Which digits are significant?
All non-zero digits count. Zeros between non-zero digits count. Leading zeros never count — 0.0043 has two significant figures. Trailing zeros after a decimal point do count, because writing them is a claim about precision: 4.30 has three. Trailing zeros in a whole number are ambiguous, which is precisely why 4,500 is unclear and 4.5 × 10³ is not.
Why do significant figures matter?
Because they communicate how precise a measurement actually is, and stop calculations manufacturing precision that was never measured. If you measure a room as 3.2 by 4.1 metres, the product is 13.12 on a calculator — but reporting 13.12 claims accuracy to a hundredth of a square metre from measurements accurate to a tenth of a metre. The honest answer is 13 square metres.
What are the rules for calculations?
Two different rules, which is the part people miss. For multiplication and division, the result takes the same number of significant figures as the least precise input. For addition and subtraction, it takes the same number of decimal places as the input with the fewest — not significant figures, decimal places. Applying the multiplication rule to a sum is a very common mistake.
When should I round — during or at the end?
At the end. Rounding intermediate steps compounds error through the calculation, sometimes shifting the final digit. Carry extra digits through the working and round only the reported answer. In multi-stage work, keep at least two guard digits beyond your intended precision. This is the opposite of what many students are taught to do by hand.
How do exact numbers and constants affect the count?
They do not limit it. Counted quantities are exact: 12 items is exactly 12, not 12 to two significant figures. Defined conversions are exact too — 1 inch is precisely 2.54 cm by definition. So when multiplying a measurement by an exact factor, the measurement alone determines the significant figures. Treating an exact conversion as a two-figure limit needlessly discards precision.
What is round-half-to-even and why is it used?
When the digit to be dropped is exactly 5 with nothing after it, always rounding up introduces a small systematic upward bias across many values. Round-half-to-even, also called banker's rounding, rounds to the nearest even digit instead: 2.5 becomes 2, 3.5 becomes 4. Over many calculations the biases cancel. Most scientific and financial software uses it by default, which is why a spreadsheet sometimes disagrees with what you expected.