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Numbers: Fractions
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Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet ... added to the general theory, as have numerous contributors to the applications of the subject.
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Once you're learned about fractions, there is another major classification of numbers: the ones that can't be written as fractions. Remember that fractions (... known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all those numbers that can be written as non-repeating, non-terminating decimals are non-rational, so they are called the "irrationals". Examples would be sqrt(2) or pi (from geometry). The rationals and irrationals are totally separate number types; there is no overlap.
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Without getting technical, real numbers are all numbers that can be written as a possibly never repeating decimal fraction. For example, all rational numbers are real. Their decimal representations do repeat. Decimal fractions whose representation do not repeat are irrational.
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Rational numbers are numbers which can be written as fractions. This means that they can be written as a divided by b, where the numbers a and b are integers, and b is not equal to 0.
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Fractions