Significant figures (also called significant digits) represent the digits in a number that carry meaningful precision based on how the measurement was made. A measurement of 4.56 cm has three significant figures, implying the measurement is precise to the nearest 0.01 cm. Writing it as 4.560 cm implies four significant figures — the trailing zero asserts that the measurement was precise enough to know the last digit is zero, not just rounded to it.
The rules for counting significant figures: all non-zero digits are significant; zeros between non-zero digits are significant (e.g., 1,003 has four significant figures); trailing zeros in a decimal number are significant (4.560 has four); leading zeros before the first non-zero digit are not significant (0.0045 has two significant figures — the leading zeros just place the decimal).
Significant figures in multiplication and division
In multiplication and division, the result should have the same number of significant figures as the least precise input. If you multiply 4.56 (three significant figures) by 1.4 (two significant figures), the result should be rounded to two significant figures: 4.56 × 1.4 = 6.384, rounded to 6.4. The false precision of reporting 6.384 implies a precision that the input data didn't support.
Most calculators display all available decimal places regardless of input precision, which is why significant figure discipline is a human responsibility, not something the calculator enforces. In physics, chemistry, and engineering labs, reporting too many significant figures is considered an error — it misrepresents the quality of the underlying measurement.
Significant figures in addition and subtraction
For addition and subtraction, the rule is different: the result should be rounded to the same decimal place as the least precise input, not to the same number of significant figures. Adding 123.4 (precise to 0.1) and 5.678 (precise to 0.001) gives 129.078, which rounds to 129.1 — one decimal place, matching the precision of the less precise input.
This distinction matters when mixing addition/subtraction with multiplication. Follow the order of operations, applying the appropriate rounding rule at each step rather than waiting until the end. Rounding only at the final step introduces intermediate rounding errors and gives a slightly more accurate result, but intermediate steps should still be tracked with mental awareness of the effective precision.