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Simplifying this further gives us the solution x = −3. It is easily checked that none of the zeros of x ( x + 1)( x + 2) – namely x = 0 , x = −1 , and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.
The two sequences {Τ 2n−1} and {Τ 2n} might themselves define two convergent continued fractions that have two different values, x odd and x even. In this case the continued fraction defined by the sequence { Τ n } diverges by oscillation between two distinct limit points.
For example, 1 / 4 , 5 / 6 , and −101 / 100 are all irreducible fractions. On the other hand, 2 / 4 is reducible since it is equal in value to 1 / 2 , and the numerator of 1 / 2 is less than the numerator of 2 / 4 . A fraction that is reducible can be reduced by dividing both the numerator ...
For example, the numerators of fractions with common denominators can simply be added, such that + = and that <, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what 5 12 + 11 18 {\displaystyle {\frac {5}{12}}+{\frac {11}{18}}} equals, or whether 5 12 {\displaystyle {\frac {5 ...
Every fraction 2 / y requires two terms in its greedy expansion if and only if y ≡ 1 (mod 2); the simplest such fraction is 2 / 3 . A fraction 3 / y requires three terms in its greedy expansion if and only if y ≡ 1 (mod 6), for then −y mod x = 2 and y(y + 2) / 3 is odd, so the fraction remaining after a ...
Now, a and b are both odd and will remain odd until the end of the computation; While a ≠ b do If a > b, then replace a with a – b and divide the result by two until a becomes odd (as a and b are both odd, there is, at least, one division by 2). If a < b, then replace b with b – a and divide the result by two until b becomes odd.
The odd greedy algorithm cannot terminate when given a fraction with an even denominator, because these fractions do not have finite representations with odd denominators. Therefore, in this case, it produces an infinite series expansion of its input. For instance Sylvester's sequence can be viewed as generated by the odd greedy expansion of 1/2.
For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a×b=a×c, then the multiplicative term a can be canceled out if a≠0, resulting in the equivalent expression b=c; this is equivalent to dividing through by a.