Nature has 2.71828… fingers

Napoléon Bonaparte, Château de la Malmaison - photo by Jean-Marie Hullot

Napoleon introduced the metric system to Europe which is based upon the fact that we have ten fingers.  The decimal system goes back much further being used by the Chinese thousands of years ago of which the abacus in a well known example.

The following table lists the prefixes used in the metric system, take from Appendix A of Physical Chemistry: (click to enlarge)

Although I do not have 12 fingers, there are many advantages of base twelve rather than base 10.  The number 12 is divisible by 2, 3, 4 and 6 making mental arithmetic much easier than base 10 (divisible only by 2 and 5).

Of course after February 1971 the old base 12 of British currency became decimalised, replacing the 240 old pennies of the Pound Sterling with 100 new pennies.  Before this there were 20 shillings to the pound, 12 pennies to the shilling, two half pennies (ha’pennys) to a penny and two farthings to a ha’penny .  That is there used to be 960 farthings to the pound.  Today a pint of beer costs about 2,000 farthings.

When I was almost 5 years old, I attended my first school, Arundel House, in Surbiton, Surrey.  One day on my way, I was happy to have a farthing in the pocket as I passed a confectionary shop.  I loved ice cream sandwiches (a wedge of ice cream between two wafers) and so I boldly walked into the shop, only to find that it was too expensive at sixpence (my weekly allowance!).  I said to the lady behind the counter,

How much are the wafers please?

Eight for a penny luv.

she replied

After some deep thought I said,

I’ll have two please.

But I digress.  Today we have the decimal system and it is here to stay, but what about Nature?  Nature does not have ten fingers, but rather the base for natural processes is the natural number e (Note that “e” is the symbol used, not “e” which is the charge of the electron).  The value of this number is

e = 2.71828…

We talk of exponential and natural logarithmic processes, so that log_e is special and denoted by “ln” while all the other logarithms have their base specified, for example in the decimal system, log₁₀.

The reason that the natural number e has this peculiar value is because Nature evolves continuously so that every time an atom decays or a cell divides, that new entity is instantly part of Nature.

To make this point, contrast the way banks give interest.  If you have $1 in the bank at an interest rate of say, 10%, then after one year you have

Suppose the bank next door gives the same interest rate but compounds it every six months.  Then after one year you have

But you could have monthly compound interest, and this gives

Or daily interest of

For Nature the limit is taken leading to continuous compounding,

and if you put x =1 the value of the natural number is

(You can do this on your calculator using the y^x function and values of x = 1, 2, 12, 365, etc. ). Indeed it would be difficult to have a system of units based on e rather than 10.

The half life of any exponential process does not depend upon how much you start with.  I mentioned in the blog on the Chemical Kinetics of Sobering Up, that first order kinetics is the most common, and that is exponential growth or decay.  The half life of exponential processes does not depend upon how much you start with, so after one half life, you have 0.5 left if you started with 1.0, and 500 left if you started with 1,000.  The formula for the half life is given by

where k is the rate constant, or if you want the interest rate.  A useful rule of thumb:  if you want to know how long it takes your money to double, divide 70 by the interest rate:

Approximate time to double your money =

At a rate of 10% your money will double in 7 years .  If you invest $1,000 in a tax free pension account at age 20 at 10% per year, then when you are 55 years old, that $1,000 will have grown to $32,000!

Consider however the human population growth at about 1.2% per year.  It will take about 60 years to double, so in 2070 the population would be about 12 billion and in 2130 it could be 24 billion.

Something is going to have to give.