Unit Conversion & Metric Prefixes Comprehensive Cheat-Sheet

Distance / Length

• Core U.S. customary relationships
  • 12\;\text{in} = 1\;\text{ft}
  • 3\;\text{ft} = 1\;\text{yd}
  • 5280\;\text{ft} = 1\;\text{mi}
• Metric ↔︎ U.S. bridges
  • 1\;\text{in} = 2.54\;\text{cm} (exact, by convention)
  • 1\;\text{mi} = 1.609\;\text{km}
• Pure-metric hierarchy (linear)
  • 1\;\text{m} = 100\;\text{cm}
  • 1\;\text{km} = 1000\;\text{m}
• Conceptual takeaway: Converting within U.S. or within metric usually involves simple integer factors ((3, 12, 100, 1000)), whereas cross-system conversions rely on defined constants ((2.54\;\text{cm/in}), (1.609\;\text{km/mi})).

Mass / Weight

• Base metric facts
  • 1\;\text{kg} = 1000\;\text{g}
• U.S. customary facts
  • 1\;\text{lb} = 16\;\text{oz}
  • 1\;\text{ton} = 2000\;\text{lb}
• Metric ton(n)e (a.k.a. tonne)
  • 1\;\text{metric\;ton} = 1000\;\text{kg}
  • 1\;\text{metric\;ton} \approx 2204\;\text{lb}
• Cross-system anchor
  • 1\;\text{kg} \approx 2.2\;\text{lb} (widely used rounding for quick estimates)

Volume / Capacity

• Metric relationships
  • 1\;\text{L} = 1000\;\text{mL} (liquid measure)
  • 1\;\text{mL} = 1\;\text{cm}^3 (links volume to linear cm)
  • 1\;\text{m}^3 = 1000\;\text{L}
• U.S. customary liquid chain
  • 1\;\text{gal} = 4\;\text{qt}
  • 1\;\text{qt} = 2\;\text{pt}
• Metric ↔︎ U.S. bridge
  • 1\;\text{gal} \approx 3.785\;\text{L}

Time

• Standard calendar / clock facts
  • 1\;\text{hr} = 60\;\text{min}
  • 1\;\text{min} = 60\;\text{s}
  • 1\;\text{day} = 24\;\text{hr}
• Larger periods
  • 1\;\text{year} \approx 365\;\text{days} (average)
  • Including leap-year adjustment: 1\;\text{yr} = 365.25\;\text{days}
  • 30\;\text{days} \approx 1\;\text{month} (rough planning approximation)
  • 1\;\text{century} = 100\;\text{yr}
  • 1\;\text{millennium} = 1000\;\text{yr}
  • Leap year special case: 366\;\text{days}

Metric Prefixes (Powers of 10)

• Large-scale (greater than one)
  • tera- : 10^{12}
  • giga- : 10^{9}
  • mega- : 10^{6}
  • kilo- : 10^{3}
  • hecto- : 10^{2}
  • deca- : 10^{1}
• Sub-unit (smaller than one)
  • deci- : 10^{-1}
  • centi- : 10^{-2}
  • milli- : 10^{-3}
  • micro- : 10^{-6}
  • nano- : 10^{-9}
  • pico- : 10^{-12}
• Significance: Prefixes allow immediate scaling of base units without changing the unit symbol—e.g., 5\;\text{MW} = 5 \times 10^{6}\;\text{W}.

Conversion-Factor Notation (Scientific-Form Examples)

• 1\;\text{GW} = 1 \times 10^{9}\;\text{W}
• 1\;\text{MJ} = 1 \times 10^{6}\;\text{J}
• 1\;\text{cm} = 1 \times 10^{-2}\;\text{m}
• 1\;\text{ns} = 1 \times 10^{-9}\;\text{s}
• 1\;\mu\text{m} = 1 \times 10^{-6}\;\text{m}

Worked Example Conversions (from transcript)

• Linear length: 4\;\text{ft} \rightarrow \text{in}
  \frac{4\;\text{ft}}{1} \times \frac{12\;\text{in}}{1\;\text{ft}} = 48\;\text{in}
• Mass: 350\;\text{g} \rightarrow \text{kg}
  \frac{350\;\text{g}}{1} \times \frac{1\;\text{kg}}{1000\;\text{g}} = 0.35\;\text{kg}

Strategy & Best-Practice Tips

• Dimensional-analysis mindset: Always set up a chain of fractions so that units cancel sequentially; the leftover unit is the desired one.
• Use powers of 10 for mental arithmetic: Recognize that moving from kilo- to mega- is a shift of 10^{3} (three orders of magnitude).
• Approximation ethics: When communicating engineering results, specify whether you used the exact factor (e.g., 1\;\text{mi} = 1.60934\;\text{km}) or a rounded classroom factor (e.g., 1.61).
• Real-world relevance:
  • Medicine relies on \mu\text{g} and \text{mL} precision—misplacing a metric prefix can be lethal.
  • Computing & telecom use giga-, tera-, nano-, etc., to label data rates and transistor dimensions.
• Philosophical insight: The metric system’s base-10 structure mirrors humanity’s decimal counting, enhancing global standardization and reducing cognitive load compared with the ad-hoc U.S. customary system.

Historical / Contextual Nuggets

• Leap-year rule embeds astronomy: The extra 0.25 days per year corrects for Earth’s orbital period (~365.2422 days).
• The international inch-to-centimeter definition (exact 2.54 cm) was agreed upon in 1959 by English-speaking nations, exemplifying global compromise for scientific consistency.

Quick Reference (One-Line Mnemonics)

• "A liter’s a cubic decimeter—lock volume to length."
• "Kilo-, hecto-, deca-—down the hill to deci-, centi-, milli-."
• "A pint’s a pound the world around" (~16\;\text{oz} ≈ 1\;\text{lb} fresh water) – helpful kitchen rule of thumb.

Timestamp Acknowledgment

• All examples and figures sourced from video at 23{:}19 as cited in transcript.