Calculating with numbers: Decimal numbers
Calculation rules for significance in multiplication and division
When calculating with decimal numbers, you must always write the result with the correct number of significant digits and, if necessary, round \to this number of meaningful digits. The example below illustrates this.
A rectangular room has the following dimensions: \(2.5 \times 3.5 \mathrm{\;m}.\) What is the area?
It holds: \( 2.5 \times 4.5= 11.25\), but this number has more significant digits than any of the factors. The accuracy cannot increase with a calculation, so we round to 2 significant digits, in this case 11.
The requested area is therefore equal to \(11. \mathrm{\;m}^2.\)
You can also interpret the above as follows:
Because the measured size of the chamber is specified with numbers with a precision of 1 decimal, the size of the chamber is a minimum of \(2.4 \times 4.4 \mathrm{\;m}\) and a maximum \(2.6 \times 4.6 \mathrm{\;m}.\) The area is somewhere between \(2.4 \times 4.4 = 10.56\) and \(2.6 \times 4.6 = 11.96\). Looking at these results, it seems reasonable to round to two significant figures.
In calculations, however, you should not round off intermediate results, but continue to calculate with extra significant digits (usually 1 or 2) and only then round off the obtained final result using the general rule below:
The rule following applies to multiplication and division:
The result of a calculation has as many digits as the data with the fewest number of significant digits.
