4 Rounding to $n$ significant figures

This process is similar to rounding to decimal places but there are some subtle differences.

To round a number to $n$ significant figures we look at the ${\left(n+1\right)}^{th}$ digit in the decimal expansion of the number.

• If the ${\left(n+1\right)}^{th}$ digit is 0, 1, 2, 3 or 4 then we round down : that is, we simply chop to $n$ places, inserting zeros if necessary before the decimal point. (In other words we neglect the ${\left(n+1\right)}^{th}$ digit and any digits to its right.)
• If the ${\left(n+1\right)}^{th}$ digit is 5, 6, 7, 8 or 9 then we round up : we add $1$ to the $n^{th}$ decimal place and then chop to $n$ places, inserting zeros if necessary before the decimal point.

Examples are given on the next page.

$\begin{array}{ccccc}\hfill \frac{1}{3}& \hfill =\hfill & 0.3333\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \frac{8}{3}& \hfill =\hfill & 2.66667\hfill & \hfill & \text{rounded to 6 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \pi & \hfill =\hfill & 3.142\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 2136& \hfill =\hfill & 2000\hfill & \hfill & \text{rounded to 1 significant figure}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 36.78& \hfill =\hfill & 37\hfill & \hfill & \text{rounded to 2 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 6.2399& \hfill =\hfill & 6.240\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \end{array}$

Sometimes the phrase “significant figures" is abbreviated as “s.f." or “sig.fig."

Example 3

Write down each of these numbers, rounding them to 4 significant figures:

$0.12345$ , $-0.44444$ , $0.5555555$ , $0.000127351$ , $25679$ , $123.456789,$ $3456543$

Solution

$0.1235$ , $-0.4444$ , $0.5556$ , $0.0001274$ , $25680$ , $123.5$ , $3457000$

Task!