The Pythagorean Theorem

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What is the Pythagorean Theorem?

The Pythagorean theorem involves a right triangle, its hypotenuse (c) and other two sides a and b. A right triangle is a triangle with one 90-degree angle. You can refer to a right angle as the corner of a piece of paper. The hypotenuse is the longest side of the triangle.

The Pythagorean theorem states that when c is the hypotenuse and a and b are the legs in a right triangle, then:

Who was Pythagoras?

Pythagoras of Samos (572-407 B.C.E.) is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras' writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

The son of Mnesarchus, Pythagoras is arguably the most interesting figure in the history of mathematics. He was probably born between the 50th and 52nd Olympiads, thus it is generally agreed upon he was born around 580 B.C.E. He traveled a great deal until he settled in Crotona where he gained many followers. Among these disciples, he formed a brotherhood, united by common philosophical beliefs and pursuits. The Pythagoreans, as they were come to be known, was both a religious order and a philosophical school. They were bound by oath not to speak of the doctrines and discoveries of their school; the fact that no extant works of Pythagoras or the Pythagoreans exist, it is difficult to identify the contributor of any portion of their creed, thus it is necessary to acknowledge the Pythagoreans as a collective body and not as individual contributors. It should be noted that the Pythagoreans were in custom to attribute all their doctrines to their master. Pythagoreans made many discoveries in mathematics, music, astronomy and science.

The Origins of the Theorem

The Greeks

In the sixth century B.C. a great mathematician named Pythagorean and his followers, the Pythagorean made many contributions to mathematics. They helped build the foundations of geometry. They were the first to describe numbers as "odd" and "even." It was they who discovered that the earth is round by observing the shadow the earth cast on the moon. They also discovered the relation between the length of a musical string and the pitch of its note.One of the most famous proofs credited to him is the Pythagorean Theorem. The theorem states that the sum of the squares of the two shorter sides of a right triangle equals the square of the one side, which is called the hypotenuse.

The Chinese

Claims have been made that China was well along the mathematics as early as 15,000 B.C. Although this is not proven, we know for sure that the Chinese had gone quite far in mathematics at least as early as the Babylonians and Egyptians. One of their discoveries was the Pythagorean theorem shown below.

Proving the Pythagorean Theorem

Many cultures have proven the Pythagorean theorem through the ages.Here is one type of proof:

To prove that in a right triangle the square of hypotenuse is equal to the sum of the squares of the sides, the U.S. president James Garfield decided to construct the trapezoid shown on the right. It is a trapezoid, consisting of two right triangles with sides a, b, and c and a right isosceles triangle with two equal sides of length c. It is possible to verify that the angle AOB, which is just 180-(<1)-(<2), must be equal to 90 degrees, since the sum of angles one and two plus 90 degrees in each right triangle is 180 degrees.

President Garfield then calculated the area of the trapezoid in two different ways. He was aware that the area is simply the product of its base, a+b, times one half the sum of its sides, 1/2(a=b). We have not proved this, but you can easily show that it is true by building a rectangle from two trapezoids.

The area can also be calculated by adding the areas of each of the three triangles, ab/2, ab/2, and c²/2. As a result,

(a+b)×(a+b)/2= ab + c²/2

By using the equation: (a+b)²=a²+2ab+b², President Garfield could expand the left side of the equation and rediscover the equality found almost 2500 years earlier by Pythagoras.