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There
is great, ancient and sustaining wisdom in the exponential
symbol. Its Latin origins are exponere – to expound, to make
explain, to make clear, to make manifest, to elucidate, to clear
of obscurity, to unfold or to the illustrate the meaning of. The exponential
symbol, especially when combined with the function symbol,
provides us with profound insight into the nature of change and how the
universal potential is actualised.
(The
function
symbol is employed here to convey the idea that two quantities have a
relationship and changes in one affect changes in the other. They have a
functioning relationship. For instance, the size of the human population
varies with time. If we know the relationship, then our knowledge of one
quantity enables us to know the what the other quantity is.)
Exponential
functions can unfold the story of our universe for us,
provide great meaning and enable us to predict the consequences
of our actions. Change in the universe does not happen uniformly and
occurs in differing patterns of waves, each with its unique rhythms. The
only thing that is constant is change. Exponential functions is a use of
symbols that can enable us to elucidate our place in the universe and
explain how many changes occur.
"The
greatest shortcoming of the human race is our inability to understand
exponential functions"- Albert
A Bartlett
A
ten year old child can understand the essence of exponential functions
(see sample learning exercises below) and it is probable that this
shortcoming is a failure of our spirit stemming from our capacity to
deny stewardship/change. We simply do not want to know the implications
of our actions and this denial is reflected in our institutions.
This
self-deceit is how we enable a handful of merchant bankers to generate
trillions of dollars of credit with little or no relationship to our
assets. This self-deceit is why the human population has grown over
three fold in a century while in the same time we have destroyed over
half of the easily extracted mineral oil on which most of our systems
are now based. This self-deceit is how we justify bequeathing debt with
onerous interest on our children.
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The
T2 or 70/n Rule
It
is the right of every child to know this simple rule:
The
doubling time of a quantity subject to constant growth can
calculated by dividing 70 by n where n is the rate of increase.
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They
should know how easy this calculator is. See how it works to answer the
following questions:
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How
long will it take a population to double if it is increasing
at 7% a year?
Answer:
70 divided by 7 = 10 years.
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How
long will it take for the cost of something to double if inflation
is occurring at 10% a year?
Answer:
70 divided by 10 = 7 years.
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Exponential
change can occur at an increasing or a decreasing rate. In this example
the rate of change doubles with each generation.
human
human human
human human human human
human human human human human human human human
( One person begets two children who beget four grandchildren who beget
eight great grandchildren…)
Compare
this to a constant or linear rate of change:
human
human
human
human
( One person begets one
child who begets one grand child who begets one great grand child…)
Graph.
(to be added)
This grand denial of
stewardship/change is manifest in our school curricula, which associates
the exponential symbol with mathematical symbols. Thus most
people graduate associating the symbol with mind-numbing impenetrable
complexity, the domain of a few myopic geeks. They do not associate the
symbol with art, language, history, geopolitics, biology, economics and
all the variant change of the universe(s).
When the underlying patterns of
change are revealed to us we are better able to lead lives that
are in harmony with them. We are better connected to the universe. We
can be more at one with the greater the ebb and flow - whether it be in
understanding how microbe or human populations expand and shrink or how
interest rates accumulate and usury drives people into debt and misery
or how solar, weather, ocean and other systems form and dissipate
or how the cumulative impacts of our abuse of our carbon potential with
our discovery, extraction and destruction of resources such as mineral
oil and gas affect our children...
In summary
Conserve the exponential
symbol by employing it throughout the range of our discourse so our
children embrace their roles as stewards within change in all its
variety. Thus great and
sustainable ideas can spread exponentially throughout our communities.
Examples
of exponential change
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Lilies
on a pond
Imagine
a pond with water lily leaves floating on the surface. The lily
population doubles every day and if left unchecked it will smother
the pond in thirty days, killing all the living things in the
water. Day after day the plant seems small and so it is decided to
leave it to grow until it half-covers the pond, before cutting it
back. What day will that be?
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(Answer
– the 29th day and then there will be just one day to save
the pond.)
Source: wiki
exponential growth
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Wheat
on a chess board
When
the creator of the game of chess (sometimes named Sessa or Sissa,
a legendary brahmin)
showed his invention to the ruler of the country, the ruler was so
pleased that he gave the inventor the right to name his prize for
the invention. The man, who was very wise, asked the king this:
that for the first square of the chess board, he would receive one
grain of wheat, two for the second one, four on the third one and
so forth, doubling the amount each time. The ruler, who was not
strong in math, quickly accepted the inventor's offer, even
getting offended by his perceived notion that the inventor was
asking for such a low price, and ordered the treasurer to count
and hand over the wheat to the inventor. However, when the
treasurer took more than a week to calculate the amount of wheat,
the ruler asked him for a reason for his tardiness.
The
treasurer then gave him the result of the calculation, and
explained that it would be impossible to give the inventor the
reward. The ruler then, to get back at the inventor who tried to
outsmart him, told the inventor that in order for him to receive
his reward, he was to count every single grain that was given to
him, in order to make sure that the ruler was not stealing from
him.
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"The
amount of wheat is approximately 80 times what would be produced in one
harvest, at modern yields, if all of Earth's arable land could be
devoted to wheat. The total of grains is approximately 0.0031% of the
number of atoms in 12 grams of carbon-12 and probably more than 200,000
times the estimated number of neuronal connections in the human brain
(see large
numbers)."
Source: wiki
wheat and chessboard problem
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Square
number
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Grains
on each square
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Total
grains on board
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1
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1
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1
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2
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2
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3
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3
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4
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7
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4
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8
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15
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5
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16
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31
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6
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32
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63
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7
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64 |
127
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8
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128
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255
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64
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2
(power 64)
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2
(power 63) minus 1
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The
total number of grains is probably more than all the grains that humans
have ever cultivated in the history : 18,446,744,073,709,551,615 grains
of wheat.
Source wiki
Second Half of the Chessboard
Teachers
know best and can create many games based on this exponential function
. An example might be to have a
child print a message of hope 32 times on a page. They copy and paste
the original thought once to give two copies. They copy these two copies
to give four and so on.
The student
then cuts the page in two and gives each half (16 messages) to two
children. They in turn cut their half page in two and give each quarter
page (8 messages) to two more children. Those four children cut their
quarter page in two and pass on 4 messages to two other children. At
this point fifteen of a class of 32 has read the message and yet each
student has only had to communicate it to two others. Played once a week
students will begin to sense the power of exponential functions
Links to Dr
Albert Bartlett
Article:
"Forgotten
Fundamentals of the Energy Crisis"
Audio
Dr.
Albert Bartlett: Arithmetic, Population and Energy
Video
(8 parts)
The Most IMPORTANT Video You'll
Ever See (part 1 of 8)
Enjoy
the rewards of being a conservator of the potential of our greatest
symbols.
Page last updated:
Aug 2010
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