Definition:Congruence (Number Theory)/Modulo Zero

Definition

Let $x, y \in \R$.

The relation congruence modulo zero is defined as:

$x \equiv y \pmod 0 \iff x \bmod 0 = y \bmod 0 \iff x = y$

and:

$x \equiv y \pmod 0 \iff \exists k \in \Z: x - y = 0 \times k = 0 \iff x = y$

This definition is consistent with the general definition of congruence modulo $z$ for any $z \in \R$.

Also see

• Equivalence of Definitions of Congruence

• Congruence Modulo $m$ is Equivalence Relation

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $11$