Definition:Common Denominator
Definition
Consider the expression:
- $\dfrac a b + \dfrac c d$
where $a$, $b$, $c$ and $d$ are any expressions whatsoever which evaluate to a number such that neither $c$ nor $d$ evaluate to zero.
In order to be able to perform the required addition, it is necessary to put the expressions $\dfrac a b$ and $\dfrac c d$ over a common denominator.
Hence the operation is:
- to multiply both the numerator (top) and denominator (bottom) of $\dfrac a b$ by $d$
and in the same operation:
- to multiply both the numerator (top) and denominator (bottom) of $\dfrac c d$ by $b$
in order to obtain the expression:
- $\dfrac {a d} {b d} + \dfrac {b c} {b d}$
Hence one may perform the operation as:
- $\dfrac {a d + b c} {b d}$
and either evaluate or simplify appropriately.
Lowest Common Denominator
Let $\dfrac a b$ and $\dfrac c d$ be fractions.
The lowest common denominator of $\dfrac a b$ and $\dfrac c d$ is the lowest common multiple of the denominators of $\dfrac a b$ and $\dfrac c d$:
- $\lcm \set {b, d}$
Examples
Multiples of $12$
Multiples of $12$ are all common denominators of:
- $\dfrac 1 2$, $\dfrac 1 4$, $\dfrac 1 6$
Hence these fractions can be expressed as:
- $\dfrac 6 {12}$, $\dfrac 3 {12}$, $\dfrac 2 {12}$
$2$, $3$ and $7$
The fractions:
- $\dfrac 1 2$, $\dfrac 1 3$, $\dfrac 3 7$
have common denominators:
- $42$, $84$, $126$, $168$
and so on.
The lowest common denominator of $\dfrac 1 2$, $\dfrac 1 3$, $\dfrac 3 7$ is $42$.
$3$, $8$ and $6$
The fractions:
- $\dfrac 1 3$, $\dfrac 7 8$, $\dfrac 5 6$
all have a common denominator of $24$.
Hence:
\(\ds \dfrac 1 3 + \dfrac 7 8 - \dfrac 5 6\) | \(=\) | \(\ds \dfrac 8 {24} + \dfrac {21} {24} - \dfrac {20} {24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {8 + 21 - 20} {24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 9 {24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 3 8\) |
Also see
- Results about common denominators can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): common denominator
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): common denominator
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): common denominator
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): common denominator
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): common denominator