Equivalence of Definitions of Distance to Nearest Integer Function

Theorem

The following definitions of the distance to nearest integer function $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ are equivalent:

Definition 1

$\norm \alpha:= \min \set {\size {n - \alpha}: n \in \Z}$

Definition 2

$\norm \alpha:= \min \set {\set \alpha, 1 - \set \alpha}$

where $\set \alpha$ is the fractional part of $\alpha$.

Proof

Let $\alpha \in \R$, $n \in \Z$.

From Real Number is between Floor Functions:

$\floor \alpha \le \alpha < \floor \alpha + 1$

For any $n < \floor \alpha \le \alpha$:

\(\ds \size {n - \alpha}\)

\(=\)

\(\ds \alpha - n\)

Definition of Absolute Value

\(\ds \)

\(>\)

\(\ds \alpha - \floor \alpha\)

\(\ds \)

\(=\)

\(\ds \size {\alpha - \floor \alpha}\)

Definition of Absolute Value

For any $n > \floor \alpha + 1 > \alpha$:

\(\ds \size {n - \alpha}\)

\(=\)

\(\ds n - \alpha\)

Definition of Absolute Value

\(\ds \)

\(>\)

\(\ds \floor \alpha + 1 - \alpha\)

\(\ds \)

\(=\)

\(\ds \size {\floor \alpha + 1 - \alpha}\)

Definition of Absolute Value

Thus:

\(\ds \min \set {\size {n - \alpha}: n \in \Z}\)

\(=\)

\(\ds \min \set {\size {\floor \alpha - \alpha}, \size {\floor \alpha + 1 - \alpha} }\)

Other $n$'s are disregarded by above

\(\ds \)

\(=\)

\(\ds \min \set {\alpha - \floor \alpha, \floor \alpha + 1 - \alpha}\)

\(\ds \)

\(=\)

\(\ds \min \set {\set \alpha, 1 - \set \alpha}\)

Definition of Fractional Part

which shows that the definitions are indeed equivalent.

$\blacksquare$