December 07, 2017 Thursday
Bedtime Story
Solving Polynomials by Addition Only
For today’s bedtime story, you will need to
refer to the four columns of numbers that we had briefly gone through last
night.
Note that the numbers in the last column
are always constant.
This is crucial for the working of the
difference engine.
Any polynomial equation of degree n, the
column number n + 1 will always be constant.
In our case, since it was a quadratic
equation, meaning n being 2, the third column was constant with values of 4.
Just in case you have doubt, that fourth
column is actually the third column since the first column is merely the values
being fed in.
The column that we constructed goes from
left to right.
It can also be constructed from right to
left.
The main beauty of this table is that now
that we know the value of the first and the fourth column, we can calculate the
values of second and third columns for the next value just by addition.
For example in this case, if we want to
calculate p(5), we need not actually solve the polynomial equation.
We can bring down the value 4 in the fifth
row.
Then the value of the third column is
obtained by adding 4 to 11 to get 15.
Then the 15 of the third column thus
obtained is added with 22, the number that was available in the second column
of the p(4) row.
This gives the value 37 which is the
solution to p(5) of this particular quadratic function.
The value 37 also becomes the value that
will be placed in the first column of the fifth row and will or can be used to
calculate the next order of the polynomial function namely p(6).
You can try mentally calculating p(6)
You will need to add 4 which is the number
of the last column to the number 15 which was placed in the third column of the
fifth row.
This will get us 19.
This 19 will then need to be added to 37 of
the second column which will get us 56.
This 56 is the solution of the polynomial
p(6).
This process can be repeated over and over
to get the values of subsequent polynomial functions.
So you see mon ami, we are solving a polynomial
equation without actually solving it but simply by adding few specific numbers
in a repeated pattern.
If this were to be done by the difference
machine, the requirements for the machine would be the ability to add two
numbers from one loop to the next and have the ability to store two numbers, in
this case those two numbers being the last numbers in the first and second
columns.
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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