Physics · Ch 1 — Units and Measurement

Rules for Arithmetic Operations with Significant Figures

1.3.1

Rules for Arithmetic Operations with Significant Figures

When you perform calculations with measured values, the result cannot be more precise than the least precise measurement you started with. The final result must reflect the uncertainties in the original data.

Multiplication and Division

In multiplication or division, the final result should have as many significant figures as the original number with the fewest significant figures.

Example: Mass = 4.237 g (four significant figures), Volume = 2.51 cm³ (three significant figures).

\text{Density} = \frac{4.237 \text{ g}}{2.51 \text{ cm}^3} = 1.68804780876 \text{ g/cm}^3

It would be absurd to report 11 decimal places when the volume measurement has only three significant figures. The result should be rounded to three significant figures:

\text{Density} = 1.69 \text{ g/cm}^3

Another example: Speed of light = $3.00 \times 10^8$ m/s (three significant figures). One year = $3.1557 \times 10^7$ s (five significant figures). A light-year is:

3.00 \times 10^8 \text{ m/s} \times 3.1557 \times 10^7 \text{ s} = 9.47 \times 10^{15} \text{ m}

The result has three significant figures, matching the speed of light (the least precise input).

Addition and Subtraction

In addition or subtraction, the final result should have as many decimal places as the number with the fewest decimal places .

Example: Add 436.32 g, 227.2 g, and 0.301 g.

Arithmetic sum: 663.821 g. But 227.2 g is correct to only one decimal place. The result should be rounded to one decimal place:

$663.8 \text{ g}$ …

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