A calculator gives many digits from two measured values, but most of them are not supported by the measurements. Products keep the fewest significant figures; sums keep the fewest decimal places.
A tabletop is measured as 4.237 m long and 1.2 m wide. The calculator gives the area as 4.237 × 1.2 = 5.0844 m².
The width, 1.2 m, has two significant figures. It is known only to about the nearest 0.1 m. An area calculated from it cannot be known to 0.0001 m². The calculator treats both inputs as exact, so it reports digits the measurements do not support.
The answer must be rounded to the precision the measurements allow. There are two rules: one for multiplication and division, and a different one for addition and subtraction.
The result keeps as many significant figures as the input with the fewest significant figures.
For the tabletop, 4.237 m has four significant figures and 1.2 m has two. The result keeps two: 5.0844 m² becomes 5.1 m².
A car travels 100.0 m in 8.0 s. Its speed is 100.0 ÷ 8.0. What is the speed, correctly rounded?
The calculator gives 12.5 m/s. 100.0 has four significant figures; 8.0 has two.
Division keeps the smaller count, two. The speed is 13 m/s.
The time was measured only to about 0.1 s in 8 s, roughly 1%. The speed cannot be known more precisely than that, so the third digit is not reliable.
The result keeps as many decimal places as the input with the fewest decimal places. This rule counts decimal places, not significant figures.
12.587 m + 3.4 m = 15.987 m on a calculator. 3.4 m has one decimal place, so the sum is rounded to one decimal place: 16.0 m.
Each measured number is uncertain in its last written digit. There are two ways to describe that uncertainty.
The absolute uncertainty is an amount in the same unit as the measurement. 12.3 cm is uncertain by about 0.1 cm.
The relative uncertainty is the absolute uncertainty divided by the value, often given as a percentage. 0.1 cm out of 12.3 cm is about 0.8%.
Why does addition use decimal places, while multiplication uses significant figures?
Addition. Take a = 12.3, uncertain in the tenths place, and b = 1.045, uncertain in the thousandths. Write a = 12.3 + e and b = 1.045 + f, where e and f are the unknown errors. Then a + b = 13.345 + e + f. The error e is about 0.1, so the sum is uncertain in the tenths place, and every digit after it is unreliable. Round to 13.3.
In a sum, absolute uncertainties add. The largest absolute uncertainty sits at the coarsest decimal place, so that place sets where the sum stops being reliable.
Multiplication. Take a rectangle 2.3 cm long and 1.045 cm wide. The length lies between 2.25 and 2.35 cm: an uncertainty of about 0.05 in 2.3, roughly 2%. The width lies between 1.0445 and 1.0455 cm: about 0.05%.
Using the lowest and highest lengths: 2.25 × 1.045 = 2.351 and 2.35 × 1.045 = 2.456. The area is somewhere between about 2.35 and 2.46 cm². The product 2.3 × 1.045 = 2.4035 claims far more precision than that. Rounded to two significant figures, the area is 2.4 cm².
In a product, relative uncertainties add. The 2% from the length dominates the 0.05% from the width. Fewer significant figures means a larger relative uncertainty, so the input with the fewest significant figures sets the precision of the product.
In a calculation with several steps, do not round after each step. Each rounding adds a small error, and the errors accumulate. Keep one or two extra digits through the intermediate steps, and round only the final answer.
1When measured values are multiplied or divided, the result keeps…
2When measured values are added or subtracted, the result keeps…
3A tabletop is 4.237 m by 1.2 m. Its area, correctly rounded, is…
4100.0 m travelled in 8.0 s gives a speed of…
5For 12.587 m + 3.4 m, the answer is rounded to ____ decimal place(s).
6When should a multi-step calculation be rounded?
7Why do addition and multiplication use different rounding rules?
Next: describing a quantity by its dimensions.
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