Significant figures

In our everyday world, we measure things with precision. This is where significant figures come in. They help us communicate how precise our measurements are. Use this resource to learn how to identify and apply significant figures.

A significant figure (or "sig fig") is a digit in a measurement that gives us an idea of how precise the measurement is. It tells us which digits are reliable and meaningful and give us an indication of how accurate the instrument or method used is.

All the digits in a measured value are called the significant figures. The last digit is often an estimate.

If a number has more significant figures, this suggests that a quantity has been measured more precisely, If it has fewer significant figures, this indicates less certainty in the measurement.

Rules for significant figures

There are \(5\) rules to help you identify the number of significant values in a measured value.

1. Any non-zero digit is significant. The position of a decimal point makes no difference.

   For example, \(15.7\) and \(1.57\) both have \(3\) significant figures.

2. Zeroes between numbers are significant.

   For example, \(1.05\) has \(3\) significant figures, and \(200.708\) has \(6\).

3. Zeroes at the right-hand end of whole numbers are not significant, unless otherwise stated.

   For example, \(2860\) has \(3\) significant figures, while \(70\) has only \(1\).

4. Zeroes at the left-hand end of decimal numbers are not signicant.

   For example, \(0.28\) has \(2\) significant figures, and so does \(0.0039\).

5. Zeroes at the right-hand end of decimal numbers are significant.

   For example, \(12.0\) has \(3\) significant figures and \(0.48300\) has \(5\).

Example 1 – identifying the number of significant figures

Find the number of significant figures in \(0.000\,3\).

According to rule \(4\), zeroes at the left-hand end of decimal numbers are not signicant, so \(0.000\,3\) has only \(1\) significant figure.
\[0.000\,\underline{3}\]

Exercise – identifying the number of significant figures

1. State the number of significant figures in each of the following values.
1. \(345\)
2. \(17\,642\)
3. \(0.003\,3\)
4. \(0.000\,306\)
5. \(870\)
6. \(20\,000\)
7. \(140.600\)
8. \(710.0\)
9. \(0.040\,80\)
10. \(0.005\,0\)

Significant figures and scientific notation

Using scientific notation is a helpful way to manage significant figures, especially with very large or small numbers. It clearly expresses the precision of a measurement by keeping only the significant digits. Scientific notation is also handy when you need to round a value to a specific number of significant figures.

Consider the number \(34\,218\,043\), which has \(8\) significant figures. We want to round it to \(3\) significant figures.

One way we could do this is by writing it as:
\[34\,200\,000\]

but this can lead to ambiguity about the precision of the number; someone else reading the number may not know if the zeroes that come after the \(2\) are significant.

Instead, we can write it in scientific notation:
\[3.42\times10^{7}\]

which clearly shows \(3\) significant figures.

Significant figures for measurements

When recording measurements, the number of significant figures depends on the scale of the instrument.

Consider this ruler.

Let's say a measurement is taken between \(6.4\) and \(6.5\) centimetres.

• We can say with certainty that the measurement is greater than \(6\) and less than \(7\) centimetres.
• We can say with certainty that the measurement is greater than \(6.4\) and less than \(6.5\) centimetres.
• We CANNOT say with certainty that the measurement is greater than \(6.43\) and less than \(6.45\) centimetres.

The result would be recorded as \(6.44\) centimetres, to \(3\) significant figures. The first two digits are certain and the last is a good estimate.

Writing \(6.44\) implies \(6.44\pm0.005\), i.e. between \(6.435\) and \(6.445\), unless otherwise stated.

It is important to report your measurements to the appropriate number of significant figures. This will help you to avoid:

• overstating or understating the precision of the result
• making calculating errors
• forming misleading conclusions about your results.

Reporting your values to the correct number of significant figures helps to maintain the integrity of the data you generate.