Thursday, 16 August 2012

Explaining Exponential Growth - its basic features...

One can find examples of Exponential Growth (EG^) in almost all fields of life.  EG^ mathematically describes how some fundamental attribute of a system will change with time.  Systems with EG^ can show non-intutive bizarre behaviours and makes exponential growth a difficult concept to comprehend and accept its conclusions.  
Mysterious as it may sound, an understanding of EG^ is crucial to appreciate what is going on in our daily lives and to plan for the future.

From calculating the interest earned on your capital to using Moore's Law to predict the number of transistors on a chip, or to calculate the power output in a nuclear reactor, EG^ is there to help. Its twin brother, exponential decay (ED^) only differs in the direction of change but shares all the concepts of EG^.

There are three ways to describe EG^ - they are equivalent but one is used in preference to others depending on the situation:  In EG^

1. Rate of change (% Change) depends on the amount present. e.g. 
Annual interest rate is 5% (interest earned per year is 0.05 times the money at the start of the year). 
Inflation index which measures the % change in the cost of a basket of goods over a year, was 2.5%. 
Global population increased by 1.4% per year in the 20th Century.
E.Coli colony increases is size at 3.5% per minute.  etc...

2.  The quantity present doubles after a certain time.
(Doubling Time) 
If you leave your money in a bond, it will double in 14 years
Inflation will double the cost of goods in 28 years.
Global population doubled every 50 years in the 20th Century.  
E Coli colony doubles in size in 20 minutes.  etc...

3.  Doubling steps: 
Accumulate something with the added amount doubling in successive steps.

The only important parameter in exponential growth is
% rate of change or doubling time. 
Rate of change and doubling time are measured in the same units of time - be it years, minutes, seconds, centuries, nanoseconds or whatever. 
They are simply related as follows (called the rule of 70):

% rate of change = 70 divided by the doubling time, and of course
The doubling time = 70 divided by the % rate of change

If we start with 1 unit and the doubling time is T then at time

10T the number of units will be 1,000 (actually 1024)
11T                                          2,000
20T                                              1,000,000
30T                                              1,000,000,000
40T                                                 1,000,000,000,000 
41T                                                 2,000,000,000,000
What quantity (population, money, no of transistors) and  the unit of time (years, seconds, minutes etc.) you choose  depends on the problem.

The thing to note is that at 10T after one doubling time, the growth was 1,000;
while at 40T after one doubling time, the increase was 1,000,000,000,000 - a billion times greater. 
This is true of all systems showing exponential growth - this is simple maths.

Also note that in each doubling time, the increase is as much as has happened in all the previous doubling steps. 
So when we say that energy consumption will double in 40 years; it means that in the next 40 years we shall consume as much energy as we have used since the beginning!  To make it clearer:
From 1970 to 2010 we used as much energy as we had used from 1800 to 1970.  (1800 is used as a reference point - energy used before 1800 was very small)

If we were to plot the quantity against time then the graph will show very little change in the beginning but after 10 or 20 doubling times the numbers would have grown to thousands of times bigger and the graph will show an almost vertical swing.

Another interesting property of EG^ graph is that it looks the same no matter what section of the graph you are looking at.  For example, in the picture, you can change the x-axis (time axis - doubling steps) from the current range of 0 to 14 to a range from 20 to 34.  The y-axis (quantity) will be changed to start at 1000,000 (the actual value at 20 doubling times) and go to 4,000,000,000 (at a doubling time of 34).  The curve on the graph will be exactly the same. 
Isn't that wonderful?

Here is a chart of how world population grew in the past 

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