Properties of Numbers¶
Recall (doc/postdoc). Not a first course.
Job: field/ring axioms you use without naming them; zero-product and cancellation later fail without integral domains.
Core¶
For \(a,b,c\) in a commutative ring \(R\) (e.g. \(\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}\)):
| Name | Law |
|---|---|
| Commutative | \(a+b=b+a\), \(ab=ba\) |
| Associative | \((a+b)+c=a+(b+c)\), \((ab)c=a(bc)\) |
| Distributive | \(a(b+c)=ab+ac\) |
| Identities | \(a+0=a\), \(a\cdot1=a\) |
| Add. inverse | \(a+(-a)=0\) |
| Mul. inverse | \(a\cdot a^{-1}=1\) iff \(a\in R^\times\) |
Not commutative/associative: subtraction, division.
Zero-product (integral domains, e.g. \(\mathbb{R}\)): \(ab=0\Rightarrow a=0\) or \(b=0\). Fails in \(\mathbb{Z}/6\mathbb{Z}\) (\(2\cdot3=0\)).
Cancellation: \(ac=bc\) and \(c\) not a zero-divisor \(\Rightarrow a=b\).
Used later for¶
- Rings and fields: axiom lists, domains, fields
- Elementary algebra: legitimate equation moves
- Linear algebra: scalar field axioms
- Number theory: zero-divisors mod \(n\)
Check¶
Why does \(ab=0\Rightarrow a=0\lor b=0\) fail in \(\mathbb{Z}/8\mathbb{Z}\)?
\(2\cdot4\equiv0\pmod8\) with neither factor \(0\) — zero-divisors; not an integral domain.