Significant figures are the formal language scientists use to communicate measurement precision. Every measurement has uncertainty, and sig figs encode where that uncertainty sits — a length reported as 2.50 cm asserts precision to the hundredths place, while 2.5 cm asserts only tenths. The rules for counting, rounding, and propagating sig figs catch students off guard because zeros play different roles depending on their position. Understanding the four counting rules and the two propagation rules eliminates the guesswork.
Why Zeros Cause All the Trouble
Non-zero digits are easy: they are always significant. The hard cases are zeros, because a zero can either be carrying precision information or merely holding a place in the decimal system. Leading zeros (the three in 0.00450) are pure place-holders — they tell you the number's magnitude but not its precision, and writing 4.50 × 10⁻³ communicates the same value with no ambiguity. Trailing zeros are the trickiest: in 1500, the trailing zeros could mean 'measured precisely to the unit' (4 sig figs) or 'rounded to the hundreds' (2 sig figs). The convention — that trailing zeros count only when a decimal point is explicitly present — exists to remove this ambiguity. 1500 has 2 sig figs by convention; 1500. has 4; 1.500 × 10³ has 4 unambiguously. Sandwiched zeros (the 0 in 307) are always significant because they cannot be place-holders — there's nothing for them to hold a place for when non-zero digits surround them on both sides.
Propagation Rules: Different for × and +
A common student mistake is treating addition and multiplication identically. They are not. For multiplication and division, the result inherits the minimum sig-fig count: 4.56 × 1.4 produces 6.4, because 1.4 has only 2 sig figs and you cannot create precision out of nothing. For addition and subtraction, the result inherits the minimum decimal-place count, not the minimum sig-fig count. 100.0 + 1.234 = 101.234 by arithmetic, but should be reported as 101.2, because 100.0 is known only to the tenths place — adding the more-precise 1.234 cannot make 100.0's hundredths digit knowable. Mixing these rules (e.g. applying the multiplication rule to an addition) is the single most common source of sig-fig errors in introductory chemistry and physics, and it produces answers that are either dishonestly precise or unnecessarily imprecise.
Scientific Notation as Disambiguation
Scientific notation is not just a way to write very large or very small numbers — it is also a way to make sig-fig counts unambiguous. Writing 1500 leaves the trailing zeros' status to convention; writing 1.5 × 10³ makes it explicit that there are 2 sig figs, and writing 1.500 × 10³ makes it explicit that there are 4. Practicing scientists default to scientific notation in publications precisely for this reason. The calculator handles e-notation (6.022e23, 2.50e-3, 1.500e+5) by counting sig figs in the coefficient only. The exponent does not affect sig figs because it represents magnitude, not measurement precision. When in doubt about a sig-fig count, rewrite the number in scientific notation — if the count is still ambiguous, the number was written ambiguously to begin with.