What Nobody Sees
©   Barry Davies   2011

I like to propose to people that we calculate in our heads two year's growth on   100£   at   1%   per year. We get   1£   the first year, and another   1£   and a pence the second year - for a total of   102.01£.


When I decided to write down the exercise just described, I could not stop at two years and wrote down the table immediately below. After the row that has only   100   in it, each row contains the exact theoretical growth/interest for one compounding. However, I no longer want to think of the numbers as representing money. And the compoundings need not represent years. My purpose below is to look at the mathematical properties of this kind of rolled over growth - regardless of what is growing. After the column that has only   100   in it, each column contains growth amounts for the previous column and from previous rows. In the third compounding, we have   1   from the   100,   and we have   .02   from the two previous   1's,   and   .0001   from the previous   .01   and so on. The factors apply to the numbers below the 'factor' row, and I think this makes clear the meaning of this scientific notation for small numbers - where the power of ten has a minus sign: There is an implicit decimal point after each number in the last row, and we move that decimal point to the left by the number of places indicated by the power of ten, and pad with zeros.


At this point, I would like to point out
(1) that modern humans are about 100,000 years old, and
(2) that tool making has been going on for about   2,000,000   years.

Therefore:


Before the end of this presentation, I will want to look both   100,000   and   2,000,000   years into the future - under the assumption that we will be at least as long lasting as those more primitive modern humans and tool makers.


With   100,000   compoundings,
the first two columns sum to the first two numbers below,
and all the rest of the columns sum to the third number.

Anyone who needs to contemplate the national debt of the USA should be comfortable with scientific notation.

If we have all the digits, we move the dicimal point   434   places to the right.
But we only need the first one or two digits to say that the number is larger than the   434th   power of ten,
which is a one followed by   434   zeros.

The number below is the integer part of the sum of all columns beyond the second.
The fractional part is a little more than one half.


The digits have been arranged so that it is easy to see that there are   434   after the initial   1.


When Whinnie the Pooh received his small helping of honey from Piglet, he said "I did mean a little bit larger small helping." Well, we need really to get a firm understanding of how big is the   434th   power of   10   and for this purpose I would like to look at a little bit smaller large number. It is a   1   followed by   135   zeros. I have expressed the little bit smaller large number as the product of three even smaller numbers, which will be discussed in the next three frames.


At the top of this page, I used the British pound, but I will not use the British (and German) system of naming numbers - except for this anecdote: I was in Germany when the global financial crisis was unfolding. The American law makers were debating their "trillion" dollar bailout for Wall Street, and the German head line said "billion" - which is their name for the American's trillion shown below. In the American system, the name comes from the number of groups of three zeros which follow the first group of three zeros. To avoid confusion when dealing internationally, it might be best just to use the powers of ten. In any case, if we multiply this number by   14   we get something that is still smaller than the US national debt.


To appreciate how big the next number is,
multiply the mass of the earth by   16   and two-thirds, and then
imagine how much more massive that is than one cubic meter of water.


A   1   with   100   zeros is called a googol ...


... and in my smaller large number the googol was in parentheses and multiplied by an American trillion. To appreciate how big is this US trillion googols, just imagine how much more voluminous the known universe is than a hydrogen atom.


And now we go back to my little bit smaller large number ...

... and the known universe as a multiple of the hydrogen atom is to the   135th   power of ten
as a cubic meter of water is
to   16   and two-thirds times the earth!

Universe busting is the name I like to call numbers this size or larger.


But look how much smaller my little smaller large number is than the larger large number we had before! We can cube the   135th   power of ten and still get something that is very much smaller than the   434th   power of ten, which is smaller than our   100,000   compoundings number - which was obtained above from the expression shown here: 100   times the   100,000th   power of   1+1/100.

And the sideways "v" always points to the smaller number.
That is to say, the big number is always at the big end.
AND(!) we can always read the symbol EITHER from left to right OR from right to left - whichever we prefer.


Since the "universe busting" explosion occurs in "all the other columns"
it seemed a good idea to look at when all the other columns surpass the first two.



I have a friend who argues for the reimposition of usury limits.
Based on these figures, I would suggest that the limit should be   2%.


Next,
I wanted to look at what the impact might be on sustainability
if we truncate the interest to the first two columns


The impact of truncation seems encouraging for high growth rates, such as 20%:


Here are some usefull scientific notation representations for some American big and small number names.



Here is where my little bit smaller large number came from.

This grows an American (trillionth of a trillionth of a trillionth) of a hundredth
at one basis point per year
for   4,000,000   years.


And here is what we get at ten basis points for   2,000,000   years.


BUT AGAIN (with the American number names):
Truncation to the first two columns seems to radically improve sustainability.


Finally, I have included last what started out as my first frame. How would it be if our lungs worked like our economies? We would not be able to exhale without excruciating pain! And if our economies worked like our lungs? That's how it should be. And sometimes I think today's experts are like medieval mechanical engineers. Things are set up so that abject misery and failure follow in the absence of a perpetual motion (growth) machine, and they are classically striving to build such a machine - oblivious to the fact that it is classically impossible. "And that's all I have to say about that." - Forest Gump.