Introduction
The Pythagorean Theorem is one of the oldest mathematical proofs. This more than famous theorem is named after Greek philosopher and mathematician, Pythagoras. The theorem is not the only mathematical concept he was interested in. He also explored the principles of mathematics and the idea of proof. As a result of his ambitions, there was an amalgamation of mathematics and astronomy. Therefore, by searching for a divine truth through mathematics, he made many pivotal discoveries in different areas of math and science that are both mathematically sound and continue to be applied today.
Pythagoras
While Pythagoras helped solve some of the world’s hardest questions, there's still a copious amount of questions regarding his life. His life is clouded by legends and obfuscation, but he lived approximately around 570-495 BC and died around the age of 75. One question that hasn’t rested even after ~1,500 years after his death is who was his teacher? A popular theory is that the Egyptians taught him geometry, the Chaldeans astronomy, Phoenicians taught him arithmetic, and the Magi–principles of religion and practical maxims for the conduct of life. In his adulthood, he accomplished many significant contributions to both mathematics and scientific branches, as well as music, astronomy, and medicine.
Pythagorean Theorem
In mathematics, the Pythagorean theorem is a fundamental relation in Euclidean geometry among three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be written out as: a2+b2=c2, when c is the length of the hypotenuse and a and b are the lengths of the triangle’s other two sides.
Proof by Rearrangement
This theorem has been given a copious amount of proofs–possibly the most for one theorem. All of the proofs are so differing, including geometric and algebraic proofs. The first proof is named “Proof by rearrangement”. This takes place when two squares each contain four identical triangles, the only difference between the two triangles is the way the triangles are arranged. Thus, the white space encompassed by the two large triangles must have the equal area. Equating the area of the white space yields the Pythagorean theorem.
Law of Quadratic Reciprocity
This proof is based on the proportionality of the sides of two similar triangles upon that the ratio of any two analogous sides of similar triangles is the same regardless of the triangle sizes. This right triangle, with the right angle located at C, draw the altitude from C and say H is the intersection. It divides the length of the hypotenuse (longest side) into parts. The new, smaller triangle is still similar to the older, larger triangle because of they both have a right angle. This requires the triangle postulate which is when the sum of the angles in a triangle is two right angles, and is equivalent to parallel postulate: BC/AB=BH/BC and AC/AB=AH/AC. This is the first result that equates the cosines of the angles. The role of this proof in the history of mathematics is the subject of much speculation. Not many mathematicians used it, but instead created their own.
Pythagorean Triples
This formula is made of three positive integers, a well known example of this is (3,4,5). The Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with only the non integer side lengths do not form Pythagorean triples. For example a triangle with sides that a = b = 1 and c = √2 is indeed a right triangle, but not a Pythagorean triple because √2 is not an integer. In addition, 1 and √2 do not have any integers in common because √2 is irrational. Pythagorean triples have been known since the beginning of time.
Law of Cosines
While the history of Pythagoras is very limited, all historians agreed that he was the first person to claim that the Earth is a sphere. His view on the cosmos was very elementary. The Earth orbited a central fire (the sun) and so did all of the stars and other planets. He also claimed its because of the Earth’s orbit that Earth has a night and day. Pythagoras was a fundamental key to unlocking the distances between each star and planet. This is the Law of Cosines: c2=a2+b2-2ab cos C, this is when sides a, b, and c of the triangle with angle c being the opposite. This equation comes from the knowledge that not all angles in space are a perfect ninety degrees, so this is what’s needed to be taken into consideration.
Euclidean Geometry
This is a mathematical system that consists in assuming and deducing many theorems from a set of appealing amioms. Even through this type of mathematics had been in the books for years, it wasn't until Euclid (hence the name) showed how proportions could manage a spot between comprehensive, deductive, and logical systems. Euclidean Geometry dates back to the 19th century, being discovered along with many of the consistent non-Euclidean geometries are known. This type of geometry is a perfect examples of synthetic geometry, meaning it proceeds logically from axioms to proportions without the use of coordinates.
Conclusion
In this brief, four page report about the Pythagorean theorem, it covered some of the laws to proofs. Obviously this is just scratched the surface of this complex topic. There are still so many different types of mathematics to be discovered and expressed. One of the oldest mathematical proofs, the Pythagorean Theorem is still one of the most popular and most used proofs around the world.