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 Exponential Functions
(A wonderful use of symbols that reveals
 the nature of change in our universe.)


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sustainable uses
  of key symbols
  - including

energy  efficiency








Humans have a great capacity to deny change. We have the ability to construct very sophisticated systems of beliefs and norms to deny our mortality and our roles as stewards in change. Our schools amplify this tendency by failing to imbue in us a profound and sustaining sense of how exponential functions work and how they help reveal the true nature of change. 
This failure puts humanity at grave risk.

Thought derived from 
The Sustainability Principle of Energy



Do you know what exponential means? 
Could you teach the basics of exponential functions? 
Does your “environmental education” programme communicate these basics? 
And could it be that if our schools gifted our children with an understanding of how exponential functions work then there would be no need for the Environmental Education industry? Could it be that if our children understood exponential functions they would mature as wiser beings and adopt life styles in harmony with the greater change around them?

And the question has to be asked: why is it that so few adults understand exponential functions when a child of five can understand the basic notions?  

Origins and power of the exponential symbol

As always there is great, ancient and sustaining wisdom in powerful symbols such as exponent. 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.

And this is precisely what exponential functions can do. They can unfold the story of our universe for us and provide great meaning. Change in the universe does not happen uniformly and occurs in differing patterns of waves, each with its unique rhythms. Exponential functions is a use of symbols that can enable us to elucidate our place in the universe and explain how many changes occur.

When the underlying patterns 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 weather systems form  and dissipate  or how we discover, extract and destroy resources such as mineral oil and gas…..

This use of the function symbol is employed to convey the idea that two quantities have a relationship and changes in one affect changes in the other. They have a functioning relationship.

Exponential functions connect us to our history and to our greater environment. They can help us avoid catastrophe by teaching us how to conserve limited resources and setting of destructive debt and weather spirals. They can inspire us by showing us how great and sustainable ideas can spread throughout our communities.  

Simple lessons for younger students

Here are two examples of how a child of ten or younger can begin exploring our exponential beings.  


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?  


(Answer – the 29th day and then there will be just one day to save the pond.)  
Source: wiki exponential growth

(Teachers know best and can create many games based on this principle . 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 the exponential function


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.  


"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

Square number Grains on each square Total grains on board
1 1 1
2 2 3
3 8 7
4 16 15
5 32 31
66 64 6
7 128 127
8 256 255
64 2 (power 64) 2 (power 63) minus 1

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.  
ource wiki Second Half of the Chessboard

Uses of the Exponential Symbol

The exponential function is invaluable for communicating how to lead sustainable lives in that it reveals how wars are born out of inflation, how usery destroys lives (as we are seeing with the current global “credit squeeze” *) and how our misuse of, for instance, our carbon potential can put us at grave risk. 

In the case of mineral oil and gas we have used up half the known reserves over the centuries. However the increase in the rate of our consumption of the resources means that remaining supplies may last as little as a decade – assuming the extraction rate increases with demand.

*Note March 2008: The current inflation pressures and associated “credit squeeze” or “subprime mortgage crisis”  is really a manifestation of what happens when a society confuses energy with one of the forms it can take and does not understand simple exponential functions. 
In this case we have employed the energy symbol as in “energy= mineral oil/gas” and consumed the resource as though it is as bounteous as energy. It is not and if we continue to use the resource at current rates it will be depleted soon. Its market price has risen 1000% this century ($US9.98 in 1999 to $US108 a 42 gallon barrel in 2008). The root cause of our inflation is our abuse of the energy symbol and our carbon potential.
What we are really experiencing is a “carbon use crisis” in general and, in particular, a “mineral oil/gas abuse crisis”.  


The T2 or 70/n Rule

It is the right of every child that they should 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. 


Use this easy calculator and see how it works to answer the following questions:

How long will it take a population to double if it is increasing  at 7% a year?

 Answer: 70 divided by 7 = 10 years.



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.


Albert A Bartlett expounds in wonderful discussion the revelatory power of exponential functions using three media. He provides, in addition to much of the above, the example of bacteria multiplying in a glass jar to illustrate in a very simple and clear way what happens when exponential growth occurs in a finite environment. 

His thesis: "The greatest shortcoming of the human race is our inability to understand exponential functions"

Article: "Forgotten Fundamentals of the Energy Crisis"

Audio Dr. Albert Bartlett: Arithmetic, Population and Energy  (70839 hits March 11 2008)

Video (8 parts) 

And probably the most boring. But then again, when I told that to my students and had them give me feedback, most said that if you followed along with what the presenter (a professor emeritus of Physics at Univ of Colorado-Boulder) is saying, it's quite easy to pay attention, because it is so compelling." (43640 viewings March 11 2008)

Another useful and short video is  Are Humans Smarter Than Yeast?  

An appreciation of exponential functions is invaluable for communicating climate and tectonic processes. For instance see in the footnotes the charts of the Beaufort Scale (wind speed and wind pressure) and the Richter Scale (earthquake force and acceleration.)

