January 16, 2017 Monday
Bedtime Story
Mathematical Induction (vs. Inductive Reasoning)
Axiom of Induction or in general, mathematical induction is a
technique of mathematical proof that is used to prove a statement that concerns
a well-ordered set.
One of the most famous well-ordered set is the set of natural
numbers.
The 9th axiom of Piano is exactly this: an axiom of
induction that is based on second-order logic.
It is important to stress at the very onset that one ought not to
confuse mathematical induction with inductive reasoning on which the sciences
of physics, chemistry and to some extent biology are based upon.
Mathematical induction is actually a deductive form of reasoning
and is used as a proof (in contrast to inductive reasoning which is inherently
uncertain and is all about probability of a certain conclusion being true).
It is believed that both Plato and Euclid used mathematical
induction, Euclid specifically using it to prove that the number of primes are
infinite.
Then it was Bhāskara II of Bijapur, Karnataka who somewhere around
1150 AD used this mathematical induction in his chakravala method (cyclic
algorithm) to solve indeterminate quadratic equations.
The indeterminate quadratic equations look something like this:
ax2 + bx = y2
Incidentally, Bhāskara II was also the first to discover the
principles of differential calculus.
Mathematical induction is a form of direct proof.
But what does one understand by a direct proof?
A direct proof is a method of establishing a truth or falsehood of
a statement using a combination of established facts.
By established facts, I mean well-known axioms, lemmas or
theorems.
This is in contrast to indirect proof which tends to start with
some form of hypothesis or many hypotheses, following which steps are taken to
eliminate most until one inevitable conclusion remains and that has to be the
truth.
Examples of such indirect proofs include proof by contradiction
and proof by infinite descent.
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.
Advertisements
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:
No comments:
Post a Comment