May 13, 2018 Sunday

Bedtime Story 

Note E of Ada Lovelace - Part 4

Tonight we shall continue with the Note E of Ada Lovelace where she is describing the mechanics of the multiplication operation (A + A₁cosθ + A₂cos2θ + A₃cos3θ…).(B + B₁cosθ).

As I explained to you last night, it is rather simple and the product will appear as:

BA + BA₁cosθ + BA₂cos2θ + BA₃cos3θ…

        B₁Acosθ + B₁A₁cosθ.cosθ + B₁A₂cos2θ.cosθ + B₁A₃cos3θ.cosθ…

So let us see how the Engine will handle them by placing them in different variable columns:

 

The variable belonging to each coefficient is written below it, as we have done in the diagram, by way of memorandum.

The only further reduction which is at first apparently possible in the preceding result, would be the addition of V₂₁ to V₃₁ (in which case B₁A should be effaced from V₃₁).

The whole operations from the beginning would then be-

First series of            Second series of      Third series, which

Operations                Operations              Contains only one

                                                             (final) Operation

¹V₁₀x¹V₀=¹V₂₀    ¹V₁₁x¹V₀=1V₃₁      ¹V₂₁x¹V₃₁=²V₂₁,

                                                    And V₃₁ becomes=0

¹V₁₀x¹V₁=¹V₂₁    ¹V₁₁x¹V₁=1V₃₂

¹V₁₀x¹V₂=¹V₂₂    ¹V₁₁x¹V₂=1V₃₃

¹V₁₀x¹V₃=¹V₂₃   ¹V₁₁x¹V₃=1V₃₄

We do not enter into the same detail of every step of the processes as in the examples of Notes D and G, thinking it unnecessary and tedious to do so.

The reader will remember the meaning and use of the upper and lower indices, etc, as before explained.

To proceed: we know that

  

(3)

               

Consequently, a slight examination of the second line of (2) will show that by making the proper substitutions, (2) will become

 

These coefficients should respectively appear on

V20           V21              V22                V23                V24   

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:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg