March 24, 2018 Saturday
Bedtime Story
Jacob Bernoulli's "Summae Potestatum"
Jacob Bernoulli's "Summae Potestatum"
Last night I had left you with some
equations wherein Jacob Bernoulli is describing the sum on n integers raised to
the powers p as polynomial functions.
The letter n on the left hand side
represents the upper limit of the range of summation which will vary from 1 to
n depending on the number chosen.
The general formula comes later wherein
Jacob uses constant coefficients A, B, C, D.
It is these constant coefficients in the
general formula which are the famous Takakazu/Bernoulli numbers.
Today the Bernoulli numbers are denoted by
capital letter B and since the value of B will vary for each number n, they are
labeled as B1, B2, B3, B4 and so on
or more generally they are labeled as Bk Bn.
In this case, the general formula for
summation becomes:
ck
The term above is called falling factorial powers and is equal to
c(c-1)(c-2)...(c-k+1).
What is you see above is the modern version of the general formula for summation of integers that Bernoulli had envisaged.
This is how Jacob Bernoulli had written it
down in his book – The Art of Conjecture:
His notations were obviously a bit
different that today’s but the idea does not get lost in translation.
The capital letters that you see at the
bottom A, B, C, D are the Bernoulli numbers.
I shall list the first 20 Bernoulli numbers
for you.
We shall see how the first 20 Bernoulli
numbers look in the nights to come and then having understood what these
numbers are, we shall be armed with enough knowledge to return to the Note G of
Ada Lovelace.
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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