Pythagoras of Samos (580BC – 500AD) was a Greek philosopher and mathematician, best known for the Pythagoras theorem, which bears his name. The modern version of Pythagoras' theorem is stated in its algebraic form:
a^2 +b^2 = c^2
Where a, b, c, are the length of three sides of the triangle ABC as shown in Figure 1. AC and BC are called the legs and AB, the side opposite the right angle, called the hypotenuse. To the Greeks, the theorem was a statement about areas of squares. Starting with the right angle ABC, they constructed squares upon the hypotenuse and the legs. The theorem stated in words is:
- "The area of the square on the hypotenuse exactly equalled the sum of the areas of the squares on the legs."
Pythagoras' theorem is a remarkable decomposition of one square's area into that of two smaller ones.
Pythagoras’ Theorem Has Many Proofs
An early twentieth century professor named Elisha Scott Loomis collected and published 367 of them in a book called The Pythagorean Proposition.
Some of these proofs, which Loomis classifies as algebraic, geometric, dynamic, or quarternionic, are only minor variants of others, but their existence makes a point most clearly: The theorem has occupied mathematicians from classical times to the present.
The many proofs illustrate the agility of mathematicians in attacking the same problem from different angles. It should be noted that the early mathematicians, such as the Greeks did not have algebraic symbols, no formulas and no exponents.
Whether regarded as algebraic or geometric, the theorem is of supreme mathematical importance.
Three Selected Proofs
A number of proofs are worthy of examination. However, there are three, which are different and stand out:
- "The Chinese Proof" as suggested by ancient Chinese treatise
- "A Similarity Proof" as popularised by the 17th century English mathematician, John Wallis
- "A Trapezoidal Proof" as discovered in 1876 by U.S. politician and later President, James A. Garfield.
Pythagorean Converse is True
Converse is a term from logic that has precise meaning. Beginning with the statement "If A, then B", if one interchanges hypothesis and conclusion, one gets the related statement, "If B, then A".
This is the converse of the original. Some converses are true, but some are false. In the case of Pythagoras' theorem, the converse is:
If c^2 = a^2 +b^2, the triangle ABC is a right triangle.
The Pythagoras theorem's converse is true, as Euclid proved in the final proposition of the first book of the Elements. It established that a triangle is right angled if and only if the square of the hypotenuse is the sum of the squares on the other two sides.
Pythagorean Triples
A Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing. A triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).
References:
- "Hypotenuse", in The Mathematical Universe, William Dunham. John Wiley, NY, 1994.
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