common denominator

Lowest common denominator

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. It is the smallest positive integer that is a multiple of the denominators. For instance, the LCD of

left{ frac{5}{12}, frac{11}{18} right}

is 36 because the least common multiple of 12 and 18 is 36. Likewise the LCD of

left{ frac{5}{6}, frac{1}{4} right}

is 12. Using the LCD (or any multiple of it, such as the product of the denominators) as a denominator enables addition, subtraction or comparison of fractions:

frac{5}{6} - frac{1}{4} = frac{10}{12} - frac{3}{12} = frac{7}{12};

frac{1}{2} - frac{1}{3} = frac{3}{6} - frac{2}{6} = frac{1}{6};

frac{7}{9} < frac{19}{24}text{ since }frac{112}{144} < frac{114}{144}.

The lowest common denominator of two vulgar fractions can be found by calculating the least common multiple of their denominators.

Some K-12 math standards such as the latest revision of the NCTM math standards and reform mathematics textbooks created since the 1990s de-emphasize or omit coverage of the LCD entirely in favor of finding any common, but not necessarily the lowest common denominator, or by using less powerful methods such as fraction strips or "benchmark" fractions. The "cross-multiply" method of comparing fractions effectively creates a common denominator by multiplying both denominators together.

Figurative uses

The term is used figuratively to refer to the "lowest"—least useful, least advanced, or similar—member of a class or set which is common to things that relate to members of that class. For instance, ASCII characters are the lowest common denominator for computers, in that this set is very limited, but practically every modern computer can interpret binary data into these characters.

Another figurative use is as a rhetorical device in criticism of mass media. When a media outlet has been charged with appealing to the "lowest common denominator", it means they have targeted the lowest, meanest, crudest, most basic and perhaps prurient of all possible hopes and dreams of their intended audience.

A third figurative use is to describe negotiations and agreements which only cover the points where everybody's previous positions coincide.

Note that, in some of these cases, the concept being expressed is actually closer to the related-but-different mathematical concept of greatest common divisor.

See also

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