Exact Numbers, Significant Figures, SI Units & Metric Prefixes

Exact vs. Measured Numbers

• Measured numbers

  • Obtained by comparing a physical quantity to a standard (ruler, balance, clock, etc.).
  • Always include uncertainty because of instrument limits and human reading error.
  • Precision communicated by the number of significant figures (sig figs).

• Exact numbers

  • Defined or counted, not obtained by measurement; therefore carry zero uncertainty.
  • Treated as having infinite sig figs and infinite decimal places.
  • Purpose: ensure they never become the limiting term when applying sig-fig rules.

Why exactness matters

• Mis-treating an exact value as approximate causes unnecessary rounding and loss of precision (e.g.
  • Using "12" in the inch–foot relationship as two-sig-figs instead of exact).
• Exact numbers should never dictate rounding; measured quantities should.

Three Main Sources of Exact Numbers

1. Counted Values

   • Simply tallying entities (e.g. 84 students, $100$ electrons, $3\,500$ pennies).
   • Counting can be wrong but is never uncertain—there is no fractional ambiguity.

2. Defined (or Declared) Values

   • Established by agreement; convert within one system or between systems by decree.
   • Examples
     • $1\ \text{foot}=12\ \text{inches}$ (exact, by definition).
     • $60\ \text{seconds}=1\ \text{minute}$.
     • Metric prefixes: $1\ \text{L}=10^3\ \text{mL}$, $1\ \text{kg}=10^3\ \text{g}$.
   • Remarkable case: U.S. inch re-defined by Congress to be exactly $2.54\ \text{cm}$; British inch differs slightly.

3. Simple Fractions & Mathematical Constants in Equations

   • Fractions such as $\frac{1}{2},\ \frac{4}{3}$ emerge from geometry/algebra, not measurement.
   • Ex: Sphere volume $V=\frac{4}{3}\pi r^3$ — $\frac{4}{3}$ and $\pi$ are treated as exact.
   • Converting radius/diameter: $r=\tfrac{1}{2}d$; the factor $\tfrac{1}{2}$ is exact, so diameter may keep all sig figs.
   • Physical constants measured to enormous precision (effectively exact in typical coursework)
     • Speed of light: $c=2.99792458\times10^{8}\ \text{m·s}^{-1}$ (often rounded to $2.998\times10^{8}$ in exams).
     • Avogadro’s number: $N_A=6.02214076\times10^{23}\ \text{mol}^{-1}$ (now defined exactly via kilogram re-definition).

Consequences for Calculations & Rounding

• When multiplying/dividing, sig-fig rule: answer inherits the fewest sig figs among measured factors.
  • Exact numbers supply “infinite,” so they never limit.
• Calculator use:
  • Enter exact constants with dedicated keys (e.g. the $\pi$ button) or full precision values.
  • Avoid approximations like $3.14$ unless every measured term has (\leq3) sig figs.

SI (Système International) Fundamentals

• Evolved from the metric system; adopted for international consistency.
• Original artifacts (platinum-iridium meter, kilogram cylinder) now replaced by physical-constant definitions.
• Naming rule: If a unit is named after a person, its symbol is capitalized but the spelled-out name is not (kelvin, joule, ampere).

Seven Fundamental SI Units

• Note: kilogram uniquely includes a prefix (kilo) in its base name.
• Mole = chemist’s “big dozen” for entities; exactly $6.02214076\times10^{23}$ particles.

Derived SI Units (built from fundamentals)

• Area: $\text{m}^2$
• Volume: $\text{m}^3$ (rare in lab; often replaced by $\text{L} = 10^{-3}\,\text{m}^3$)
• Velocity: $\text{m·s}^{-1}$
• Density: $\text{kg·m}^{-3}$ (commonly reported as $\text{g·cm}^{-3}$ in chemistry)
• Energy (work):
  • From kinetic energy $E_k=\tfrac12 m v^2$ gives units $\text{kg·m}^2\text{s}^{-2}$.
  • Renamed the joule (J) so we can flexibly use prefixes (kJ, mJ, etc.).

Metric Prefixes: Scaling Units by Powers of Ten

• General idea: one big unit = many small units. Use positive exponents for the “many” side.
• Organize in symmetric "large" and "small" pairs (×$10^n$ vs ÷$10^n$).

Common Large-Side Prefixes

Common Small-Side Prefixes

• Mnemonic mindset: match large vs. small: kilo ↔ milli, mega ↔ micro, giga ↔ nano, etc.
• Treat negatives conceptually ("small side"), not operationally; avoid explicit –-exponents in conversions.

Practical Conversions: Strategy

1. Identify which version is larger (contains the prefix? check magnitude).
2. Write the equality one big unit = many small units using the positive-exponent factor.
3. Build a conversion factor (ratio) so unwanted units cancel.
4. Multiply/divide measured value; round only according to sig-fig rules for measured quantities.

Example: Convert $7.20\ \text{mm}$ to meters.

• Big unit = meter; small = millimeter.
• $1\ \text{m}=10^3\ \text{mm}$.
• $7.20\ \text{mm} \times \dfrac{1\ \text{m}}{10^3\ \text{mm}} = 7.20\times10^{-3}\ \text{m}$ (3 sig figs).

No negative exponent needed in the setup; the algebra produces it naturally.

Real-World & Historical Contexts / Examples

• Foot–inch: inch defined as $\tfrac{1}{12}$ of a foot → exact 12 in calculations.
• Seconds–minutes: $60$ exact; similar rationale.
• U.S. vs. British inch: political re-definition demonstrates human authority to create exactness.
• Meter & kilogram prototypes: stored in Paris; now replaced by light wavelength (meter) and Planck-constant-based watt balance (kilogram).
• Kelvin name rule: unit lowercase, symbol uppercase (kelvin = K); applies to joule, ampere, etc.
• Photography & femtoseconds: ultra-fast shutter speeds (10^{-15} s) allow imaging molecular motions.
• Computer storage: kilobyte, megabyte, gigabyte use prefixes on non-SI unit "byte" (8 bits).
• Pico Boulevard (Los Angeles): street name reflects $10^{-12}$ prefix; cultural imprint.

Ethical & Practical Implications

• Maintaining global standards (SI) avoids costly mismatches (e.g., engineering failures, commercial disputes).
• Proper sig-fig treatment respects measured data; careless rounding discards information and can mask experimental error.
• Recognizing the boundary between exact definitions and empirical measurement fosters transparent, reproducible science.