May 21, 2018 Monday
Bedtime Story
Note G of Ada Lovelace - Part 13
You may not recall, but we had left the
Note G at the point when Ada Lovelace brought in the subject of repetitive
usage of the very same seventy-five Variable-cards for the computation of every
succeeding Number of Bernoulli.
That was on the Wednesday night of May, 09,
2018 (Note G- Part 12).
The program or more rightly algorithm for
computation of Bernoulli numbers would involve cycle of a cycle (nested loops
in modern terms of computer programming) of Variable-cards which was covered in
great detail in the Note E that we had very recently covered in great detail.
It is this idea of cycles that was
explained in Note E using very original mathematical notations that will
finally result in the production of Numbers of Bernoulli.
Lets us see how.
We are now almost at the end stage of Note
G of Ada Lovelace, and overall very near to finishing off with the Notes of
hers.
“If we now apply the notation for cycles,
as explained in Note E., we may express the operations for computing the
Numbers of Bernoulli in the following manner:-
(1…7),(24,25)……………gives B1=1st
number;(n being=1)
(1…7),(8…12),(24,25) gives B3=2nd
number;(n being=2)
(1…7),(8…12),(13…23),(24,25) gives B5=3rd
number;(n being=3)
(1…7),(8…12),(13…23),(24,25) gives B7=4thnumber;(n
being=4)
…………………………………………………………………………………………
…………………………………………………………………………………………
(1…7),(8…12),∑(=1)n-2(13…23)(24,25)gives B2n-1=nthnumber;(n
being=n)
Again,
represents the total operations for
computing every number in succession, from B1 to B2n-1
inclusive.
In this formula we see a varying cycle of
the first order, and an ordinary cycle of the second order.
The later cycle in this case includes in it
the varying cycle.
On inspecting the ten Working-Variables of
the diagram, it will be perceived, that although the value on any one of them
(excepting V4 and V5) goes through a series of changes, the office which each
performs is in this calculation fixed and invariable.
Thus V6 always prepares the numerators of
the factors of any A; V7 the denominators.
V8 always receives the (2n-3)th factor of A2n-1,
and V9 the (2n-1)th.
V10 always decides which of two courses the
succeeding processes are to follow, by feeling for the value of n through means
of a subtraction; and so on; but we shall not enumerate further.”
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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