mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...

equating two expressions that each are a sum of rational expressions – which includes simple

fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

Example

Consider the equation :

\frac x 6 + \frac y  = 1.

The smallest common multiple of the two denominators 6 and 15''z'' is 30''z'', so one multiplies both sides by 30''z'': :

5xz + 2y = 30z. \,

The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original. For when we substitute and in the last equation, both sides simplify to 0, so we get , a mathematical truth. But the same substitution applied to the original equation results in , which is mathematically meaningless.

Description

Without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...

, we may assume that the

right-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.least common denominator, which is the

least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...

of the . This means that each is a

factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...

of , so for some expression that is not a fraction. Then :

\frac = \frac = \frac D \,,

provided that does not assume the value 0 – in which case also equals 0. So we have now :

\sum_^n \frac = \sum_^n \frac D = \frac 1 D \sum_^n R_i P_i = 0.

Provided that does not assume the value 0, the latter equation is equivalent with :

\sum_^n R_i P_i = 0\,,

in which the denominators have vanished. As shown by the provisos, care has to be taken not to introduce

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...

s of – viewed as a function of the

unknown Unknown or The Unknown may refer to: Film * ''The Unknown'' (1915 comedy film), a silent boxing film * ''The Unknown'' (1915 drama film) * ''The Unknown'' (1927 film), a silent horror film starring Lon Chaney * ''The Unknown'' (1936 film), a ...

s of the equation – as spurious solutions.

Example 2

Consider the equation :

\frac+\frac-\frac = 0.

The least common denominator is . Following the method as described above results in :

(x+2)+(x+1)-x = 0.

Simplifying this further gives us the solution . It is easily checked that none of the zeros of – namely , , and – is a solution of the final equation, so no spurious solutions were introduced.

References

* {{cite book , title=Algebra: Beginning and Intermediate , edition=3 , author=Richard N. Aufmann , author2=Joanne Lockwood , page=88 , publisher=Cengage Learning , year=2012 , isbn=978-1-133-70939-8 Elementary algebra Equations