On exponentiality, a really useful rule of thumb is the Rule of 70. I first read of it in a James Cameron column in the Guardian and, being an MBA student at the time, did the calculations and found that it holds roughly true except at the margins (very low or high numbers).

Basically, the doubling time of a percentage rate of increase (or halfing time of decrease) can be found by dividing 70 by the percentage in question … thus why, as John says here, 7% growth means a doubling every 10 years.

A.      

I can recommend Prof Albert A Bartlett's seminal "Population, Arithmetic, and Energy" lecture.  It's particularly pertinent to the global warming situation, where we want to think that cutting fossil fuels will solve the whole problem and allow sustainability into the future.  This is just wishful thinking... but let Prof Bartlett explain.

http://www.youtube.com/watch?v=F-QA2rkpBSY&feature=list_related&playnext=1&list=SP6A1FD147A45EF50D

The first 4 parts are concerned with the exponential function, and it's implications - e.g. that 7% growth means a doubling every ten years.

Part 5 starts here:
http://www.youtube.com/watch?v=t-X6EpvWWu8&NR=1

Quoting from 2.25 minutes into Part 5 of
[quote]
"The more optimistic the prediction the greater is the probability that it is based on faulty arithmetic or on no arithmetic at all."

Another quote, after seeing what rubbish is written:  "We can't let other people do the thinking for us"  8 minutes into part 6.

Part 7, 1:50 minutes in, he quotes Winston Churchill: "Sometimes we have to do what is required".

Part 8, 2 minutes in:
"If ever there was a time that the human race needs people who can think, it's right now.  It's our responsibility as citizens in a democracy to think."

"But to be successful with this experiment of human life on earth we have to understand the laws of nature as they are encountered in the study of the sciences and mathematics."  "Thinking is very upsetting - it tells us things that we'd rather not know."

Aldous Huxley: "Facts do not cease to exist because they are ignored".

H L Mencken: He believed it is the nature of the human species to reject what is true but unpleasant and to embrace whait is obviously false but comforting.

With respect to overpopulation, Dr Martin Luther King Jr said that "overpopulation is soluble by means we have discovered and with resources we possess.  What is lacking is not sufficient knowledge of the solution, but universal consciousness of the gravity of the problem and the education of the billions who are its victims".

Prof Bartlett concludes: "The greatest shortcoming of the human race is our inability to understand the exponential function"

Cheers,

John