## Sunday, September 21, 2008

### PYTHAGORAS' THEOREM

Hye...okey...let's talk about my favorite subject in school...mathametic...this is one of the many subject that i love...pythagoras theorem...
PYTHAGORAS: FATHER OF NUMBERS

* Born around 570 B.C. on Samos, an island lying off the western coast of Asia Minor
* Founder of the Pythagorean school (surprise, surprise!)
* He was a vegetarian
* Was also a philosopher as well as a pretty good mathematician
* Famous for the Pythagoras'Theorem (who woulda guessed it?)

PYTHAGORAS' THEOREM
This theorem is great due to the fact it can be easily proved and is easy to understand.The theorem basically states that for a right angled triangle, the square of the hypotnuse is equal to the sum of the squares of the other 2 sides:
So if we know any 2 sides, we are able to calculate the length of the remaining side:
By rearranging the formula:
This is my favourite proof, developed by the Indian astronomer Bhaskara (1114-1185):
The area of the big square with sides a + b = (a + b)²
This is equal to the sum of the area of the internal square (c²) plus the sum of the 4 triangles (4 x ½(a x b))Equating these:
(a + b)² = (4 x ½(a x b)) + c²(expanding)
a² + 2ab + b² = 2ab + c²(subtracting 2ab from both sides)
a² + b² = c²
There are many other proofs, click here to see 72 of them!
A brilliant piece of maths i think you will agree. Much better than Fermat's Last Theorem, of which the proof is only understood by a select few........

USES OF THE THEOREM
We can apply this theorem to many everyday situations:
* Construction workers finding lengths of supports
* Finding the argument of a complex number
* Calculating distances in cartesian coordinates
* Navigation of the sea, land and air
* Finding angles of elevation and depression
* Checking if a traingle is right angledOk, so not really 'everyday' for most of us, but it is still extremely useful.

CONSEQUENCE OF THE THEOREM - PYTHAGOREAN TRIPLESA
Pythagorean triple consists of three positive integers a, b, and c, such thata² + b² = c².
This is commonly written (a, b, c), and a well-known example is (3, 4, 5):3² + 4² = 5²9 + 16 = 25
There are infinitely many Pythagorean triples.
The simpliest proof for this being any multiple of a Pythagorean triple is again a Pythagorean triple, ie, if (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k, therefore there will be infinitely many.
More examples of Pythagorean triples:
(5, 12, 13)
(7, 24, 25)
(9, 40, 41)
(11, 60, 61)

GENERATING PYTHAGOREAN TRIPLES
An effective way to generate Pythagorean triples is by Euclid's formula.
This was recordered in his book Elements around 300BC:
If m and n are two positive integers with m > n, then:
* a = m² - n²
* b = 2mn
* c = m² + n²

PROPERTIES OF PYTHAGOREAN TRIPLES
* Exactly one of a and b is odd, c is always odd
* Exactly one of a and b is divisible by 3
* Exactly one of a, b, c is divisible by 5
* ab is always divisible by 12
* abc is always divisible by 60
* Exactly one of a, b, (a + b), (a - b) is divisible by 7
* Every integer greater than 2 is part of a Pythagorean triple
* There is no Pythagorean triple in which the hypotenuse is equal to either 2a or 2b

Simply amazing!
So thanks a lot Pythagoras, we owe you 1!