Category:Definitions/Congruence (Number Theory)
This category contains definitions related to congruence in the context of number theory.
Related results can be found in Category:Congruence (Number Theory).
Let $z \in \R$.
Definition by Remainder after Division
We define a relation $\RR_z$ on the set of all $x, y \in \R$:
- $\RR_z := \set {\tuple {x, y} \in \R \times \R: \exists k \in \Z: x = y + k z}$
This relation is called congruence modulo $z$, and the real number $z$ is called the modulus.
When $\tuple {x, y} \in \RR_z$, we write:
- $x \equiv y \pmod z$
and say:
- $x$ is congruent to $y$ modulo $z$.
Definition by Modulo Operation
Let $\bmod$ be defined as the modulo operation:
- $x \bmod y := \begin{cases}
x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$
Then congruence modulo $z$ is the relation on $\R$ defined as:
- $\forall x, y \in \R: x \equiv y \pmod z \iff x \bmod z = y \bmod z$
Definition by Integer Multiple
Let $x, y \in \R$.
Then $x$ is congruent to $y$ modulo $z$ if and only if their difference is an integer multiple of $z$:
- $x \equiv y \pmod z \iff \exists k \in \Z: x - y = k z$
Subcategories
This category has the following 2 subcategories, out of 2 total.
P
R
Pages in category "Definitions/Congruence (Number Theory)"
The following 20 pages are in this category, out of 20 total.
C
- Definition:Congruence (Number Theory)
- Definition:Congruence (Number Theory)/Incongruence
- Definition:Congruence (Number Theory)/Integer Multiple
- Definition:Congruence (Number Theory)/Integers
- Definition:Congruence (Number Theory)/Integers/Integer Multiple
- Definition:Congruence (Number Theory)/Integers/Modulo Operation
- Definition:Congruence (Number Theory)/Integers/Remainder after Division
- Definition:Congruence (Number Theory)/Modulo Operation
- Definition:Congruence (Number Theory)/Modulo Zero
- Definition:Congruence (Number Theory)/Modulus
- Definition:Congruence (Number Theory)/Notation
- Definition:Congruence (Number Theory)/Remainder after Division
- Definition:Congruence (Number Theory)/Residue
- Definition:Congruence Modulo Integer
- Definition:Congruence Modulo Zero