  ### 4.1 Decimal places

The decimal part of the number can be thought of as a series of fractions, with the denominator being ten for the first decimal place to the right of the separator, 100 for the second decimal place, 1000 for the third and so on. Take the decimal number 67.329:

tens units separator tenths hundreths thousanths
Decimal place:     . First Second Third
Number 6 7 . 3 2 9
Value 60 7 .
 3 10
 6 100
 9 1000

Any fractional value smaller than one tenth will have the number zero in the first decimal place, and any fractional value smaller than one hundreth will have zeros in the first and second decimal places. Thus:

 1 = 0.04 25

and:

 1 = 0.004 250

Where the decimal result of a calculation does not yield an exact value, the result will normally be expressed as a certain number of decimal places, with the last number being rounded up if the number following it would have been between 5 and 9. So:

 Three Decimal places Two decimal places 6.921 6.92 6.924 6.92 6.925 6.93 6.928 6.93

Obviously, expressing a value to a specified number of decimal places implies something about the precision of the measurement (see the separate helpsheet on precision if you don't understand this). Say we measure the height of seven seedlings using a 30 cm ruler graduated in centimetres and millimetres. If the seven measurements are 4.7, 6.3, 5.2, 7.1, 6.9, 3.5 and 5.9 cm, we can add these together and divide by 7 to calculate the mean (average) value. A typical result for this on a calculator is 5.6571429 cm. The final '9' represents a length of just 9 nanometres, or nine millionths of a millimetre (you could fit just 300 helium atoms side-by-side into this distance). Clearly, writing the result of the mean to many more decimal places than the original measurements (which were to the nearest millimetre) is absurd, and the average should have been expressed to one decimal place as 5.7 cm (see also helpsheets on measurements and descriptive statistics).

You may be asked to express the result of a calculation to a certain number of decimal places, or you may need to decide for yourself what is appropriate. It is often sensible to state this if you are providing a written answer (e.g. 'result is 3.942 to three decimal places'). The Format>Cells menu in a spreadsheet will allow you to set the number of decimal places that are displayed (the spreadsheet will round the result for you).