HISTORY OF PYTHAGOREAN THEOREM : HISTORY OF PYTHAGOREAN THEOREM GHADA SALEM 1
Pythagorean's Theorem : Pythagorean's Theorem 2
PYTHAGOREAN THEOREM : PYTHAGOREAN THEOREM 3
WHO IS BEHIND THE THEROM: WHO IS BEHIND THE THEROM PYTHAGORAS(569-500 B.C.E.) was born on the island of Samos in Greece. Traveled to Egypt to learn mathematics, and Philosophy 4
THE PYTHAGOREANS : Pythagoras believed that "Number rules the universe ,“ and gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities. THE PYTHAGOREANS 5
secrecy of PYTHGOREANS : The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them. The Pythagoreans were very secretive and they all took oaths to ensure that their discoveries remained with the Pythagorean society. secrecy of PYTHGOREANS 6
Hippasus and Irrational numbers : Hippasus and Irrational numbers 7
Pythagoreans reaction to Irrational numbers : Pythagoreans reaction to I rrational numbers 8
Proofs of Pythagorean theorem ( 400 hundred known proofs): Geometric approach 1 Numerical approach 3 4 5 5 12 13 9 40 41 Pythagorean triples Proofs of Pythagorean theorem ( 400 hundred known proofs) 9
PROOFS WITH Geometric approach: PROOFS WITH Geometric approach Pythagoreans : Book II of Euclid's Elements . Euclid proof.docx 10
The 3-4-5 Right Triangle In Ancient Egypt: The 3-4-5 Right Triangle In Ancient Egypt Rope stretchers laying the foundations for the Pyramids 11
famous Hindu mathematician: famous Hindu mathematician 1- He used two proofs for the Pythagorean theorem. 2- In the first Theorem He used 4 congruent triangles and two congruent square Bhaskara 12
First Proof: Area of the blue triangles = 4(1/2) ab Area of the yellow square = Area of the big square = 4(1/2) ab + = 2ab+ -2ab + = + Since, the square has the same area no matter how you find it A = = + First Proof 13
More GEOMETRIC PROOFS: More GEOMETRIC PROOFS http:// www.cut-the-knot.org/pythagoras/index.shtml 14
Geometric proofs: Geometric proofs Based on creating geometric shapes like squares, rectangles , triangles , trapezoids…etc. then forming two equations of areas of congruent geometric shapes , these equations will equal each other and by some algebraic cancelation end up with the Pythagorean theorem. 15
Garfield which was the twentieth president of the United States: Garfield which was the twentieth president of the United States 16
Garfield proof: First, let us make the trapezoid. You start with a triangle of sides a, b, and c where the sides a and b meet to form a right angle. Garfield proof 17
GARFIELD PROOF : Then put a second triangle below the first such that side a is an extension of the other triangles b side. GARFIELD PROOF 18
GARFIELD PROOF : Next connect the end of side a at the top with side b on the bottom to create the trapezoid. GARFIELD PROOF 19
GARFIELD PROOF : T he area of this trapezoid will be the same as the sum of the areas of the three triangles inside the trapezoid GARFIELD PROOF 20
GARFIELD PROOF : GARFIELD PROOF + + = = 2ab + 21
Pythagorean Triples…..: Pythagorean Triples….. Pythagorean Triples and Generalized Fibonacci Numbers: Using the Fibonacci-type series of 4 numbers: 1, 3, 4, 7 : Multiply the middle two numbers and double the result, here 3 and 4 multiply to give a product of 12 and this doubles to give 24 : the first side of our Pythagorean Triangle Multiply the two outer numbers ( 1 and 7 ) giving 7 : the second side of the Pythagorean triangle Add the squares of the middle two numbers ( 3 and 4 ) to get the third side: here 3 2 + 4 2 gives 25: the hypotenuse of the Pythagorean triangle of 24, 7, 25. 22
PYTHAGOREAN TRIPLES : PYTHAGOREAN TRIPLES odd A next B=A+2 1 / A + 1 / B Hypotenuse 1 3 4/3 5 3 5 8/15 17 5 7 12/35 37 7 9 16/63 65 The Two unit - Fractions method of generating Pythagorean Triples 23
The Two - Fractions method : The Two - Fractions method Take any two fractions (or whole numbers) whose product is 2 and 6 Add 2 to each fraction: and 8 Cross multiply to turn both into whole numbers and 24 These are two sides of a Pythagorean triangle: To find the of third side just apply the Pythagoras Theorem 7 2 +24 2 = 625 √625 = 25 24
The m , n formula for generating Pythagorean Triples: The m , n formula for generating Pythagorean Triples You can generate Pythagorean Triples if you can find two integers m & n t hat satisfy this equation ( m 2 – n 2 ) 2 + (2 m n) 2 = ( m 2 + n 2 ) 2 ( Euclid generated this formula) Once we have found one triple, we have seen that we can generate many others by just scaling up all the sides by the same factor. 25
some of the Pythagorean's triples : some of the Pythagorean's triples 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 13 84 85 15 112 113 17 144 145 19 180 181 21 220 221 26
Some of the properties of the Pythagoras Triples:: Some of the properties of the Pythagoras Triples: one side is a multiple of 3 one side is a multiple of 4 so the product of the two legs is always a multiple of 12 and the area is therefore always a multiple of 6 one side is a multiple of 5 so the product of all three sides is always a multiple of 3×4×5=60 27
The Pythagorean primitive triples: : The Pythagorean primitive triples: A Primitive Pythagorean Triple is a Pythagorean triple a , b , c with the constraint that 1- gcd ( a , b )=1 , which implies gcd ( a , c )=1 and gcd ( b , c ) =1. Example: a =3, b =4, c =5. Exactly one of a , b is odd, c is odd. This follows from: At least one leg of a primitive Pythagorean triple is odd since if a , b are both even then gcd ( a , b )>1 . primitives.docx 28
Plotting the Primitive Pythagorean triangle : Plotting the Primitive Pythagorean triangle 29
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REFRENCES : REFRENCES 32