Arithmetic Operations with Significant Figures

You now know how to count significant figures in a single number. But what happens when you combine measured values through calculations? If you measure an object’s mass to four significant figures and its volume to three, your calculator will happily spit out a density with eleven decimal places. Can you trust all those digits?

Of course not. A result built from approximate measurements cannot suddenly become perfectly precise. The output of a calculation must honestly reflect the uncertainty sitting inside the input values. Two clean rules govern how this works, and they are different for different types of operations.

The Core Idea

Every measured value carries a built-in limit on its precision. When you perform arithmetic with such values, the result inherits the weakest link. No amount of calculation can manufacture precision that the original measurements never had. The rules below make this idea concrete.

Rule 1: Multiplication and Division: Count Significant Figures

When you multiply or divide measured quantities, the final result keeps as many significant figures as the input with the fewest significant figures.

Here is why this makes sense: in a product or quotient, every digit of one number interacts with every digit of the other. The least precise input contaminates the entire result, so you trim the answer to match that weakest input.

Worked Example: Density Calculation

Suppose you measure:

• Mass = 4.237 g (four significant figures)
• Volume = $2.51 \text{ cm}^3$ (three significant figures)

A calculator gives:

$\text{Density} = \frac{4.237}{2.51} = 1.68804780876... \text{ g cm}^{-3}$

Recording all those digits would be absurd. The volume, which is the less precise measurement, has only three significant figures. So the density must also be rounded to three significant figures:

$\text{Density} = 1.69 \text{ g cm}^{-3}$

Worked Example: The Light Year

The speed of light is given as $3.00 \times 10^8 \text{ m s}^{-1}$ (three significant figures). One year equals $3.1557 \times 10^7$ s (five significant figures). A light year is distance = speed $\times$ time:

$\text{Light year} = 3.00 \times 10^8 \times 3.1557 \times 10^7 = 9.4671 \times 10^{15} \text{ m}$

The speed has fewer significant figures (three), so the answer is rounded to three:

$\text{Light year} = 9.47 \times 10^{15} \text{ m}$

Rule 2: Addition and Subtraction: Count Decimal Places

When you add or subtract measured quantities, the final result keeps as many decimal places as the input with the fewest decimal places.

Notice the switch: for addition and subtraction, you count decimal places, not significant figures. The reason is that in a sum or difference, digits line up column by column. If one number is uncertain starting at the tenths column, the sum is also uncertain from the tenths column onward, regardless of how many significant figures the other numbers have.

Worked Example: Adding Masses

Add three measured masses: 436.32 g, 227.2 g, and 0.301 g.

Straight arithmetic gives:

$436.32 + 227.2 + 0.301 = 663.821 \text{ g}$

Now look at the decimal places in each input:

• 436.32 has two decimal places
• 227.2 has one decimal place
• 0.301 has three decimal places

The fewest is one decimal place (from 227.2 g), so the result is rounded to one decimal place:

$\text{Sum} = 663.8 \text{ g}$

Worked Example: Subtracting Lengths

$0.307 \text{ m} - 0.304 \text{ m} = 0.003 \text{ m} = 3 \times 10^{-3} \text{ m}$

Both inputs have three decimal places, so the result also has three decimal places. That gives 0.003 m, which has just one significant figure.

A Common Mistake to Avoid

It is tempting to apply the multiplication/division rule (count significant figures) everywhere, including for addition and subtraction. This is wrong and leads to results that do not correctly convey measurement precision.

Consider the addition example above. If you mistakenly applied the significant-figures rule, you would note that 0.301 has the fewest significant figures (three), and round the sum to three significant figures: 664 g. But that does not correctly convey the measurement precision. The correct answer, using the decimal-places rule, is 663.8 g. Similarly, in the subtraction example, applying the wrong rule would give $3.00 \times 10^{-3}$ m (three significant figures), which falsely implies much greater precision than the measurement actually has.

The key distinction:

• Multiplication and division — the rule is about significant figures (the total count of meaningful digits)
• Addition and subtraction — the rule is about decimal places (which column the uncertainty starts in)

Keeping these two rules separate is essential for reporting results that honestly represent the precision of your measurements.