July 2020
umasundresh
DICE FACT
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ONE THOUSAND FACTS
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FUNDAMENTAL THEOREM OF ARITHMETIC
This theorem says that every positive number (except 1) is either a prime number or can be written uniquely as a product of prime numbers. (i.e.,) we can always break a positive integer into prime factors.
Euclid gave an almost complete proof over 2000 years ago.
| 2 | PRIME |
| 3 | PRIME |
| 4 | 2 X 2 |
| 5 | PRIME |
| 6 | 2 X 3 |
| 7 | PRIME |
| 8 | 2 X 2 X 2 |
| 9 | 3 X 3 |
| 10 | 2 X 5 |
| … | … |
Carl Friedrich Gauss was the first who provided the first proof in 1801.
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TWIN PRIME CONJECTURE
There are infinitely many prime numbers ‘p’ such that ‘p+2’ is also prime. Such a kind of pairs of prime numbers are called as twin primes.
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ZERO CONCEPTS
- Zero is the number which is neither positive or negative
- Mayans were the first to use zero as a number.
- Brahmagupta’s Brahmasputha siddhanta was the first work to tell the world about  the rules of using zero.
- Leonardo Fibonacci worked hard to popularize zero in Europe.
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PYTHAGORAS THEOREM
Pythagoras theorem holds an important place in geometry.
The theorem states that
There are hundreds of different proofs of the Pythagoras theorem.
One of the following is proved by US President Garfield.
Steps to understand the construction of Trapezium:
Build another triangle like the first one, however, this time side ‘b’ will radiate outward in a straight line from the initial side ‘a’.
Side ‘a’ construct from ‘b’ which is also parallel to the initial side ‘b’.
Side c connecting the endpoints of the new a and new b.
We need to find the unknown angle ‘x’.
Angle of the straight line sum upto 180°
Therefore, 90°-ϴ +x + ϴ = 90°
Now join the end points of the sides ‘b’ and ‘a’.
View the illustration as a trapezium.
Now it is easy to understand the proof given by Garfield.
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MATHEMATICAL INDUCTION
Induction is the method of considering certain set of statements as true and coming up with the general conclusion.
We can also compare this with the recursion in computers.
The description is:
1. Show the first element of the series is true
2. Then assume any one of the element in the series is true and prove the very next element is true.
3. Conclude for all.
The falling dominoes speaks about this:
1. The first domino falls
2. If any one of the domino falls, the next will fall
The conclusion is : All fall down.
Mathematically we says that,
First prove for n=1
Assume the result is true for n=k, and prove the result is true for n=k+1.
Then generalize it for all
one of the easiest induction problem anyone can solve is:
Sum of the positive integers =n(n+1)/2
(i.e.,) 1+2+3+…+n=n(n+1)/2
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INFINITY
We all know that modern mathematical definition of infinity in set theory was well developed in 19th century.
But over 3 thousand years ago, the yajur veda described infinity as
“IF YOU REMOVE A PART FROM INFINITY OR ADD A PART TO INFINITY, STILL WHAT REMAINS IS INFINITY”.
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SURREAL NUMBERS
These are the elements of a huge set of numbers that includes all real numbers as well as infinite numbers and infinitesimal numbers.
* Infinite numbers: Larger in value than any positive integer
*Infinitesimal numbers: Smaller than any positive real number.
