February 07, 2018 Wednesday
Bedtime Story
Continuing with Menabrea - 13
We are continuing with the treatise of
Menabrea:
“Sketch of the Analytical Engine” that was
translated into French by Ada Lovelace and published along with her notes in
1842.
“These new cards may follow the first, but
may only come into play contingently upon one or other of the two circumstances
just mentioned taking place.
Let us consider a term of the form abn;
since the cards are but a translation of the analytical formula, their number
in this particular case must be the same, whatever be the value of n; that is
to say. Whatever be the number of multiplications required for elevating b to
the nth power (we are supposing for the moment that n is a whole number).
Now, since the exponent n indicates that b
is to be multiplied n times by itself, and all these operations are of the same
nature, it will be sufficient to employ one single operation-card, viz. that
which orders the multiplication.
But when n is given for the particular case
to be calculated, it will be further requisite that the machine limit the
number of its multiplications according to the given values.
The process may be thus arranged.
The three numbers a, b and n will be
written on as many distinct columns of the store; we shall designate them V0,
V1 and V2; the result abn will place itself on the column V3.
When the number n has been introduced into
the machine, a card will order a certain registering-apparatus to mark (n-1),
and will at the same time execute multiplication of b by b.
When this is completed, it will be found that
the registering-apparatus has effaced a unit, and that it only marks (n-2);
while the machine will not again order the number b written on the column V1 to
multiply itself with the product b2 written on the column V3, which
will give b3.
Another unit is then effaced from the
registering-apparatus, and the same processes are continually repeated until it
only marks zero.
Thus the number bn will be found
inscribed on V3, when the machine, pursing its course of operations, will order
the product bn by a; and the required calculation will have been
completed without there being any necessity that the number of operation-cards
used should vary with the valus of n.
If n were negative, the cards, instead of
ordering the multiplication of a by bn, would order its division;
this we can easily conceive, since every number, being inscribed with its
respective sign, is consequently capable of reacting on the nature of
operations to be executed.
Finally, if n were fractional, of the form
p/q, an additional column would be used for the inscription of q, and the
machine would bring into action two sets of processes, one for raising b to the
power of p, the other for extracting the qth root of the number so obtained.
Again, it may be required, for example, to
multiply an expression of the form axm + bxn by another
Axp + Bxq, and then to reduce the product to the least
number of terms, if any of the indices are equal.
The two factors being ordered with respect
to x, the general result of the multiplication would be Aaxm+p + Abxn+p
+ Baxm+q + Bbxn+q.”
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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