How do exact numbers affect rounding?

Exact numbers, such as counted discrete objects or defined constants, have infinite significant figures. They never limit the precision of your final answer.

What happens if a calculation involves both addition and multiplication?

Determine the precision of the intermediate result using the appropriate rule (addition or multiplication), but do not round yet. Carry the unrounded number forward, while noting its effective significant figures for the final operation.

How should I round if the first dropped digit is exactly 5?

Standard engineering and physics conventions dictate rounding to the nearest even digit (e.g., 2.45 becomes 2.4). Check your specific course guidelines, as some introductory classes simplify this by always rounding up.

How to round to the correct number of significant figures

To round to the correct number of significant figures, identify the limiting term in your calculation. For multiplication and division, the result has the same number of significant figures as the term with the fewest significant figures. For addition and subtraction, the result has the same number of decimal places as the term with the fewest decimal places. Apply this method at the very end of a multi-step calculation. Keeping extra digits during intermediate steps prevents the accumulation of rounding errors.

The setup

Determine the mathematical operations required for the calculation. Identify the significant figures or decimal places for every measured input value. Separate measured values from exact numbers, such as integers in formulas or defined conversion factors.

The steps

Compute the exact mathematical result keeping all calculator digits. 2. If the last operation was multiplication or division, find the input with the fewest total significant figures. 3. If the last operation was addition or subtraction, find the input with the fewest decimal places. 4. Round the raw result to the limit identified in step 2 or 3. If the first dropped digit is 5 or greater, round the last retained digit up.

Checking the result

Verify that your final answer does not claim more precision than the least precise input. If your rounded result ends in a zero that is significant, ensure you write it explicitly, using scientific notation if necessary to clarify trailing zeros.

Common errors

Rounding intermediate steps is a frequent error that leads to final results being off by a small margin. Another common mistake is applying the multiplication rule (counting total figures) to addition operations, which strictly depend on decimal places.

Worked example

Calculate the volume of a rectangular block with lengths $L = 12.3$ cm, $W = 4.56$ cm, and $H = 2.1$ cm, and report it to the correct number of significant figures.

Volume $V = L W H$ . Calculate the raw product: $12.3 imes 4.56 imes 2.1 = 117.7848$ cm $^3$ . Identify the significant figures for each factor: $L$ has 3, $W$ has 3, and $H$ has 2. The operation is multiplication, so the final result is limited by the factor with the fewest significant figures, which is $H$ (2). Round $117.7848$ to 2 significant figures. The third digit is 7, so round up. $V = 120$ cm $^3$ , or explicitly $1.2 imes 10^2$ cm $^3$ .

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References: OpenStax University Physics, Volume 1 · Halliday, Resnick, and Walker, Fundamentals of Physics