The Pythagorean Theorem
Pythagorean theory was one of the oldest theorems in mathematics. The Pythagorean Theorem is one of the foundations of geometry today. The Pythagorean Theorem was a work by Pythagoras. Pythagoras is a Greek mathematician who founded the Pythagorean School of Mathematics in Greece. Pythagoras is very popular for his mathematical works but he was alleged to take credit of the works of his student. Still, Pythagoras is regarded as one of the greatest mathematician of all time (Crane, 2008).
Pythagoras ace mathematical contribution is said to be the Pythagorean Theorem. Pythagoras was so excited about the discovery of the theorem that he sacrificed an ox to thank the gods for giving him such discovery. However, the theorem still has some flaws. One of the flaws of the theorem is that some of the numbers seems that it does not follow rationality. An example is the square root of the integer 2. Pythagoras’ students realized that there is no exact square root for 2 and said that the root of two is indeed irrational.
The problem of expressing two as ratio of two integral multiples has been a pain in the butt of Pythagoras and it is rumored that anyone who tried to reveal the secret was drowned in the sea (Morris, 1997). Pythagorean Theorem One of the basic arguments of the Pythagorean Theorem is that “the area of the square that is built upon the hypotenuse of a right triangle is equal to the sum of the areas of the square upon the remaining sides. ” If a, b and c are sides of a right triangle with c as the hypotenuse, then Pythagorean theorem can be mathematically stated as a2 + b2 = c2.
The Pythagorean Theorem is a credited work for Pythagoras but it is said that Babylonians knew about the theorem long before Pythagoras knew it (Morris, 1997). Proof of Pythagorean Theorem Fig. 1: Illustration of the Pythagorean Theorem Source: http://www. myastrologybook. com/PythagoreanTheorem16c. gif In the illustration, the proof is formed with the use of different sizes of squares connected at their corners to form a right triangle. One of the proofs of Pythagorean Theorem can be found in Book I of Euclid’s elements.
Proof: Let a, b and c be the sides of the right were c is the hypotenuse of the triangle as seen on figure 2. We draw a rectangle with c as the side of the rectangle like in figure 2. The area of the triangle is 1/2bh, where b is the base and h is the height. Then the area of the triangle in the figure will be 1/2ab. Atriangle = 1/2ab The area of the four triangles in the figure will be 2ab. Fig. 2: Proof figure. Source: http://www. briar. demon. co. uk/george/Pythagoras. gif The length of the side of the small square in the center will be (b – a).
Then the area of the small square in the center will be (b – a)2. Asmall square = (b – a)2 = b2 – 2ab + a2 The area of the whole circle will be: Awhole circle = c2 = 4(Atriangle) + Asmall square By substitution, c2 = 2ab + b2 – 2ab + a2 c2 = a2 + b2 Thus, c2 = a2 + b2 (Bellevue Community College Mathematics Department, 2008). Pythagorean problem One unique property of the Pythagorean Theorem is that there exist numbers a, b and c which are elements of integer such that a2 + b2 = c2. These set of integers that can satisfy the Pythagorean equation is known as the Pythagorean triple.
Pythagorean triples are unique in such a way that it can be solved using separate formula. The following formula first emerged in Book X of Euclid’s Elements (Morris, 1997). a2 + b2 = c2 a = 2mn b = m2 – n2 c = m2 + n2 The variables n and m should be positive integers and m > n. Some of the Pythagorean triples are as follow: (3, 4, and 5), (5, 12, and 13), (7, 24 and 25), (9, 40 and 41) and (11, 60 and 61) (Morris, 1997). Pythagorean Theorem is not only about a2 + b2 = c2. The Pythagorean Theorem should be taught to students for them to see algebraic and geometric connections of the theorem.
The Pythagorean Theorem can denote more than just the equation and plugging in of values for the equation. References Morris, S. (1997). The Pythagorean Theorem. Retrieved November 13, 2008 from http://jwilson. coe. uga. edu/EMT669/Student. Folders/Morris. Stephanie/EMT. 669/Essay. 1/Pythagorean. html Bellevue Community College Department of Mathematics. (2008). The Pythagorean Theorem. Retrieved November 13, 2008 from http://scidiv. bcc. ctc. edu/math/pythagoras. html Crane, G. (2008). The Pythagorean Theorem. Retrieved November 13, 2008 from http://www. perseus. tufts. edu/GreekScience/Students/Tim/Pythag%27sTheorem. html
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