December 05, 2017 Tuesday
Bedtime Story
Mathematics of Difference Machine
We shall continue with the quote from the
1953 book “Faster than Thought” written by Bertram Bowden that describes best
how Babbage got the idea of making mechanical machines to mass-produce
mathematical tables.
“Three or four of their mathematicians
decided how to compute the tables, half a dozen more broke down the operations
into simple stages, and the work itself, which was limited to addition or
subtraction, was done by eighty computers who knew only these two arithmetical
processes.
Here, for the first time, mass production
was applied to arithmetic, and Babbage was seized by the idea that the labors
of the unskilled computers could be taken over completely over by machinery
which would be quicker and more reliable.”
It was then in 1819 that the mathematician
polymath began to seriously work on a mechanical machine that would be able to churn
out solutions to polynomial functions.
Now I was left thinking why make a machine
that would solve polynomial functions for making mathematical charts of
logarithmic and trigonometric functions.
On some researching I came to find out that
most mathematical functions including logarithmic and trigonometric functions
can be approximated by polynomials.
You see, at the very heart of even the
first mechanical computing device, even a theoretical one, there lay pure
mathematics.
Let me briefly take you through the
mathematics of what in future would be known as the difference machine.
Babbage used two mathematical concepts for
designing his machine: finite difference and divided differences.
Let me first state what a finite difference
is and then we shall discuss at length divided difference that was used for the
machine.
For two numerical variables a and b, finite
difference is represented by the formula f(x + b) – f(x + a).
While that may not mean much to most people
like us, it seems it is a very useful concept for mathematicians.
This is so because the derivatives of
finite differences are crucial in finite difference methods as they help to
provide solutions to differential equations.
Divided differences, on the other hand, is
a recursive division process.
When used as an algorithm, it can help to
arrive at the coefficients of polynomial equations.
The mathematics and it’s notation would be
very alien to the mind of any average ape, but still I will briefly mention how
divided differences works out.
You know that polynomial equations can be
tackled both algebraically as well as geometrically as in graph forms.
Polynomials have in their equations or
functions variables in the form of x, y, z and so on and coefficients.
Polynomials are widespread and encountered in
several areas of different sciences and mathematics.
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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