Almost every child will have to face a world in which usury abounds and it is in the interests of money traders (banks etc) to promote ignorance of the truths that exponential functions reveal. Many householders paying interest only on their mortgage at 7% fail to understand that the mortgage doubles every seven years. Many people paying interest only on their car loans at 20% fail to appreciate that the size of the loan doubles every three and half years.

The failure of our schools in countries like the USA and New Zealand  to communicate the nature of exponential functions is resulting in societies that are generating massive economic/ecological debt. Check out the US National Debt Clock and be mindful this is only a fraction of the debt being generated. Check out the New Zealand household debt/disposable income graph - rising from about 50% before the "economic reforms" of the 1980s to of disposable income to

Classic money riddle

You are given two choices for an increasing weekly allowance:

Option one: Your allowance begins with 1 cent and doubles each week.

Option two: Your allowance begins with $1 and increases by $1/week

Which option do you choose if you want a large allowance quickly? Check the chart and see how wise your decision is.



Option 1

Option 2

























































   Chaos Theory

With an understanding of the fundamentals of exponential functions it is possible to enjoy glimpses into the amazing revelations of chaos theory (an unhelpful name in that the theory actually finds order amidst the seeming chaos of weather and climate systems, cloud, galaxy, leaf and mountain formations, fluid dynamics, population changes, forest fires, wars, disease, systems that follow the laws of quantum mechanics and general relativity and the dynamics of the action potentials in our neurons.

You will probably know Edward Lorenz’s discussion entitled Predictability: Soes the Flap of a Butterfly’s Wings in Brazil set off a tornado in Texas? To quote wiki:

"The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different."

The patterns of change described by chaos theory are not the same as those produced by simply doubling a number in that they describe closed systems in which both explosive  and implosive change occurs. However an insight into the rate at which change in the amount of change occurs  and a sense of the wonderful complexity of change which is our universe.

This is why our teachers of Mathematics are truly Environmental Educators for they provide us with visions of change in which we can make great connections with our universe and have a more powerful sense of its vitality. Our children can never be given too many opportunities to play with exponential functions.

Summary thought

All teachers become environmental educators of the finest order when they imbue in our children a knowledge and love of exponential functions.


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Wind speed and wind pressure

The wind pressure can be approximated by: Pressure = ½ x (density of air) x (wind speed)2 x (shape factor)

  • The density of air is about 1.25 kg/m3.
  • The shape factor (drag coefficient) depends on the shape of the body. It has order of magnitude 1 and is dimension less.
  • The wind speed must be expressed in m/s. In that case the pressure has units kg/m/s2, i.e. N/m2.

See this table:


Wind speed (m/s)

Wind pressure (N/m2)


Lower limit

Upper limit

Upper limit




















































> 660

Since the scale of Beaufort and the wind speed are not related linearly, it is not possible to express the wind pressure as:
Pressure = (constant) x (wind speed in Bft)2.

Source  KNMI Hydra Project

Earthquake force and acceleration

When trying to understand the forces of an earthquake it can help to concentrate just upon the up and down movements. Gravity is a force pulling things down towards the earth. This accelerates objects at 9.8 m/s2. To make something, such as a tin can, jump up into the air requires a shock wave to hit it from underneath travelling faster than 9.8m/s2. This roughly corresponds to 11 (Very disastrous) on the Mercalli Scale, and 8.1 or above on the Richter Scale. In everyday terms, the tin can must be hit by a force that is greater than that which you would experience if you drove your car into a solid wall at 35 khp (22 mph).  

Richter Scale

Richter Scale


Mercalli equivalent


< 1 cm/s2



2.5 cm/s2





10 cm/s2



25 cm/s2



50 cm/s2



100 cm/s2



250 cm/s2





500 cm/s2



750 cm/s2


> 8.1

980 cm/s2


  Source Geography Site UK

Other NZ household debt accumulation data can be viewed in this Reserve Bank Paper (2004)

Sample symbol source


ex·po·nent  ( k-sp n nt, k sp n nt)


1. One that expounds or interprets.

2. One that speaks for, represents, or advocates: Our senator is an exponent of free trade.

3. Abbr. exp Mathematics A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, symbol, or expression, denoting the power to which that number, symbol, or expression is to be raised. Also called power.


Expository; explanatory.

[Latin exp n ns, exp nent-, present participle of exp nere, to expound; see expound.]

ex·po·nen·tial  ( k sp -n n sh l)


1. Of or relating to an exponent.

2. Mathematics

a. Containing, involving, or expressed as an exponent.

b. Expressed in terms of a designated power of e, the base of natural logarithms.

exponential  ( k sp -n n sh l)

Relating to a mathematical expression containing one or more exponents. Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents. A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.


[Middle English expounden, from Anglo-Norman espoundre, from Latin exp nere : ex-, ex- + p nere, to place; see apo- in Indo-European roots.]













Without symbols there is no civilisation.


Symbols enable us to communicate and reflect reality.


By maximising the potential of symbols we can enjoy the greatest harmony with all.


Failure to conserve symbols and  the flawed  uses of symbols destroys civilisations.


Symbol use born of compassion works to sustain humanity.