Significant figures in calculations

Significant figures are important in calculations because they help keep the precision of measurements accurate in the results. When you do math operations, the number of significant figures in your values decides how precise the final answer should be. Use this resource to learn the rules for using significant figures correctly in calculations.

The rules for significant figures depend on whether you are adding and subtracting, or multipling and dividing.

Addition and subtraction

When you add and subtract numbers with a different number of significant figures:

1. Align the numbers by their decimal points.
2. Identify the number with the fewest decimal places.
3. Round off the other numbers so that they all have the same number of decimal places as this number.
4. Complete the calculation.
5. The answer will automatically have the correct number of significant figures.

Rounding numbers

Remember the rules for rounding numbers.

1. If the last digit is greater than \(5\), round up the second last digit by \(1\).

   For example, \(5.78\) is rounded up to \(5.8\).

2. If the last digit is less than \(5\), round down the second last digit by \(1\).

   For example, \(1.304\) is rounded down to \(1.30\).

3. If the last digit is \(5\), round off the second last digit to the nearest even number.

   For example, \(4.85\) becomes \(4.8\) (not \(4.9\)) and \(4.55\) becomes \(4.6\) (not \(4.5\)).

Exercise – rounding numbers

1. Round off each of the following numbers to one less significant figure, then to two less significant figures.
1. \(7.668\)
2. \(0.085\,4\)
3. \(21.092\)
4. \(255.6\)
5. \(35.3\)
6. \(6.75\)
7. \(2.85\)

Example – using significant figures in addition and subtraction

Add \(6.703\), \(2.49\) and \(11.736\,8\) and give your answer to the correct number of significant figures.

First, let's align the numbers.
\[\begin{align*} 6&.703\\
2&.49\\
11&.736\,8
\end{align*}\]

\(2.49\) has the fewest number of decimal places: \(2\). We need to round the other numbers to \(2\) decimal places. \(6.703\) would become \(6.70\) and \(11.7368\) would become \(11.74\). We can then add the numbers.
\[\begin{array}{r}
6.70\\
2.49\\
\underline{+11.74}\\
20.93
\end{array}\]

The answer, given to the correct number of significant figures, is \(20.93\).

Exercise – using significant figures in addition and subtraction

1. Calculate the following and give your answers to the correct number of significant figures.
1. \(6.9+0.35+12.625\)
2. \(27.1+13+64.51\)
3. \(8.72-3.001-0.2\)
4. \(67.4-18.37+0.55\)
5. \(0.003+0.0125-0.003\,78\)

Multiplication and division

When you multiply and divide numbers with a different number of significant figures:

1. Identify the number with the fewest significant figures.
2. Complete the calculation.
3. Round the final answer to the same number of significant figures as the number identified.

Example 1 – using significant figures in multiplication and division

Multiply \(5.2\) by \(6.3\) and give your answer to the correct number of significant figures.

Both numbers have \(2\) significant figures. We multiply as is:
\[5.2\times6.3=33\]

The answer is \(33\), to \(2\) significant figures.

Exercise – using significant figures in multiplication and division

1. Calculate the following and give your answers to the correct number of significant figures.
1. \(8.3\times0.25\)
2. \(0.2\times1.3\)
3. \(1.13\times3.5\times0.964\)
4. \(6.71\times3.4\)
5. \(3.000\times91\div72.60\)
6. \(450\times3\div0.671\)