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An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). [1] In other words, a fraction a / b is irreducible if and only if a and b are coprime ...
Cancelling out is a mathematical process used for removing subexpressions from a mathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects. [1] For example, a fraction is put in lowest terms by cancelling out the common factors of the ...
Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. 1 / 8 = 1 / 2 3 . In Unicode, precomposed fraction characters are in the Number Forms block.
The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32 / 13 = 2 / 1 and its inverse are the only solutions in base 4 with two digits. [2]An example of anomalous cancellation with more than two digits is 165 / 462 = 15 / 42 , and an example with different numbers of digits is 98 / 392 = 8 / 32 .
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.
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, replacing $ 23.4476 with $ 23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414. Rounding is often done to obtain a value that is easier to report and communicate than the original.
For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical".