January 17, 2017 Tuesday
Bedtime Story
Using Mathematical Induction for Natural Numbers
Proof by mathematical induction proceeds in two steps.
Let us study these 2 aspects with its most fundamental application
which is:
The proof of any statement concerning the set of natural numbers.
I wish to take you back to my bed-time story that I wrote about
the little Carl Gauss when he was just eight (there are many other similar
anecdotes regarding this wunderkind).
The teacher had asked the students to add all the numbers from 1
to 100 so that the teacher could relax a bit keeping the students busy.
Gauss had the answer in no time.
He was extremely quick in figuring out an unique pattern that
could be made use of in coming to the solution.
He simply multiplied 101 hundred times and divided the product by
2.
(100 X 101)/2 = 2
We can generalize this trick of little Gauss to make a statement
that defines natural numbers.
0 + 1 + 2 + 3 + 4 +…+ n = n(n+1)/2
P(n) = n(n+1)/2
A statement can be made that this formula P(n) is true for all
natural numbers.
It can be easily proved by mathematical induction.
The proof goes as follows:
The first step is:
1 The base case: is a proof that a statement holds true for the
first element of a well-ordered set.
In our case, or Gauss case the base case is P(0) = 0
That this formula holds true for n = 0
The second step is:
2. Inductive Step: this step is the proof that if the statement
made in the base case holds true for a natural number n, then it also holds
true for n + 1.
In our case, that would amount to saying:
If P(k) holds (as it did for P(0)), then P(k+1) must also hold
good.
We shall study the proof of it in the nights to come.
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.
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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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