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The Pythagorean Theorem is a geometrical expression used often in math and physics. It used to
2 2 2
find the unknown side of a right triangle. The exponential form of this theorem a + b = c . That is
the equation you use when you are looking for the unknown side of a right triangle, and it is what
I’ll demonstrate on the attached exhibit.
The upside down capital L in the bottom of the left hand corner indicates that sides A & B are
the legs of the triangle. Since we know side A = 5 inches and B = 3 inches we may fill that in to
2 2 2
or equation for step one. (1) 5 + 3 = c What the theorem will help us find is the c side of this
triangle. 2. 25 + 9 = c All we do is distribute 5 to the second power and 3 to the second power as seen is step two. Next, we add these two numbers together to get 34, 25+9=34, in step three. 3. 25+9=34 Then, in step four we find the square root of 34. 4. 34 In step five we see that 5.83 is the unknown side of the right triangle. 5. c= 5.83 We found this answer by using the Pythagorean Theorem as taught in geometrical form.
This theorem may also be summed up by saying that the area of the square on the hypotenuse, or opposite side of the right angle, of a right triangle is equal to sum of the areas of the squared on the legs.
The Pythagorean Theorem was a studied by many people and groups. One of those people being Euclid. Sometimes the Pythagorean Theorem is also referred to as the 47th Problem of Euclid. It is called this because it is included by Euclid in a book of numbered geometric problems. In the problem Euclid studied he would always use 3, 4, and 5 as the sides of the right triangle. He did this because 5 x 5 = 3 x 3 + 4 x 4. The angle opposite the side of the legs was the right angle, it had a length of 5. The 3:4:5 in the right triangle was known as a Pythagorean triple or a three digits that could be put in a right triangle successfully. These three numbers were also whole numbers and were used in the Egyptian string trick, which I will talk about later. This Pythagorean triple, 3:4:5, are the smallest integer series to have been formed, and the only consecutive numbers in that group that is important. These numbers can be, and often were, studied from a philosophical stand point.
The symbolic meanings of the 3:4:5 triple told by modern writers such as Manly P. Hall say 3 stands for spirit, 4 stands for matter, and 5 stands for man. Using Hall’s study the symbolism of this arrangement is as follows: “Matter” (4) lays upon the plane of Earth and “Spirit” (3) reaches up to the Heaven and they are connected by “Man” (5) who takes in both qualities.
A process similar to that of Euclid's 47th Problem was the Egyptian string trick. Egyptians were said to have invented the word geometry (geo = earth, metry = measuring.) The Egyptians used the 3:4:5 right triangle to create right triangles when measuring there fields after the Nile floods washed out there old boundary markers. The Egyptians used the same theory of Euclid,
5 x 5 = 3 x 3 + 4 x 4, to get there boundaries marked correctly.
Although Euclid and the Ancient Egyptians studied the theorem, the true inventor of it ( or the person most people believed invented it first ) was Pythagoras of Samos and his group the Pythagoreans. Pythagoras was a man born in 580 B.C. on the island of Samos, in the Aegean Sea. It is said Pythagoras was a man that spent his life traveling the world in search of wisdom. This search for wisdom led him to settle in Corona, a Greek colony in southern Italy, in about 530 B.C. Here Pythagoras gained famous status for his group known as the Brotherhood of Pythagoreans. This group devoted there lives to the study of mathematics. The group, as led by Pythagoras, could be described as almost cult-like because that it had symbols, rituals, and prayers. The group was also cult-like because of there odd ways of not writing down any of there discoveries. It was also said that Pythagoras himself sacrificed a hecatomb, or an ancient Greek ritual of 100 oxen, when he discovered the Pythagorean Theorem.
The group was also said to have vowed to secrecy. One day the Pythagoreans discovered irrational numbers. They referred to these numbers as “algon” or unutterable. They were so shocked by these numbers they killed a member of the group that mentioned them in public.
The group believed in many things had to do with numbers. They said “all things are numbers,” and also “numbers rule the universe,” Pythagoreans believed that numbers were divine. He also thought numbers one through ten, those of a decade, were especially sacred. Pythagoreans also thought that numbers had characteristics: 2 was female, 3 was male, odd numbers were good, and even numbers were evil. This belief by the Pythagoreans led to many discoveries including the Pythagorean Theorem.
The Pythagoreans first discovered numbers could be associated with shapes. Numbers six, ten, and fifteen were all triangular numbers because they can be arranged in equilateral triangles.
This study of numbers and shapes eventually led to the discovery of many different and important theory having to do with geometry. Although, nobody is really positive who invented the theorem, Pythagoras or the Pythagoreans? This is unknown because of there vow so secrecy and there neglect of writing there discoveries down. So now nobody is even sure if Pythagoras had anything to do with the discovery.
The puzzle of who invented the Pythagorean Theorem, Pythagoras or his followers, is so confusing because the group studied such a wide variety of different topics. Studies of the group includes many geometric proofs, astronomy, and music. The Pythagoreans believed that all these things had to do with numbers.
If you would ask my opinion on the theorem, I would have to say the original inventors were the Egyptians. They used the theorem, but the Pythagoreans were the first ones to write about and describe it more thoroughly. I think that the theorem was an important discovery for the future . I say this because the theorem was studied by so many great thinkers. The theorem is complex but simple because it is easy to use with right angles after you learn it, but it also has many philosophical meanings and parts to it. In all I think the Pythagorean Theorem is a confusing, but an important part to the past, present, and future of geometry.
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