the account would hold $1024. If an ancestor had

deposited $1 in your account 280 years ago, that would give 20 doubling periods and

the account would contain $1,048,576.

Every student who has had a least one semester of calculus has studied the exponential

function. That means several million Americans have studied the exponential function.

But our public discourse suggests that they all forgot about it after they passed their

final exams. Understanding the consequences of exponential population growth is

probably the most important piece of understanding required to avert the end of

civilization. But you don’t have to study calculus to put your arms around an

understanding of what is involved.

The easiest way to look at the exponential function is to think in terms of doubling

times. All exponential function look alike in the following sense. If you drew a graph of

the exponential function on a rubber sheet that you could stretch or compress

horizontally or vertically, you could stretch one version of the exponential function to

exactly any other. At 7% interest compounded monthly it takes about ten years for the

account to double. At 14% it would take five years. At 3.5% it would take 20 years.

Human population growth is not constant in time and it is not uniform all over the

world. If the population grows at one rate for a while and then at another rate for a

while, the doubling period for each rate would be different. But a period over which it

doubles at one rate following by a doubling period for the other rate is still two

doubling periods. Pay attention to the doubling periods.

Estimating the rate of human population growth is not simple. Longevity has little to do

with it. The recent increase in expected life span has increased the number of people but

it is nothing but a bump in the numbers. It is the portion of the population that is fertile

that determines the birth rate.

My source for the following facts is Wikipedia. The human population reached one

billion in the early 19th century. The population grew by another billion by the early 20th

century, a doubling period of about 123 years. It topped four billion in 1974, a doubling

period of less than 50 years. The population is expected to top eight billion by 2025, a

doubling period of 51 years. The recent jump in population growth is widely attributed

to improved agricultural technology, medicine, and industry. But these resources are

not increasing exponentially. The Earth has now all the water, all the air, all the steel it

will ever have. In fact, many resources like petroleum, timber, and arable land are

decreasing.

Bringing the standard of living of the developed countries to the less developed

countries is a mixed blessing. These improvements would motivate a reduction in the

growth rate, but increasing their living standards would place greater demands on the

limited resources the Earth has to offer.

The consequences of different rates of growth is different parts of the world are

alarming. Suppose there were two population groups of equal size but the doubling

period in one population was twice as long as the doubling period for the other. At the

end of one doubling period for the group growing at the slower rate the faster group

would have gone through two doubling periods and their population would be twice as

large. At the end of two doubling period for the group with the slower growth rate, the

faster group would have gone through four doubling. The slow growth group would

now consist of 20% if the population. At the end of 10 doubling periods the slow

growth group would comprise about 0.1% of the population. The United States cannot

tackle this problem alone.

The use of the word “exponential” in common usage to mean a sudden increase is a

mistake and it is misleading. Exponential growth is not sudden. It is inexorable.

Nothing physical can increase exponentially forever. It will stop. The current average

population density is about 132 persons per square mile. The most densely populated

country currently is Macao with 48,000 people per square mile. It would take a mere

eight doubling periods to reach that average density for the entire world. We have used

up three doubling periods between 1804 and 2025. In three more doubling periods the

world population density would exceed that of Haiti today.

Population growth will stop for one of two reasons. It will stop because we have the

wisdom to curb the growth. Or it will stop because of forces beyond our control. The

longer we argue unproductively about it the more difficult the first option will be.

Evar D. Nering

January 2011

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