Significant Figures: How Much of Your Number Is Actually Real

Every non-zero digit is significant. Zeros BETWEEN non-zero digits are significant — 1005 has four. Leading zeros never are: 0.0032 has two, because the zeros only place the decimal point. Trailing zeros are the ambiguous case, and they are ambiguous because of how we write numbers rather than because of anything real. Does 1500 mean "about fifteen hundred" or "exactly 1500"? Written as 1.5 x 10^3 it clearly has two digits; as 1.500 x 10^3 it clearly has four. Scientific notation exists partly to settle this. Counted things and defined constants are EXACT and have unlimited digits. Twelve bolts is exactly twelve, and 1 inch is exactly 25.4 mm by definition. Neither ever limits a calculation.

Writing 24.3 mm claims you know the tenths and are unsure in the hundredths. Writing 24.30 mm claims you know the hundredths. They are different measurements made with different instruments, and the trailing zero is doing real work. So read your instrument to the smallest division and then ESTIMATE one digit beyond it — halfway between two marks is a real observation, and it is the last honest digit you have. A steel rule marked in millimetres supports tenths of a millimetre estimated by eye; it does not support hundredths. A digital display looks like it settles this and often does not. A caliper showing 24.31 is displaying hundredths, but if it wanders between 24.29 and 24.33 when you re-close it, the hundredths digit is noise. The repeat spread, not the display, tells you where the real number ends.

Round the uncertainty first, to one or two digits, then round the value to the same decimal place. "49.3 plus or minus 0.4 cm3" is right; "49.32175 plus or minus 0.4" is announcing digits the uncertainty has already denied. Give the units every time, and use scientific notation when the digit count would otherwise be ambiguous. A number without units is not a measurement, and unit confusion has destroyed real spacecraft. And keep the RAW readings in the notebook, at full precision, alongside the rounded result. Rounding is for presenting; the raw numbers are the evidence, and somebody — possibly you — will want to recalculate them differently later.