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Definition: the exponential symbol

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.

 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.

They should know how easy this calculator is. 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.

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…)

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

 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

 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 4 7 4 8 15 5 16 31 6 32 63 7 64 127 8 128 255 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.
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

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