Editing Lowest common denominator

{{Short description|Lowest common multiple of the denominators of a set of fractions}}
{{More citations needed|date=May 2023}}
In [[mathematics]], the '''lowest common denominator''' or '''least common denominator''' (abbreviated '''LCD''') is the [[least common multiple|lowest common multiple]] of the [[denominator]]s of a set of [[fraction]]s. It simplifies adding, subtracting, and comparing fractions.

== Description ==
The lowest common [[denominator]] of a set of [[fraction]]s is the lowest number that is a [[multiple (mathematics)|multiple]] of all the denominators: their [[least common multiple|lowest common multiple]]. 
The product of the denominators is always a common denominator, as in:

: <math>\frac{1}{2}+\frac{2}{3}\;=\;\frac{3}{6}+\frac{4}{6}\;=\;\frac{7}{6}</math>

but it is not always the lowest common denominator, as in:

: <math>\frac{5}{12}+\frac{11}{18}\;=\;\frac{15}{36}+\frac{22}{36}\;=\;\frac{37}{36}</math>

Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a common denominator, but calculating with that denominator involves larger numbers:
: <math>\frac{5}{12}+\frac{11}{18}=\frac{90}{216}+\frac{132}{216}=\frac{222}{216}.</math>

With variables rather than numbers, the same principles apply:<ref name=brooks/>

: <math>\frac{a}{bc}+\frac{c}{b^2 d}\;=\;\frac{abd}{b^2 cd}+\frac{c^2}{b^2 cd}\;=\;\frac{abd+c^2}{b^2 cd}</math>

Some methods of calculating the LCD are at {{Section link|Least common multiple|Calculation}}.

== Role in arithmetic and algebra ==
The same fraction can be expressed in many different forms. As long as the ratio between numerator and denominator is the same, the fractions represent the same number. For example:

: <math>\frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}</math>
because they are all multiplied by 1 written as a fraction:
: <math>\frac{2}{3}=\frac{2}{3}\times\frac{3}{3}=\frac{2}{3}\times\frac{6}{6}=\frac{2}{3}\times\frac{72}{72}=\frac{2}{3}\times\frac{100,000}{100,000}.</math>

It is usually easiest to add, subtract, or compare fractions when each is expressed with the same denominator, called a "common denominator". For example, the numerators of fractions with common denominators can simply be added, such that <math>\frac{5}{12}+\frac{6}{12}=\frac{11}{12}</math> and that <math>\frac{5}{12}<\frac{11}{12}</math>, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what <math>\frac{5}{12}+\frac{11}{18}</math> equals, or whether <math>\frac{5}{12}</math> is greater than or less than <math>\frac{11}{18}</math>. Any common denominator will do, but usually the lowest common denominator is desirable because it makes the rest of the calculation as simple as possible.<ref name=worldbook/>

==Practical uses==
The LCD has many practical uses, such as determining the number of objects of two different lengths necessary to align them in a row which starts and ends at the same place, such as in [[brickwork]], [[tile|tiling]], and [[tessellation]]. It is also useful in planning [[work schedule]]s with employees with ''y'' days off every ''x'' days.

In musical rhythm, the LCD is used in [[cross-rhythm]]s and [[polymeter]]s to determine the fewest notes necessary to [[count time]] given two or more [[metre (music)|metric]] divisions. For example, much African music is recorded in Western notation using {{music|time|12|8}} because each measure is divided by 4 and by 3, the LCD of which is 12.

== Colloquial usage ==
The expression "lowest common denominator" is used to describe (usually in a disapproving manner) a rule, proposal, opinion, or media that is deliberately simplified so as to appeal to the largest possible number of people.<ref>[https://www.collinsdictionary.com/us/dictionary/english/lowest-common-denominator "lowest common denominator"], ''[[Collins English Dictionary]]'' (accessed February 21, 2018)</ref>

== See also ==
* [[Anomalous cancellation]]
* [[Greatest common divisor]]
* [[Partial fraction decomposition]], reverses the process of adding fractions into ''uncommon'' denominators

== References ==
{{reflist|refs=
<ref name="worldbook">{{cite book | url=https://archive.org/details/worldbookorgani00unkngoog | title=The World Book: Organized Knowledge in Story and Picture, Volume 3 | publisher=Hanson-Roach-Fowler Company | year=1918 | pages=[https://archive.org/details/worldbookorgani00unkngoog/page/n651 2285]–2286 | chapter=Fractions | access-date=7 January 2014}}</ref>
<ref name="brooks">{{cite book | url=https://books.google.com/books?id=axUAAAAAYAAJ&pg=PA80 | title=The Normal Elementary Algebra, Part 1 | publisher=C. Sower Company | author=Brooks, Edward | year=1901 | pages=80 | access-date=7 January 2014}}</ref>
}}

{{Fractions and ratios}}

{{DEFAULTSORT:Lowest Common Denominator}}
[[Category:Elementary arithmetic]]
[[Category:Fractions (mathematics)]]