Significant Figures & Measurement
Counting sig figs, rounding, scientific notation, SI units and prefixes, and dimensional analysis.
Exact vs measured
CORE RULEExact numbers (counts, definitions) have infinite sig figs and never limit precision. Measured values are inexact and do limit sig figs.
Counting significant figures
CORE RULERules for counting significant digits: (1) All nonzero digits are significant. (2) Zeros between nonzero digits are always significant. (3) Leading zeros (before the first nonzero digit) are never significant. (4) Trailing zeros after a decimal point are significant. (5) Trailing zeros without a decimal point are NOT significant.
Scientific notation
CORE RULEWrite as a.bcd × 10ⁿ with exactly one nonzero digit before the decimal (1.23 × 10⁴, not 12.3 × 10³). Every digit shown in the coefficient is significant.
Rounding
CORE RULEIf the first dropped digit is ≥ 5, round the last kept digit up; cascade the carry (999.6 to 3 sf → 1.00 × 10³).
Arithmetic with sig figs
METHODAdd/subtract: round to the fewest decimal places among inputs. Multiply/divide: round to the fewest sig figs. Exact numbers never limit sig figs.
Multi-step: carry all digits through every step and round only the final result. Early rounding propagates error.
Dimensional analysis
METHODMultiply by conversion factors equal to 1 with the old unit in the denominator so units cancel. Track every unit through every step.
Squared/cubic units: raise the factor to that power (1 m = 10² cm → 1 m² = 10⁴ cm²). Compound units (g/cm³ → kg/m³): convert top and bottom separately.
Common SI base units
REFERENCElength → m · mass → kg · time → s · temperature → K · amount → mol. Liter, m/s, and kg/m³ are derived units.
Temperature
CORE RULEK = °C + 273.15 (273.15 is exact). °C = (°F − 32) × 5/9. °F = °C × 9/5 + 32.
Common SI prefixes
REFERENCE| Prefix | Symbol | Value |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| deci | d | 10⁻¹ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
| femto | f | 10⁻¹⁵ |
Common conversions
REFERENCEExact (defined): 1 in = 2.54 cm · 1 lb = 0.45359237 kg · 1 mi = 1609.344 m · 1 gal = 3.785411784 L · 1 mL = 1 cm³ · 1 Å = 10⁻¹⁰ m · 1 atm = 101,325 Pa = 760 torr · 1 cal = 4.184 J (thermochemical). Approximate: 1 atm ≈ 760 mmHg.
Density as a conversion
CORE RULEDensity links mass and volume; write it in the orientation that cancels the known unit and leaves the unknown.
Watch out
WATCH OUTYou cannot add sig figs a measurement does not have. To show a whole-number result's trailing zeros are significant, use a trailing decimal (300.) or scientific notation.