Notes on Conversion Factors, Sig Figs, and Rounding

Core idea: converting using conversion factors from equalities

• An equality between two units represents the same quantity described with different units (e.g., one centimeter is equivalent to 10 millimeters). They’re numerically different, but quantity-wise the same.
• Goal: turn any known equality into a conversion factor (a ratio) to use in calculations.
• From a given equality, you can derive two conversion factors by placing the units in either direction:
  • Example from 100 centimeters = 1 meter:
  • $\frac{100\,\mathrm{cm}}{1\,\mathrm{m}}$
  • $\frac{1\,\mathrm{m}}{100\,\mathrm{cm}}$
• Which conversion factor you pick depends on what quantity you are solving for. The statement "where we’re going goes on top" means you should orient the factor so that the unit you want ends up on the numerator.
• In the second form (meters on top, centimeters on the bottom), you effectively divided both sides of the equality by a meter; this often yields a ratio useful for solving for quantities in meters, but the numeric value may look less convenient (e.g., 100/1 ≠ 1).
• Practice prompt from the transcript: derive two conversion factors for 1 liter = 1000 milliliters.
  • Two possible conversion factors:
  • $\frac{1\,\mathrm{L}}{1000\,\mathrm{mL}}$
  • $\frac{1000\,\mathrm{mL}}{1\,\mathrm{L}}$
  • The choice depends on whether you are solving for liters or milliliters.
• The instructor emphasizes active learning: write both forms yourself rather than passively listening.

Practice: write the two conversion factors for 1 L = 1000 mL

• Conversion factors:
  • $\frac{1\,\mathrm{L}}{1000\,\mathrm{mL}}$ (useful when solving for quantities in liters)
  • $\frac{1000\,\mathrm{mL}}{1\,\mathrm{L}}$ (useful when solving for quantities in milliliters)
• Note on sig figs for these factors:
  • Both directions involve metric-to-metric conversions, which are exact by definition.
  • You can treat these as having infinite significant figures in the sense that the exact relationships are exact definitions.
  • Example: 1000 milliliters in 1 liter is exact, so there are infinitely many sig figs in that relationship.
• The transcript also discusses activity for other metric–US conversions and the concept of exact vs measured quantities (see below).

Significance of sig figs in conversion factors

• Metric-to-metric conversions (e.g., liters and milliliters) are exact by definition:
  • $1000\,\mathrm{mL}=1\,\mathrm{L}$ is exact, so it has infinite sig figs.
• US customary conversions involve a mix of exact and measured quantities:
  • Example: $1\,\mathrm{ft}=12\,\mathrm{in}$ is exact by definition.
  • Many common conversions are fixed by definition, but the measured numbers you use to convert (e.g., a length in inches measured from a meter) introduce finite sig figs.
• Explicit example from the transcript:
  • The relationship that fixes one quantity and measures the other could produce an exact component (the "one") and a measured component (the numeric part like 39.4 or 62.1) with limited sig figs.
  • In the presented table, the one (the exact unit count) is infinite in sig figs; the other side (e.g., 39.4) is measured and finite in sig figs.
• Exception: certain exact conversion factors have known finite decimals that can be treated as exact for practical purposes:
  • The centimeter–inch relationship is given as an exact equivalence in the transcript: the perfect centimeter equals 2.54 centimeters per inch and the perfect inch equals 2.54 centimeters. In this specific case, 2.54 cm per inch is treated as exact, so it does not impose a sig fig limitation.
• Practical rule:
  • When you have an exact definition between units, treat that piece as having infinite sig figs.
  • The other, measured quantity in the conversion will limit the sig figs of your result.
  • If a conversion uses two exact quantities (two exact definitions), the conversion factor is exact and does not limit sig figs.
  • The overall rule is: keep all digits from your given measurements, and only round at the end according to the required sig figs.

Examples: metric–US conversions and rounding implications

• The transcript notes a common mixed scenario: converting between liters and quarts.
• Example setup: 1 L = 1000 mL (exact) and 1 qt = 946 mL (rounded value, used in practice but not exact).
• Two ways to approach this:
  • Approach A: Use a fixed liter (exact) and convert to quarts, which can yield about 1.057 qt per liter (using full precision):
  • $1\,\mathrm{L} = \frac{1000\,\mathrm{mL}}{946\,\mathrm{mL}}\,\mathrm{qt} \approx 1.057\,\mathrm{qt}$
  • If you round to three significant figures, this becomes $1.06\,\mathrm{qt}$ per liter.
  • Approach B: Use a fixed quart (946 mL) and convert to liters, which gives approximately 0.946 L per quart:
  • $1\,\mathrm{qt} = 946\,\mathrm{mL}$ (from the rounded conversion)
  • $1\,\mathrm{qt} \approx 0.946\,\mathrm{L}$
• Key point about rounding:
  • Using 1 L = 1000 mL and 1 qt = 946 mL, the derived factor 1.057 qt per L is slightly different from 1.06 qt per L when rounded to three sig figs, due to the rounding of 946 mL to three significant figures in the intermediate step.
  • The calculator-based check with both conversions shows small differences depending on which conversion factor is kept at full precision vs rounded earlier.
  • The main lesson: keep all digits you’re given in early steps, and round only at the final answer according to the required sig figs.
• The instructor notes that 946 mL per quart is a rounded figure, and using it in different directions can lead to small discrepancies; in contrast, using the exact definitions (where available) avoids this issue.
• The exception that helps: the exact centimeter–inch relationship (2.54 cm per inch) is treated as exact, so 2.54 does not limit sig figs in that particular conversion.

Takeaways and practical implications

• Always derive the two possible conversion factors from any given equality.
• For problems where you are solving for a quantity in a particular unit, place that unit on the numerator of the conversion factor you use.
• Metric-to-metric conversions are exact by definition, so they do not limit sig figs; US customary conversions often involve measured quantities, which do limit sig figs.
• When rounding, be mindful of whether you are combining multiple conversion factors; rounding one factor early can introduce small discrepancies that disappear if you retain more digits of precision until the final answer.
• Be aware of common rounding pitfalls (e.g., rounding 946 mL to 946.0 mL or 1000 mL to 1 L) and how these affect end results when chained together.
• The overall goal is to use conversion factors as flexible tools to move between units while preserving the accuracy dictated by sig figs and the exactness of certain definitions.

Quick recap: key formulas to memorize from this material

• From 100 cm = 1 m:
  • $\frac{100\,\mathrm{cm}}{1\,\mathrm{m}}$
  • $\frac{1\,\mathrm{m}}{100\,\mathrm{cm}}$
• From 1 L = 1000 mL:
  • $\frac{1\,\mathrm{L}}{1000\,\mathrm{mL}}$
  • $\frac{1000\,\mathrm{mL}}{1\,\mathrm{L}}$
• Expressing a practical liter–quart relationship (using approximate values):
  • $1\,\mathrm{L} \approx 1.057\,\mathrm{qt}$ (exact calculation with 1000 mL and 946 mL definitions; rounded to three sig figs: $1.06\,\mathrm{qt}$)
  • $1\,\mathrm{qt} = 946\,\mathrm{mL}$
  • Inverse: $1\,\mathrm{qt} \approx 0.946\,\mathrm{L}$
• Important exception to exactness in conversions:
  • 1\,\mathrm{in} = 2.54\,\mathrm{cm}} is treated as exact in this context, so $2.54\,\mathrm{cm}$ is not a limiting sig fig value in that particular ratio.