Ordering on Real Numbers from Decimal Expansion
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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 $6$