Rounding and Significant Figures

Significant figures are the digits that carry actual information. Counting starts at the first non-zero digit, so in 0.004567 the leading zeros do not count — they only hold place value — and the number has four significant figures. Three rules cover the zeros. Zeros before the first non-zero digit are never significant. Zeros between digits always are, so 2005 has four. And trailing zeros after a decimal point are significant, because writing 0.4500 is a claim about precision. Keep significant figures and decimal places separate in your mind: 0.00456 has five decimal places but only three significant figures.

To round to a number of significant figures, count that many digits from the first non-zero one, then look at the very next digit to decide. For 34567 to three significant figures: keep 3, 4, 5, and the next digit is 6 — which is 5 or more, so round up, making 345 into 346. The answer is 34600. The step people get wrong is what happens to the digits you dropped: they become zeros, not nothing. Writing 346 would make the number a hundred times too small. Always sense-check that your rounded answer is roughly the same size as the original.

Small numbers work the same way, and the leading zeros never count. For 0.004567 to two significant figures: the first significant digit is 4, the second is 5, and the next digit — 6 — decides. Six rounds up, so 45 becomes 46 and the answer is 0.0046. The leading zeros stay because they position the number. Now notice something useful: written in scientific notation that is 4.6 × 10⁻³, where the significant figures are simply the digits in front and the zeros have vanished into the power. That is one of the real reasons science uses the notation — precision becomes visible at a glance.

Scientific notation and significant figures fit together neatly: in a × 10ⁿ, the digits you write in the a part *are* the significant figures. So 3.46 × 10⁴ has three, and 4.6 × 10⁻³ has two. This also solves a problem ordinary notation cannot. Written as 2,000,000, there is no way to tell whether that is precise to one significant figure or three. But 2 × 10⁶ claims one, while 2.00 × 10⁶ claims three — the trailing zeros are deliberate. Same value, different statements about how precisely it was measured.

Two habits make rounding reliable. First, round once, at the very end. Rounding partway through a calculation lets a small error grow through every step that follows, and it is the most expensive rounding habit in measurement work — carry the full calculator value and round only the final answer. And never re-round an already-rounded number: going 2.4519 to 2.452 to 2.45 to 2.5 compounds the error, whereas rounding the original straight to three significant figures gives 2.45. Second, match the precision to the question. Money takes two decimal places; measurements usually take two or three significant figures. Giving an answer more precise than your data is a false claim about how well you know it.