The fact that any rational number has a unique representation as an irreducible fraction is utilized in various contradiction, so the premise that the square root of two has a representation as the ratio of two integers is false.
12.
In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive.
13.
Every rational number has a " unique " representation as an irreducible fraction with a positive denominator ( however \ tfrac { 2 } { 3 } = \ tfrac {-2 } {-3 } although both are irreducible ).
14.
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain : any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.
15.
As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly ( terminates ) if and only if all the prime factors of its denominator are also prime factors of the base.
16.
An irreducible fraction is one that is " visible " from the origin; the action of the modular group on a fraction never takes a " visible " ( irreducible ) to a " hidden " ( reducible ) one, and vice versa.
17.
This implies that every element of " L " is equal to an irreducible fraction of polynomials in ?, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of " K ".
18.
This implies that every element of " L " is equal to an irreducible fraction of polynomials in ?, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of " K ".
19.
:: : : : Approximations of irrationals using fractions over seven are convenient in base ten because they yield a long repeating string 1 / 7 = 0.142857142857 2 / 7 = 0.285714285714, etc ., whereas using fractions over the other digits either terminate quickly 1 / 4 = 0.25 or repeat in only one digit as when dividing by 3, 6, or 9 . This long repeating terminator with irreducible fractions over seven always show the same six numbers in the same sequence 285714285714285714; but which of those nubers begins the sequence depends on the remainder.
20.
Fortunately, we know that there is a'natural'surjection-from the class of pairs of an even denominator with an odd nominator-on the class of even fractions which are received by the original definition; Thanks to this surjection, we can receive a unique definiendum of " " even fraction " ", by replacing the previous " indefinite " definition by a " definite " definition, which is received by deviding the original definition into sub-definitions, each of which defines " the " ( unique ) " even fraction " as " the " ( unique ) irreducible fraction whose " given " denominator is even and whose " given nominator is odd ", in such a way that this devision of the original definition into sub-definitions-succeeds to preserve the original class of " " even fractions " " received by the original " indefinite " definition.
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