Definition:Modulo Subtraction

Definition

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of subtraction modulo $m$ is defined on $\Z_m$ as:

$\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$

Also denoted as

Although the operation of subtraction modulo $m$ is denoted by the symbol $-_m$, if there is no danger of confusion, the conventional subtraction symbol $-$ is often used instead.

The notation for subtraction of two integers modulo $m$ is not usually $\eqclass a m -_m \eqclass b m$.

What is more normally seen is $a - b \pmod m$.

Examples

Example: $8 - 27 \pmod {10}$

\(\ds \paren {8 - 27} \pmod {10}\)

\(=\)

\(\ds \paren {18 - 7} \pmod {10}\)

\(\ds \)

\(=\)

\(\ds 1 \pmod {10}\)

Also see

• Results about modulo subtraction can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruence modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruence modulo $n$