Every measurement you make in a lab has some uncertainty in it — a ruler marked in millimetres can't tell you a length to the nearest nanometre. Significant figures (sig figs) are the digits in a measured number that are actually known with confidence, plus one final estimated digit. Reporting the right number of sig figs tells other people how precise your measurement really was, and it stops you from claiming more precision than your instrument (or your calculator) can actually justify. This matters for the whole unit ahead: every stoichiometry answer starts from a measured mass or volume, and a calculator will happily return ten digits for . Most of those digits are noise. Significant figures are the honest way to say how much of a calculated answer you can actually trust.
When you read any scale, you record every digit the markings give you for certain, then estimate one more digit between the smallest marks. A ruler with millimetre marks lets you read cm for certain and estimate the next digit, so you record cm: three sig figs, the last one a careful guess. That single estimated digit is why the last significant figure of any measurement is the uncertain one, and why writing cm from a millimetre ruler would be dishonest: it claims a digit you never measured.
| Instrument | Smallest division | Typical reading | Sig figs |
|---|---|---|---|
| Centigram balance | 0.01 g (digital) | 4 | |
| 50 mL burette | 0.1 mL | (estimate the 0.01) | 4 |
| 100 mL graduated cylinder | 1 mL | (estimate the 0.1) | 3 |
| Thermometer | 1 °C | (estimate the 0.1) | 3 |
Counting Significant Figures
- All nonzero digits (1–9) are always significant.
- Zeros between nonzero digits are always significant, e.g. has 3 sig figs.
- Leading zeros (before the first nonzero digit) are never significant — they only locate the decimal point, e.g. has 2 sig figs.
- Trailing zeros after a decimal point are significant, e.g. has 4 sig figs.
- Trailing zeros in a whole number with no decimal point are ambiguous — could have 2, 3, or 4 sig figs. Scientific notation removes the ambiguity: (2 sf) vs (4 sf).
Rounding and Sig Figs in Calculations
To round to sig figs, keep the first significant digits and look at the next digit to decide whether to round the last kept digit up or leave it. If that next digit is 5 or greater, round up; if it's less than 5, round down (drop it). Only round your final answer — rounding after every intermediate step throws away precision you needed and can shift the last digit of your final answer. In a multi-step problem, carry at least one or two extra guard digits through every step (or keep the full value in your calculator's memory) and round once at the end.
- Multiplication / division: the answer gets the same number of sig figs as the measurement with the fewest sig figs.
- Addition / subtraction: the answer gets the same number of decimal places as the measurement with the fewest decimal places (sig-fig count doesn't matter here — decimal places do).
Why two different rules? When you add or subtract, uncertainties line up in the same decimal column: if one mass is only known to the nearest , the total can't be known to the nearest , no matter how precise the other mass is. When you multiply or divide, what carries through is the relative uncertainty (the uncertainty as a fraction of the value), and the number with the fewest sig figs has the largest relative uncertainty, so it controls the answer.
- Both factors have 2 sf, so the answer has 2 sf
- Used for density, moles from mass, molar ratios
- has 1 decimal place, the fewest
- Used for mass differences, volume changes, totals
Unit Analysis (Dimensional Analysis)
A conversion factor is a fraction equal to 1, built from an equality like — writing it as lets you multiply by it without changing the actual quantity, only its units. Chain several conversion factors together and the unwanted units cancel diagonally, leaving only the unit you want. This is the safest way to move between units in chemistry because the algebra checks itself: if the units don't cancel to what you expect, you've made a setup error before you even touch the numbers. The same method runs the rest of this unit. Molar mass (), Avogadro's number (particles per mole) and the mole ratio from a balanced equation are all conversion factors, so every stoichiometry problem is just a longer dimensional-analysis chain. To choose which way up a factor goes, put the unit you want to cancel in the denominator. If you are unsure whether to multiply or divide, write the units and let them decide. Exact defined relationships (like ) never limit your sig figs; only the sig figs in your original measured value do.