February 04, 2018 Sunday
Bedtime Story
Continuing with Menabrea - 10
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.
“Since the subtraction can only be effected
on the larger of the two numbers, it must be arranged that the disc of signs of
the larger number shall not move while the smaller of the two numbers is being
effaced from its column and subtracted from the other, whence the result will
have the sign of this latter, just as in fact it ought to be.
The combinations to which algebraical
subtraction give rise, are analogous to the preceding.
Let us pass on to multiplication.
When two numbers to be multiplied are of
the same sign, the result is positive; if the signs are different, the product
must be negative.
In order that the machine may act
conformably to this law, we have but to conceive that on the column containing
the product of the two given numbers, the digit which indicates the sign of
that product has been formed by the mutual addition of the two digits that
respectively indicated the signs of the two given numbers; it is then obvious
that if the digits of the signs are both even, or both odd, their sum will be
an even number, and consequently will express a positive number; but that if,
on the contrary, the two digits of the signs are one even and the other odd,
their sum will be an odd number, and will consequently express a negative
number.
In the case of division, instead of adding
the digits of the discs they must be subtracted one from the other, which will
produce results analogous to the preceding; that is to say, that if these figures
are both even or both uneven, the remainder of this subtraction will be even;
and it will be uneven in the contrary case.
When I speak of mutually adding or
subtracting the numbers expressed by the digits of the signs, I merely mean
that one of the sign-discs is made to advance or retrograde a number of
divisions equal to that which is expressed by the digit on the other sign-disc.
We see, then, from the preceding
explanation that it is possible mechanically to combine the signs of quantities
so as to obtain results conformable to those indicated by algebra.”
Here Ada Lovelace makes a very interesting
comment on her notes which she labels as note 5.
Menabrea in the above passages has been
tackling the problem of assigning the correct sign after the operation of an
algebraic operation.
You have to understand that Menabrea is
simply offering one of the possible ways by which a device such as the
analytical engine would tackle the issue of signs after mathematical
operations.
Menabrea is proposing especially dedicated
discs on top of columns of both the storage and the mill to track the signs.
He then taken us through each operation of
addition, subtraction, multiplication and division and explains how sign of the
product will be assigned by either advancing the sign-disc or holding it in its
position depending on the operation performed.
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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