Ordering on Real Numbers from Decimal Expansion

Theorem

Let $x, y \in \R$ be real numbers.

Let $x$ and $y$ be expressed by their decimal expansions:

$\ds x$

$=$

$\ds m \cdotp d_1 d_2 d_3 \ldots$

$\ds y$

$=$

$\ds n \cdotp e_1 e_2 e_3 \ldots$

Let $\preccurlyeq_l$ be the lexicographic ordering on $\R$ defined as:

$x \preccurlyeq_l y$ if and only if:

$m \prec n$

or:

$m = n$ and $\exists k \in \Z_{>0}: \paren {\forall j: 1 \le j < k: d_j = e_j} \land d_k < e_k$

or:

$m = n$ and $\forall j \in \Z_{>0}: d_j = e_j$.

Then:

$x \le y$

where $\le$ denotes the usual ordering on $\R$.

Proof

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Sources

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $6$