Pythagorean Theorem
Pythagoras was born in Samos, Greece around 570 BCE. From there he emigrated to Croton, Italy where most of his most important ideas and theories would develop. Early on, Pythagoras founded a society of disciples where he introduced the idea eternal recurrence into Greek thought, and it was Pythagoras’ ambition to reveal the basis of divine order. This is how Pythagoras came to mathematics, and he saw math as a purifier of the soul, and saw numbers in everything. He was convinced that the divine principles of the universe can be expressed in terms of relationships of numbers.
Over 4000 years ago, the Babylonians and the Chinese already knew that a triangle with the sides of 3, 4, and 5 must be a right triangle. They used this knowledge to construct right angles. Pythagoras studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn’t only work for the special triangles, but for any right triangle. This can be written in the form a^2 + b^2 = c^2, and today this is what is known as the Pythagorean Theorem.
The Pythagorean Theorem was one of the first times in human history that people could calculate a length or distance using only outside information. The train of thought used by Pythagoras was the first time the idea of a unset variable was used, and this idea would be used in the later development of Algebra, Trigonometry, Topology, and, eventually Calculus. The idea of A^2 + B^2 = C^2 was also one of the very first set area formulas, and set formulas are the root of Geometry. Also, in Trigonometry, the Patagonian identities such as Sin^2 + Cos^2 = 1 set up one of the main ways to prove trig equations and equalities, so it is important to higher level math as well. The Pythagorean Theorem is one of the basic roots of modern mathematical science, and, while its concept is simple in today’s terms, the value of it lies in all math studies it has helped in developing.
Pythagoras was first rejected by the mathematical society when he proposed his theorem, because the people claimed that he was merely the founder of the Pythagorean society and that many of his scientific findings were done by his members and dedicated to him. They claim this was not the work of Pythagoras, and he only gathered facts, and never a deep understanding of mathematics. It is said that Hundreds of years before Pythagoras, Egyptians discovered a way to make a right angle. In their construction of buildings, they had a rope with evenly spaced knots. They knew that if they made a triangle with it, and the sides were three, four, and five lengths, the angle between the two shorter sides would be a right angle. Also, Archeological evidence shows that the Babylonians and Chinese were also aware of this special right triangle. It is not yet confirmed whether Pythagoras was the first person to have found the relationship between the sides of the right triangles, as no texts written by him were ever found.
Proof 8
Pythagoras came up with this proof after playing around with the apple that he used in proof 7. In proof 7, he came up with BC is equal to BD, DC; therefore the figure on BC is also equal to the similar and similarly described figures on BA, AC.
He came up with proof 8 from proof 7, and even though it was ugly, it still serves as a similar purpose. Starting with the triangle 1, we add three more in the way done in proof 7: similar and similarly described triangles 2, 3, and 4. From the ratios from proof 6, (AB)(AB) + (AC)(AC) = (BD)(BC) + (DC)(BC)= (BD+DC)(BC) = (BC)(BC). From this we arrive at the sign lengths as shown in the diagram below.
It is possible to look at the final shape in 2 ways, the first being of the union of rectangle 2, and 1, 3, and 4, and the second being the union of the rectangle 1 and 2, and the triangles 3, and 4. After equating the areas it lead to, ab/c · (a² + b²)/c + ab/2 = ab + (ab/c · a²/c + ab/c · b²/c)/2. Finally after simplifying this we get to the final equation of ab/c · (a² + b²)/c/2 = ab/2, or (a² + b²)/c² = 1.
The Way I Understand
The most remarkable thing about the Pythagorean Theorem is how much it is used in our daily everyday lives. It is used anywhere from building a house, to tracking a cell phone call. The Pythagorean Theorem is all around us, and most of us don’t even realize its implications to our world. Have you ever wondered how a baseball diamond is made, or how far a catcher has to throw the ball to get from home plate to 2nd base? Let’s say that there is 90 feet between each base. By using the a^2 +b^2 = c^2 formula we can plug in 90 for a and b and get 90^2+ 90^2 = c^2, which will give us 810 + 810 + c^2. From this we can calculate further to get 127.292 feet. Often, when builders want to lay the foundation for the corners of a building, one of the methods they use is based on the Pythagorean Theorem. In laying out the footers or corners of any building the concept of a 3-4-5 right triangle is applied. When carpenters are using ladders they can use the Pythagorean Theorem, for example if you have a 35 foot ladder positioned 21 feet from the base of the building, you can use the a^2+b^2= C^2 method, plug in 21^2 + x^2= 35^2, get 441 + x^2 = 1225, then you’re left with the square root of x^2 = square root of 784^2, which will give you a length of 28, which is the how far from the ground the ladder touches the building.
Whether we know it or not the Pythagorean Theorem is part of our daily lives, and it is because of the hard work and dedication by Pythagoras that we can use this theorem to calculate many things in life that we never used to be able to do before.
