Significant Figures Calculator

Start by counting. Take 0.004506. The three leading zeros place the decimal point and claim nothing, so they are not figures. The 4, the 5, the 6 and the zero trapped between the 5 and the 6 all count: four significant figures. The last one sits in the millionths, so the number is written to six decimal places while carrying only four figures. Figures and decimal places are different quantities, and this is the number that shows it. Now round 2.675 to three figures. Keep 2, 6 and 7. The first digit dropped is a 5 with nothing after it, so this is an exact tie and the rule you use decides it. Rounding half up gives 2.68. Rounding half to even looks at the 7, finds it odd, and also goes up to 2.68 — the two rules agree here. Feed the same number to a computer and it will very likely answer 2.67, because the double it stores is 2.674999999999999822 and that is genuinely below the halfway point. Change the number to 2.665 and the same machine answers 2.67, which is correct. The float is not applying a rule; it is reporting an accident of binary storage. Multiply: 2.0 × 3.14159. The exact product is 6.28318. The factors carry two and six figures, so the answer carries two: 6.3. It does not matter that 3.14159 is known to five decimal places. Multiplying scales a relative uncertainty, and a factor known to one part in twenty drags the whole product down to one part in twenty. Add: 24.6 + 3.14159. The exact sum is 27.74159. Here the figures are irrelevant and the columns decide. 24.6 has nothing to say about the hundredths, so the answer stops at the tenths: 27.7. Counting figures instead would have given 27.7416, claiming five figures of precision from a term that has three. Subtract: 100.0 − 99.9. Both inputs carry four figures. The difference is 0.1 — one figure. Nothing has gone wrong; the leading digits were common to both numbers and cancelled, leaving only the uncertain tail. This is why a result computed as a small difference of large numbers should be treated with suspicion, and why measuring the difference directly is usually better than measuring both ends. Finally, do it in one step. Rounding 2.4449 to three figures gives 2.44. Rounding it to four first gives 2.445, and rounding that to three gives 2.45. Both look reasonable and only the first is right. Over all ninety thousand five-figure decimals between 1 and 10, staging the rounding changes the three-figure answer on exactly 4,500 of them — five per cent. Round once, at the end.