It has been stated by many mathematicians that the Pythagorean Theorem may have more known proofs than any other, as the book The Pythagorean Proposition contains 370 proofs (Maor, 2007). One of the most common used proofs for the Pythagorean Theorem is the similar triangle proof. This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles (Bell, 1999). Another highly regarded proof for the Pythagorean Theorem is Euclid’s proof. In the writings of Elements Euclid states “that the large square is divided into a left and right triangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of a square on the left-most side. These two triangles are shown to be congruent; proving this square has the same area as the left rectangle” (Maor, 2007). There are four essential criteria for the formal proof: 1) If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent. 2) The area of a triangle is half the area of any parallelogram on the same base and having the same altitude. 3) The area of a rectangle is equal to the product of two adjacent sides. 4) The area of a square is equal to the product of two of its sides (Maor, 2007).This proof which is used in Euclid’s Elements is quite distinct from the proof of similarities of triangles which is the proof that Pythagoras used (Machiavelo, 2009). Another commonly used proof to solve the Pythagorean Theorem is algebraic proofs. This theorem can be proved algebraically using four copies of a right triangle with sides a, b, & c. The final proof discussed to prove the Pythagorean Theorem is proofs using differentials. Studying how changes in a side produce a change in the hypotenuse and employing calculus is using differentials (Maor, 2007).
Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Pythagorean triples have been used by mathematicians since the Babylonian and Greek eras, they contain three positive integers and it is where the Pythagorean Theorem formula is derived from. Common examples of Pythagorean triples are (3, 4, 5) and (5, 12, 13).
There is much dispute over the discovery and history of the Pythagorean Theorem. Pythagoras, as well known and impactful as he was on mathematics, it is still unknown if he is the true founder of the theorem. There is also debate if the Pythagorean Theorem was discovered once, or many times in many places (Maor, 2007). As we discussed earlier, many of Pythagoras’ findings were already used by the Babylonians centuries earlier. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system (Maor, 2007). It is believed by most that Pythagorean triples were discovered algebraically by the Babylonians. Pythagoras then used algebraic methods to construct Pythagorean triples (Maor, 2007).
Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Pythagorean triples have been used by mathematicians since the Babylonian and Greek eras, they contain three positive integers and it is where the Pythagorean Theorem formula is derived from. Common examples of Pythagorean triples are (3, 4, 5) and (5, 12, 13).
There is much dispute over the discovery and history of the Pythagorean Theorem. Pythagoras, as well known and impactful as he was on mathematics, it is still unknown if he is the true founder of the theorem. There is also debate if the Pythagorean Theorem was discovered once, or many times in many places (Maor, 2007). As we discussed earlier, many of Pythagoras’ findings were already used by the Babylonians centuries earlier. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system (Maor, 2007). It is believed by most that Pythagorean triples were discovered algebraically by the Babylonians. Pythagoras then used algebraic methods to construct Pythagorean triples (Maor, 2007).