Decimals¶
Recall (doc/postdoc). Not a first course.
Job: base-\(10\) place-value as series \(\sum d_k 10^k\); rationals vs irrationals via eventual periodicity.
Core¶
\[
d_n\ldots d_0.d_{-1}d_{-2}\cdots=\sum_{k\le n}d_k\,10^k,\quad d_k\in\{0,\ldots,9\}
\]
| Type | Characterization |
|---|---|
| Terminating | \(=m/10^k\); equivalently denom in reduced form divides \(2^a5^b\) |
| Eventually periodic | \(\Leftrightarrow\) rational |
| Non-periodic infinite | irrational (if not rational) |
Dual representations: \(0.999\ldots=1\) (and \(0.4999\ldots=0.5000\ldots\)); same real, two expansions.
Ops: \(+/-\) align powers of \(10\); \(\times\) is Cauchy product of digit series (carry). Floating-point \(\neq\) exact \(\mathbb{R}\).
Used later for¶
- Real analysis: \(p\)-adic vs real completions; series
- Numerical analysis: IEEE float, rounding error
- Number theory: \(p\)-adic expansions, automatic sequences
- Measure theory: almost-every real has normal digits (Borel)
Check¶
Prove or refute: every terminating decimal is eventually periodic.
True — terminate with infinite \(0\)s (or dual with infinite \(9\)s); period \(0\) or \(9\).