3 Rounding to $n$ decimal places

In general, a calculator or computer is unable to store every decimal place of a real number. Real numbers are rounded . To round a number to $n$ decimal places 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. (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.

For example

$\begin{array}{ccccc}\hfill \frac{1}{3}& \hfill =\hfill & 0.3333\hfill & \hfill & \text{rounded to 4 decimal places}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \frac{8}{3}& \hfill =\hfill & 2.66667\hfill & \hfill & \text{rounded to 5 decimal places}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \pi & \hfill =\hfill & 3.142\hfill & \hfill & \text{rounded to 3 decimal places}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 2.3403& \hfill =\hfill & 2.340\hfill & \hfill & \text{rounded to 3 decimal places}\hfill \end{array}$

Sometimes the phrase ‘decimal places’ is abbreviated to ‘d.p.’ or ‘dec.pl.’.

Example 2

Write down each of these numbers rounded to 4 decimal places:

$0.12345$ , $-0.44444$ , $0.5555555$ , $0.000127351$ , $0.000005$ , $123.456789$

Solution

$0.1235$ , $-0.4444$ , $0.5556$ , $0.0001$ , $0.0000$ , $123.4568$

Task!

Write down each of these numbers, rounded to 3 decimal places:

$0.87264$ , $0.1543$ , $0.889412$ , $-0.5555$ , $45.6789,$ $6.0003$

Answer