Ideal is also the name of a late 1990s/early 2000sAmerican R&B singing group.
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In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekind's definition of ideals for rings.
An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise.
This means there is an element of the ring of integers of the class field, which is an ideal number, such that all multiples times elements of this ring of integers lying in the ring of integers of the original field define the nonprincipal ideal.
To the Greeks mathematics was essentially geometry, and the quintessence of the subject was Euclid's Elements, where the deductive methods of the Eleatic school of philosophy, the logical procedures of Aristotle and Platonic idealism came together in the notion of an axiomatic system.
It was hoped that by transforming the statements of mathematics into strings of meaningless symbols to be combined according to the rules of logic, whatever unavowed principles of reasoning had given rise to the paradoxes would be revealed.
And when Brouwer attempted to rid mathematics of ideal notions while preserving certain of its long-established aspects, he introduced ideas that seemed to many mathematicians to be just as idealistic as those he was eliminating.
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