September 12, 2019 Thursday
Bedtime Story
The Second Mathematical Problem
“Would you rather have a million dollars or
the sum of a penny doubled every day for a month?”
A quick un-thoughtful answer to the above
question would make many regret later on and which demonstrates the
counter-intuitiveness of exponential growth even if the beginning is very
humble or specially so.
The second mathematical problem concerning
geometric progression and exponential growth is the water lily problem which
poses the question to children in the following manner (it is not particularly
a problem that is suitable to be posed to adults in interviews or exams as it
is relatively much simpler in relation to the wheat/rice and the chessboard
problem):
The children are asked to imagine a pond
with water lily leaves floating on the surface.
The lily plants grow in such a manner that
the population of the floating leaves doubles in terms of its surface area
covering the pond every single day.
If left to grow by itself the leaves will
cover the entire pond in 30 days and thereby become strangle the life out of
other living forms (ignore the biological incorrectness for the sake of this
problem).
To begin with the area covered by the leaves
is small and hence it is decided it would only be sensible to cut down these
water lilies when half the pond is covered and not every day.
Now the question that is posed to children
is that when it takes 30 days for the pond to get totally covered on which day
is the pond half full so that cutting of the lilies can begin?
You will know the answer and hence I had
said this is a childish question and yet at the same time it serves as a
powerful tool to demonstrate the power of exponential growth.
Carl Sagan in his final book “Billions and
Billions: Thoughts on Life and Death at the Brink of Millennium” (1997) had the
second chapter of it titled “The Persian Chessboard”.
In it he wrote that bacterial growth in
ideal conditions exhibit exponential growth and “exponentials can’t go on
forever, because they will gobble up everything.”
In a 1972 report titled “The Limits to
Growth” that was based on a computer simulation of exponential economic and
population growth in a scenario of finite resources one of the conclusions that
was arrived at was:
“Exponential growth can never go on very
long in a finite space with finite resources”.
(At this point please recall what I had
written few nights ago - that our economic, finance and monetary system has
been inherently geared up around a system of continuous growth and relentless
money printing either out of government debts or through fractional-reserve
banking/lending which any idiot can make out are both inherently unstable).
The strange thing is that the failure of
the existing type of finance system is not merely inevitable nor need to be
predicted by economic experts but are a recurrent theme in the history of
economies.
Stay tuned to the voice of an
average story storytelling chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the International
Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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