Subtraction of Fractions
Jump to navigation
Jump to search
Theorem
Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
- $\dfrac a b - \dfrac c d = \dfrac {a D - B c} {\lcm \set {b, d} }$
where:
- $B = \dfrac b {\gcd \set {b, d} }$
- $D = \dfrac d {\gcd \set {b, d} }$
- $\lcm$ denotes lowest common multiple
- $\gcd$ denotes greatest common divisor.
Proof
\(\ds \dfrac a b - \dfrac c d\) | \(=\) | \(\ds \dfrac a b + \dfrac {\paren {-c} } d\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {a D + B \paren {-c} } {\lcm \set {b, d} }\) | Addition of Fractions | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {a D - B c} {\lcm \set {b, d} }\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fraction