Chapter 1 Review: Reciprocal Relationships, Powers of Ten, and Unit Prefixes

If a number a and b satisfy $a\cdot b = 1$, they are inverses; example: $2$ and $\tfrac{1}{2}$.
In scaling: multiplying by a factor $f$ increases a quantity, dividing by $f$ decreases it.
- \text{New} = \text{Old} \times f
- \text{New} = \frac{\text{Old}}{f}
Reciprocal of a number $a$ is \frac{1}{a}; $a\cdot a^{-1}=1$.

Positive powers increase magnitude: 10^2=100,\;10^3=1000,\;10^4=10{,}000,\;10^9=1{,}000{,}000{,}000.
Negative powers decrease magnitude: 10^{-1}=0.1,\;10^{-2}=0.01,\;\text{etc.}
Common prefixes (ascending by exponent):
- nano: $10^{-9}$, micro: $10^{-6}$, milli: $10^{-3}$, centi: $10^{-2}$, deci: $10^{-1}$
- kilo: $10^{3}$, mega: $10^{6}$, giga: $10^{9}$, tera: $10^{12}$
- hecto: $10^{2}$, deca: $10^{1}$ (less commonly used) – note the chart may start from smaller-than-base units
Example mapping: increasing powers moves to larger units; decreasing powers move to smaller units.
Typical length/volume conversions: 12\text{ in} = 1\text{ ft} and 1\text{ ft} = 12\text{ in} (quantity unchanged).

Length dimension: one-dimensional (e.g., meter, mm).
Area dimension: two-dimensional (length^2) with units like $\text{m}^2$, $\text{cm}^2$.
Volume dimension: three-dimensional (length^3) with units like $\text{m}^3$, $\text{cm}^3$.
Rule: Area is length^2; Volume is length^3; 12 inches to feet is a unit conversion, not a change in the dimension.

Not a measure of area: \text{m}^3 (cubic meter) is a measure of volume, not area.
An area measurement uses square units (e.g., \text{m}^2).

Circumference is a length (one-dimensional).
Suitable units: millimeters, centimeters, meters (not square meters).
For circle measurements, circumference relates to length, while area relates to area (two dimensions).

Increasing order (smallest to largest) of prefixes is by exponent value.
To increase a quantity, multiply by a factor; to decrease, divide by that factor.
The name of a horizontal axis is the $x$-axis; the axis names help interpret graphs.
Units must match dimension: length (1D), area (2D, units $\text{L}^2$), volume (3D, units $\text{L}^3$).