November 19, 2016 Saturday
Bedtime Story
From Primes to Integers
From Primes to Integers
Let me explain the fundamental theorem of arithmetic with an
example.
Take any number say 240.
240 = 24 x 31 x 51 = 2 x 2 x 2 x
2 x 3 x 5 = 5 x 2 x 2 x 3 x 2 x 2
The fundamental theorem of arithmetic says that every integer
greater than 1 is either a prime or a product of prime numbers.
It goes on to add that it does not matter how the factorization is
done, it will always have the same primes same number of times.
In the example above, 240 will always be a product of four 2s, one
3 and one 5.
They can be arranged differently, but the number of 2s, 3s and 5s
will remain constant.
With this theorem, you can also understand why 1 could not have
been considered a prime.
Had 1 been a prime, then the product of primes would not be unique
as 1 can be multiplied one time or infinite times and this very fundamental
theorem would be violated.
In a way, the prime numbers can be seen as a building block of the
natural numbers and going a bit further perhaps the edifice of the whole
structure of mathematics.
That the prime numbers have any utility (or not) was never the
concern of the mathematicians.
Let us leave prime numbers for now and return back to the set
theory notations.
The letter Z (actually German Zahlen) represents the integers.
Integers are those numbers that can be written without having to
use fraction.
Seen through the concept of number line, integers can be imagined
as equally spaced discrete points.
They extend to both the positive and negative side of the number
line indefinitely.
In the group theory of mathematics, the integers form the smallest
circle or the ring that contain the natural numbers.
You will be very surprised how mathematicians have devised
ingenious ways to represent numbers on paper, always attempting to give more
and more rigorous and formal definition rather than relying on commonly held
intuitive beliefs or notions.
Within the set of integers, the letter E represents the set of
even integers.
The letter O on the other hand holds all the odd integers.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night and my fellow cousin ape.
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, may I
suggest this large collection of Kids Songs:
Integers in Group Theory
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