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Metric Prefixes and Conversion

Sep 1, 2025

Overview

This lecture covers metric unit prefixes, how to convert between them, and best practices for expressing measurements using these prefixes. Step-by-step problem-solving methods are included, with worked examples shown in a math book style.

Metric vs. English Unit Systems

  • English units (inches, feet, yards, miles) have irregular conversion factors (e.g., 12 inches = 1 foot, 3 feet = 1 yard, 1,760 yards = 1 mile), making conversions inconsistent and complex.
  • The metric system uses prefixes based on powers of 10 (e.g., milli-, centi-, kilo-), making conversions straightforward and consistent.
  • Metric prefixes can be used with any unit (liters, grams, seconds, etc.).

Common Metric Prefixes and Their Meanings

  • Large prefixes:
    • tera- (T): 10¹²
    • giga- (G): 10⁹
    • mega- (M): 10⁶
    • kilo- (k): 10³
  • Small prefixes:
    • deci- (d): 10⁻¹
    • centi- (c): 10⁻²
    • milli- (m): 10⁻³
    • micro- (μ): 10⁻⁶
    • nano- (n): 10⁻⁹
    • pico- (p): 10⁻¹²
  • Capitalization matters: Mega (M) is different from milli (m).

Using Prefixes in Conversions

  • Use conversion factors based on powers of 10 to move between units with different prefixes.
  • Step-by-step method:
    1. Write what you start with on the left and what you want on the right.
    2. Set up a conversion factor so units cancel (one on top, one on bottom).
    3. Place "1" in front of the larger unit; place the appropriate power of 10 in front of the smaller unit.
    4. Multiply/divide as needed to solve.

Example 1: Convert 400 microliters (μL) to liters (L)

Step 1: Write the starting value and the desired unit.

  400 μL → ? L

Step 2: Set up the conversion factor using the definition:
  1 μL = 1 × 10⁻⁶ L

Step 3: Arrange so units cancel:

  400 μL × (1 L / 1 × 10⁶ μL)

Step 4: Cancel units and calculate:

  = 400 × (1 / 1,000,000) L
  = 0.0004 L
  = 4 × 10⁻⁴ L

Example 2: Convert 3.757 millimeters (mm) to meters (m)

Step 1:
  3.757 mm → ? m

Step 2:
  1 mm = 1 × 10⁻³ m

Step 3:
  3.757 mm × (1 m / 1,000 mm)

Step 4:
  = 3.757 / 1,000 m
  = 0.003757 m
  = 3.757 × 10⁻³ m

Example 3: Convert 1.99 megayears (My) to years (y)

Step 1:
  1.99 My → ? y

Step 2:
  1 My = 1 × 10⁶ y

Step 3:
  1.99 My × (1 × 10⁶ y / 1 My)

Step 4:
  = 1.99 × 10⁶ y

Example 4: Convert 37 kilograms (kg) to grams (g)

Step 1:
  37 kg → ? g

Step 2:
  1 kg = 1,000 g

Step 3:
  37 kg × (1,000 g / 1 kg)

Step 4:
  = 37,000 g
  = 3.7 × 10⁴ g

Example 5: Convert 0.357 micrometers (μm) to meters (m)

Step 1:
  0.357 μm → ? m

Step 2:
  1 μm = 1 × 10⁻⁶ m

Step 3:
  0.357 μm × (1 m / 1 × 10⁶ μm)

Step 4:
  = 0.357 / 1,000,000 m
  = 3.57 × 10⁻⁷ m

Example 6: Multi-step Conversion — 187.3 micrometers (μm) to kilometers (km)

Step 1:
  187.3 μm → ? km

Step 2:
  First, convert μm to m:
  1 μm = 1 × 10⁻⁶ m
  187.3 μm × (1 m / 1 × 10⁶ μm) = 1.873 × 10⁻⁴ m

Step 3:
  Then, convert m to km:
  1 km = 1,000 m
  1.873 × 10⁻⁴ m × (1 km / 1,000 m) = 1.873 × 10⁻⁷ km

Expressing and Reporting Measurements

  • Express values so the number is between 1 and 1,000 with an appropriate prefix (e.g., 65.3 kg instead of 65,300 g).
  • Avoid reporting very large or very small numbers with prefixes (prefer 2.3 mm over 0.0023 m).
  • Keep the correct number of significant figures when converting.

Example: Express 0.00374 joules (J) using a prefix

Step 1:
  0.00374 J = 3.74 × 10⁻³ J

Step 2:
  10⁻³ corresponds to milli- (m)

Step 3:
  3.74 × 10⁻³ J = 3.74 mJ

Example: Adjusting exponents to match a prefix

Given: 4.9 × 10⁴ L

Step 1: Move decimal to get a number between 1 and 1,000:
  4.9 × 10⁴ L = 49 × 10³ L

Step 2: 10³ corresponds to kilo- (k)

Step 3:
  49 × 10³ L = 49 kL

Using Calculators for Scientific Notation

  • Enter numbers in scientific notation using your calculator’s E or EXP key (e.g., 1E-6 for 1 × 10⁻⁶).
  • On TI calculators, use the SCI/ENG function to switch between scientific and standard display modes.
  • To convert a decimal answer to scientific notation, use the calculator’s SCI mode.

Practice & Resources

  • Practice single- and two-step conversions to build fluency.
  • Use the online practice tool (professorclements.com) for unlimited metric conversion exercises with step-by-step solutions.

Key Terms & Definitions

  • Metric Prefix: A word part placed before a unit to indicate multiplication by a power of ten.
  • Conversion Factor: A ratio expressing how many of one unit equals another.
  • Scientific Notation: A way to express numbers as a product of a number between 1 and 10 and a power of ten.
  • Significant Figures (Sig Figs): Digits in a measurement that are known with certainty plus one estimated digit.

Action Items / Next Steps

  • Memorize common metric prefixes and their powers of ten.
  • Practice converting between units with different prefixes, including multi-step conversions.
  • Use the online practice site for additional exercises.
  • Review calculator functions for scientific notation entry and display.

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