Arithmetic¶
Recall (doc/postdoc). Not a first course.
Job: compress number-system and elementary-operation facts as used in analysis, algebra, and NT.
Map¶
| Page | Recall role |
|---|---|
| Foundations | ops, division algorithm; PEMDAS ≠ theorem |
| Number systems | \(\mathbb{N}\subset\cdots\subset\mathbb{C}\); completion vs alg. closure |
| Properties | ring/field axioms; zero-product domains |
| Fractions | \(\operatorname{Frac}(\mathbb{Z})\) template |
| Decimals | base-\(10\) series; periodicity \(\Leftrightarrow\mathbb{Q}\) |
| Percentages | multiplicative relative change |
| Exponents | \(a^n\) laws; rational powers; branches |
| Logarithms | invert exp; discrete log later |
Density snapshot¶
\[
a=bq+r,\quad
a(b+c)=ab+ac,\quad
a^ma^n=a^{m+n},\quad
\log_b(xy)=\log_b x+\log_b y
\]
Check¶
Which extension \(\mathbb{Q}\to\mathbb{R}\) vs \(\mathbb{R}\to\mathbb{C}\) solves completeness vs algebraic closure?
\(\mathbb{R}\): order-completion (Cauchy/Dedekind). \(\mathbb{C}\): algebraic closure of \(\mathbb{R}\) (FTA), not an ordered field extending \(\mathbb{R}\)'s order compatibly with mult.