Pythagoras left Samos for Egypt in about B. Many of the practices of the society he created later in Italy can be traced to the beliefs of Egyptian priests, such as the codes of secrecy, striving for purity, and refusal to eat beans or to wear animal skins as clothing.
Ten years later, when Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon in what is now Iraq , where he met the Magoi, priests who taught him sacred rites. Iamblichus AD , a Syrian philosopher, wrote about Pythagoras, "He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians His methods of teaching were not popular with the leaders of Samos, and their desire for him to become involved in politics did not appeal to him, so he left.
Pythagoras settled in Crotona, a Greek colony in southern Italy, about BC, and founded a philosophical and religious school where his many followers lived and worked. The Pythagoreans lived by rules of behavior, including when they spoke, what they wore and what they ate.
Pythagoras was the Master of the society, and the followers, both men and women, who also lived there, were known as mathematikoi. They had no personal possessions and were vegetarians. Another group of followers who lived apart from the school were allowed to have personal possessions and were not expected to be vegetarians.
They all worked communally on discoveries and theories. The tablet details a marshy field with various structures, including a tower, built upon it. The tablet is engraved with three sets of Pythagorean triples: three whole numbers for which the sum of the squares of the first two equals the square of the third. The triples engraved on Si. These were likely used to help determine the land's boundaries.
In , Mansfield had discovered a tablet from the same period, named Plimpton , which he identified as containing another trigonometric table. But it wasn't until he saw the triples on Si. But only a very small handful can be used by Babylonian surveyors. Plimpton is a systematic study of this zoo to discover the useful shapes," Mansfield said, referring to the fact that different types of right triangles can have different interior angles.
One came from the contemporary Chinese civilization found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven. The proof of the Pythagorean Theorem that was inspired by a figure in this book was included in the book Vijaganita, Root Calculations , by the Hindu mathematician Bhaskara. Bhaskara's only explanation of his proof was, simply, "Behold". These proofs and the geometrical discovery surrounding the Pythagorean Theorem led to one of the earliest problems in the theory of numbers known as the Pythgorean problem.
Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:. The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements :. In his book Arithmetica , Diophantus confirmed that he could get right triangles using this formula although he arrived at it under a different line of reasoning. The Pythagorean Theorem can be introduced to students during the middle school years.
This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pythagorean Theorem.
Students need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials.
Through the use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than just.
The following is a variety of proofs of the Pythagorean Theorem including one by Euclid. These proofs, along with manipulatives and technology, can greatly improve students' understanding of the Pythagorean Theorem. The following is a summation of the proof by Euclid, one of the most famous mathematicians. This proof can be found in Book I of Euclid's Elements. Proposition: In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on the legs.
Figure 2. Euclid began with the Pythagorean configuration shown above in Figure 2. Then, he constructed a perpendicular line from C to the segment DJ on the square on the hypotenuse. The points H and G are the intersections of this perpendicular with the sides of the square on the hypotenuse. It lies along the altitude to the right triangle ABC. See Figure 3. Figure 3. He proved these equalities using the concept of similarity. Since the sum of the areas of the two rectangles is the area of the square on the hypotenuse, this completes the proof.
Euclid was anxious to place this result in his work as soon as possible. However, since his work on similarity was not to be until Books V and VI, it was necessary for him to come up with another way to prove the Pythagorean Theorem.
Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. The two triangles are congruent by SAS. The same result follows in a similar manner for the other rectangle and square. Katz, Click here for a GSP animation to illustrate this proof.
The next three proofs are more easily seen proofs of the Pythagorean Theorem and would be ideal for high school mathematics students.
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