Exponents¶
Recall (doc/postdoc). Not a first course.
Job: monoid homomorphism \(n\mapsto a^n\); extend to \(\mathbb{Q}\) then continuous/exp to \(\mathbb{R}\).
Core¶
For \(a\neq0\) in a field, \(n\in\mathbb{Z}\):
\[
a^0=1,\quad a^{n+1}=a\cdot a^n,\quad a^{-n}=(a^n)^{-1}
\]
| Law | Identity (where defined) |
|---|---|
| Product | \(a^ma^n=a^{m+n}\) |
| Power | \((a^m)^n=a^{mn}\) |
| Product base | \((ab)^n=a^nb^n\) |
| Roots | \(a^{1/n}=\sqrt[n]{a}\) (principal branch / positive real) |
| Rational | \(a^{m/n}=(\sqrt[n]{a})^m\) |
Fails naively: \((a^m)^n\neq a^{m+n}\); \(a^m+a^n\neq a^{m+n}\); multi-valued complex roots; \((-2)^{1/2}\notin\mathbb{R}\).
Scientific notation: \(a\times10^k\), \(1\le|a|<10\) — normalized floating point.
Used later for¶
- Logarithms: inverse of \(x\mapsto b^x\)
- Complex analysis: \(e^z\), multi-valued \(z^w=e^{w\log z}\)
- Number theory: discrete exp, orders
- Lie theory / DE: matrix exp \(e^{tA}\)
Check¶
For \(a<0\) and \(r\in\mathbb{Q}\setminus\mathbb{Z}\), when is \(a^r\) real and single-valued in school convention?
Only when reduced denom of \(r\) is odd (principal real root); even roots of negatives undefined in \(\mathbb{R}\).