The Pythagorean number theorem is the basis for modern number theory. Modern number theory is that branch of mathematics concerned with properties and relationships of numbers. This includes much of mathematics and more specifically mathematical analysis and is generally limited to the study of integers or other sets that possess properties of all integers. The Pythagoreans carried out extensive mathematical investigations concerned with numbers, mainly of odd and even numbers and of prime and square numbers, that established a scientific foundation for mathematics. Their results and beliefs are summarized into the Pythagorean number theorem.
Pythagoras was a Greek mathematician thought to have been born in about 500 B.C.. Although originally born on the island of Samos, Pythagoras was thought to have moved to Crotona in about 530 B.C. where he founded a movement known as Pythagoreanism. The disciples of this movement are believed to have formed the initial ideas for the Pythagorean number theorem. The Pythagoreans formulated a view from an arithmetical standpoint that believed the concept of the number was the key to the qualities of mankind and matter and that it was the ultimate principle of all proportion of the universe. They believed that everything was a composition of a number and that an object's existence could only be understood in that number. This was in stark contrast to the accepted thinking at the time that numbers were only of utilitarian use for solving problems in calendar construction, architecture, and commerce. The Pythagoreans believed that numbers had an importance unto themselves and that each number possessed it own special attributes. They drew distinctions between logistic, which involved the art of computation, and arithmetic, which involved number theory.
Originally the Pythagoreans treated numbers concretely as patterns but this eventually evolved into a refined concept of the number as an abstract entity. This concept of the number is what still exists today. To the Pythagoreans each number possessed its own special attributes, some of which are listed below:
1, "monad"--represents unity and is the generator of numbers.
2, "dyad"--represents diversity and opinion and is the first true female number (the Pythagoreans believed even numbers were female and odd numbers were male).
3, "triad"--represents harmony that is equal to unity plus diversity. This is the first true male number.
4--represents justice and retribution. It is the squaring of accounts.
5--represents marriage that is equal to the first female plus the first male.
6--represents creation that is equal to the first female plus the first male plus 1.
10, "tetractys"--represents universe.
In the development of the Pythagorean number theorem the Pythagoreans valued rigor and proof which led them to search for essential properties and definitions of numbers. The Pythagoreans believed a number is a collection of units. They studied prime numbers, composite or rectilinear numbers, odd and even numbers and devoted a great deal of time to the study of the "tetractys". To them the holiest of numbers was the number 10 because it had a special significance. The first four numbers also held a special significance for the Pythagoreans. They accounted for all of the possible dimensions and their sum equals 10, the holiest number, the universe. The first four numbers were the only numbers needed to represent all known objects geometrically. It is the Pythagorean veneration of the tetractys that is responsible for the present use of the base ten.
The Pythagorean number theory led to many important theorems and models. The model of the universe is based on the initial attempt by the Pythagoreans to understand cosmology in terms of mathematical principles. Since the number 10 was the number of the universe they believed that there had to be 10 heavenly bodies and that the planets orbited a central fire, the Sun. In geometry the greatest discovery of the Pythagoreans was the hypotenuse theorem, which is usually called the Pythagorean theorem. This theorem relates the hypotenuse of a right triangle to the other two sides in a purely mathematical way. The Pythagorean theorem is usually written as: a2 + b2 = r2, where a and b are the lengths of the sides of a right triangle and r is the length of the hypotenuse. Pythagorean triplets are integer solutions to the Pythagorean theorem. There also came another important theorem from the Pythagorean number theory. Basically this theory says that the diagonal of a square with sides of integral length cannot be rational. From this Pythagoras' constant, 2, was discovered. This was in conjunction with the study focused on irrational numbers and the golden ratio. The Pythagorean number theorem had an important influence on the formation of modern mathematics.
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