March 09, 2018 Friday
Bedtime Story
We saw last night that the general formula
for compounding a principal amount is to multiply the amount x with (1 + 1/n)n
where n is the interval at which the compounding is done.
In case of 6 months compounding, the
initial 100 units will need to be multiplied by 1.5 times twice, or
100 x 1.52 = 225 at the end of 1
year
1.5 comes from 1 plus 1/2
In case of every 4 months it would become
33.33 % for 4 months and in case of quarterly it would become 25% for every 3
months.
Compounding it quarterly would give us
100 x 1.254 = 244.14 at the end
of 1 year
1.25 comes from 1 plus ¼
So what we are doing is this:
100 x (1 + ¼)4
Compounding it every monthly would give us
a yield of
100 x (1+ 1/12)12 = 261.3 at the
end of one year
So now the pattern must be obvious.
Even Jacob Bernoulli saw it and came to the
conclusion that for n compounding intervals, the interest for each interval
would be 100%/n and the value at the end of the year will be 100 x (1 + 1/n)n.
True mathematicians, as I keep insisting,
never stop at a solution but keep on extending their reasoning to the extremes.
So it was with Jacob; He wondered what
would happen if you keep on making the compounding intervals shorter and
shorter.
The solution that Jacob eventually came to
in the problem of continuous compounding interest, was to find the limit of (1
+ 1/n)n as n tends to infinity.
This above statement can be translated into
a beautiful equation in the language of mathematics which I am showing below:
This is probably the world’s most basic and
familiar limiting equation of mathematics and is not hard to understand, at
least on the hind sight and after its historical explanation.
Jacob himself did not use any calculus to
solve this equation but instead used the binomial theorem to show that the
limit had to lie between 2 and 3.
Just to elaborate on the idea of limit in
this specific case for non-mathematical brains including foremost myself, let
me put this same idea in plain English sentences.
Bernoulli discovered that if he had started
with a single unit of currency and invested it in a ponzi scheme that gave him
an annual interest of 100%, then if the compounding intervals were made smaller
and smaller so that compounding computations were done at smaller and smaller
intervals, his sequence approached a limit.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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