January 19, 2017 Thursday

Bedtime Story 

Falling Dominoes Analogy of Mathematical Induction

One can imagine a long infinite row of dominoes standing on one end.

The line starts in front and extends to the right infinitely.

Now we shall use the 2-step mathematical induction:

1. The first domino falls to the right.

2. If a standing domino falls to the right, be it any which one located anywhere, then its adjacent right neighboring domino also tips and falls to the right. 

If we take these 2 statements to be given, then using mathematical induction we can conclude that all of the dominoes will tip and fall to their right.

This very simple analogy is exactly what mathematical induction is and how it works.

Now there is something more that we can glean from this domino analogy about our famous set of natural numbers.

You will surely agree that the falling dominoes would only work if they are arranged only in a very particular manner.

They must be arranged very precisely with exactly same inter-domino gap for the domino effect to “run on” along this hypothetical infinite series of stacked dominoes.

For any other arrangement of dominoes, this mathematical induction will not work.

In a very similar way, the 9^th axiom of Peano or the axiom of induction depends on something unique about the natural numbers.

That unique property of the set of natural numbers is this; that any member of this set can be reached from 0 by adding one adequate number of times.

Now if we go back to Peano axioms, it will become clear that there is only one specific type of set with a specific type of architecture that will satisfy it. 

Very much the same way as the right falling dominoes.

There is another essential point of logic involved here.

All the statements from 1 to 8 are of the first-order logic.

Only the 9^th axiom or the axiom of induction is of the second-order logic.

Hence it means that there is no way of defining the set of natural numbers using only the first-order logic statements.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

             

The arrangement of natural numbers (in contrast to say the real numbers) is akin to that of these dominoes arranged with very strict spacing. This allows the use of mathematical induction in defining them.  

                

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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