By Tom Kando
Here is a parable: An ancient Eastern Emperor, because
he was bored, ordered his court scholars
to invent an exciting game. One of them
came up with the game of checkers. The
emperor was very pleased. He told the inventor
that he would reward him with anything the man wanted. The inventor
replied that his desires were modest. He
merely asked to receive some grains of
rice, namely one grain placed on the checker board’s first square, 2 on the
second square, 4 on the third, and so on,
doubling the number until the last square of the checker board was reached.
The king laughed at this humble and
silly request, and he sent out some men
to collect the amount of rice necessary
to satisfy the request. Of course, the emperor discovered that all the rice
in all of the empire’s granaries
was insufficient to meet the inventor’s wish.
The inventor of the game of checkers was able to trick the emperor because the emperor did not understand the power of exponential growth (whether it is a good idea to trick an emperor, which can cost you your head, is besides the point).
Growth is defined as exponential
when the increase of a quantity is proportional to the size of that quantity.
Any quantity that grows by a constant percentage grows exponentially. If a
population or an economy grows by 3% per year, then each year that 3%
represents a larger absolute value,
because each year it gets added to the baseline upon which the next 3% is computed.
In the checkers story, the
exponential growth curve is exceptionally steep. The initial quantity doubles
each time. Doubling means an increase
of 100% per unit of time. Of course, how
rapid the growth is depends on what your time unit is. Something can double in
a week, or it can double in a million
years. For example: You borrow $1,000
from a loan shark, and he charges you 100%
interest every month. A year later, you owe him over a million.
Getting back to the checkers story:
Do you have any idea of how many grains
of rice the emperor would have had to
scrounge up to satisfy his inventor’s request?
Well, if you start with one
grain on the checkerboard’s first square
and you double this for each consecutive square, then this is the progression:
square #1 has one grain.
Square #2 has two grains.Square #3 has four, which is two squared.
Square #4 has eight, which is two to the third power.
And so forth.
So square #64
of the checkerboard has 2 to the 63rd power: This is
9,223,372,036,888,000,000, which is over 9 quintillion grains of rice - far more than the entire planet
could ever produce.
And here is
another interesting way to look at this:
Let’s say that
the emperor orders his imperial
farmers to produce all the rice needed
to satisfy the inventor’s request. Let’s say that the imperial rice
fields have 10,000 rice plants, and that each plant produces 100,000 grains of rice every year.
Thus, after one year, the imperial rice fields will
have produced 1 billion grains of rice.
At that rate, it will take one thousand years to produce a trillion
grains, a million years to grow a
quadrillion grains, and a billion years
for a quintillion. But since the inventor asked
for more than 9 quintillion
grains of rice, it will take over nine billion years to grow
that amount, which is the age of the Universe, all the way back to the Big
Bang.
So you see, a
silly little checkerboard with 64 squares can yield unfathomably large numbers.
.......................................
The true importance of this lies in its implications for real life. For example, Thomas Malthus warned us against the potential danger of unchecked population growth. According to the 18th century scholar, unchecked population growth is exponential, (1-2-4-8) whereas the increase in the food supply is only arithmetical (1-2-3-4). There exists therefore the danger of eventual famine and other Malthusian "population checks" such as war and disease - IF the population were to continue to grow exponentially.
Now don’t misunderstand me: I do not subscribe to neo-Malthusian doomsday prophecies. Obviously, the population has NOT continued to grow exponentially. Neo-Malthusians like Paul Ehrlich have been severely (and rightly) criticized (For example by David Deutsch, in The Beginning of Infinity).
It is the same with other exponential curves. Say, you have an increase in the number of poor people, or in unemployment, or inflation, or in global temperature, or in your body weight, or anything else. The increase may be exponential for a while. This may not require alarm, but it does suggest vigilance. Because whatever the growth coefficient may be, in due time it would lead to disaster - a 100% rate of poverty or unemployment, the type of hyper-inflation experienced in Germany in the 1020s, when a loaf of bread cost a trillion Marks, etc.
The point is that exponential growth can be so deceptive, because even if the growth coefficient is as small as 1% or 2%, it eventually leads to astronomical quantities (as the checkerboard story shows). In everyday life, we don’t usually worry about a 1% or 2% increase in something. But sometimes we should.
To be sure, growth rarely remains exponential ad absurdum. I merely point out that it BETTER slow down before it’s too late (which it usually does).
.......................................
The true importance of this lies in its implications for real life. For example, Thomas Malthus warned us against the potential danger of unchecked population growth. According to the 18th century scholar, unchecked population growth is exponential, (1-2-4-8) whereas the increase in the food supply is only arithmetical (1-2-3-4). There exists therefore the danger of eventual famine and other Malthusian "population checks" such as war and disease - IF the population were to continue to grow exponentially.
Now don’t misunderstand me: I do not subscribe to neo-Malthusian doomsday prophecies. Obviously, the population has NOT continued to grow exponentially. Neo-Malthusians like Paul Ehrlich have been severely (and rightly) criticized (For example by David Deutsch, in The Beginning of Infinity).
It is the same with other exponential curves. Say, you have an increase in the number of poor people, or in unemployment, or inflation, or in global temperature, or in your body weight, or anything else. The increase may be exponential for a while. This may not require alarm, but it does suggest vigilance. Because whatever the growth coefficient may be, in due time it would lead to disaster - a 100% rate of poverty or unemployment, the type of hyper-inflation experienced in Germany in the 1020s, when a loaf of bread cost a trillion Marks, etc.
The point is that exponential growth can be so deceptive, because even if the growth coefficient is as small as 1% or 2%, it eventually leads to astronomical quantities (as the checkerboard story shows). In everyday life, we don’t usually worry about a 1% or 2% increase in something. But sometimes we should.
To be sure, growth rarely remains exponential ad absurdum. I merely point out that it BETTER slow down before it’s too late (which it usually does).
