Definition:Congruence (Number Theory)/Integer Multiple

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Definition

Let $z \in \R$.

Let $x, y \in \R$.


Then $x$ is congruent to $y$ modulo $z$ if and only if their difference is an integer multiple of $z$:

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


Notation

The relation $x$ is congruent to $y$ modulo $z$, usually denoted:

$x \equiv y \pmod z$

is also frequently seen denoted as:

$x \equiv y \ \paren {\mathop {\operatorname{modulo} } z}$

Some (usually older) sources render it as:

$x \equiv y \ \paren {\mathop {\operatorname{mod.} } z}$


Examples

Congruence Modulo $1$

Let $x \equiv y \pmod 1$ be defined as congruence on the real numbers modulo $1$:

$\forall x, y \in \R: x \equiv y \pmod 1 \iff \exists k \in \Z: x - y = k$

That is, if their difference $x - y$ is an integer.


The equivalence classes of this equivalence relation are of the form:

$\eqclass x 1 = \set {\dotsc, x - 2, x - 1, x, x + 1, x + 2, \dotsc}$


Each equivalence class has exactly one representative in the half-open real interval:

$\hointr 0 1 = \set {x \in \R: 0 \le x < 1}$


Congruence Modulo $2 \pi$ as Angular Measurement

Let $\RR$ denote the relation on the real numbers $\R$ defined as:

$\forall x, y \in \R: \tuple {x, y} \in \RR \iff \text {$x$ and $y$}$ measure the same angle in radians


Then $\RR$ is the congruence relation modulo $2 \pi$.


The equivalence classes of this equivalence relation are of the form:

$\eqclass \theta {2 \pi} = \set {\theta + 2 k \pi: k \in \Z}$


Measurement-of-Angle.png


Hence for example:

$\eqclass 0 {2 \pi} = \set {2 k \pi: k \in \Z}$

and:

$\eqclass {\dfrac \pi 2} {2 \pi} = \set {\dfrac {\paren {4 k + 1} \pi} 2: k \in \Z}$


Each equivalence class has exactly one representative in the half-open real interval:

$\hointr 0 {2 \pi} = \set {x \in \R: 0 \le x < 2 \pi}$

and have a one-to-one correspondence with the points on the circumference of a circle.


Also see


Linguistic Note

The word modulo comes from the Latin for with modulus, that is, with measure.


Sources

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