Subtraction of Fractions

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