Congruence (Number Theory)/Examples

Examples of Congruence

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:

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

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.

Example: $8 \equiv -1 \pmod 3$

$8 \equiv -1 \pmod 3$

Example: $42 \equiv 18 \pmod 8$

$42 \equiv 18 \pmod 8$

Example: $365 \equiv 1 \pmod 7$

$365 \equiv 1 \pmod 7$

Example: $1 \cdotp 6 \equiv 0 \cdotp 6 \pmod 1$

$1 \cdotp 6 \equiv 0 \cdotp 6 \pmod 1$

Example: $5 \cdotp 74 \equiv -3 \cdotp 26 \pmod 3$

$5 \cdotp 74 \equiv -3 \cdotp 26 \pmod 3$