Percentages¶
Recall (doc/postdoc). Not a first course.
Job: affine scaling by \(1\pm p/100\); relative change is multiplicative, not additive.
Core¶
\[
p\%=\frac{p}{100},\qquad
\text{part}=r\cdot\text{whole},\quad r=\frac{p}{100}
\]
| Task | Map |
|---|---|
| Apply \(p\%\) of \(A\) | \(\frac{p}{100}A\) |
| \(A\) as \(\%\) of \(B\) | \(100\cdot A/B\) |
| Increase / decrease by \(p\%\) | \(A\mapsto A(1\pm p/100)\) |
| Undo \(p\%\) increase | \(B\mapsto B/(1+p/100)\) |
Composition: successive \(\pm p_i\%\) multiplies factors; \(+10\%\) then \(-10\%\) yields \(0.99\), not identity.
Interest (simple): \(I=Prt\). Compound: \(A=P(1+r/n)^{nt}\) — discrete exponential; continuous \(\to Pe^{rt}\).
Used later for¶
- Exp & log: compound growth \(\to\) exp
- Probability: rates, risk ratios (not probabilities themselves)
- Statistics: relative error, percent change
- Dynamical systems: discrete multiplicative maps
Check¶
Is percent change \(100(A-B)/B\) symmetric under \(A\leftrightarrow B\)?
No — \(\Delta(A,B)=-\Delta(B,A)/(1+\Delta(B,A)/100)\); use log-ratios for symmetry.