Significant figures (also called significant digits) are very important because they imply the known precision of a number.
How to Count Significant Figures
- Non-zero digits are always significant.
- Any zeros between two significant figures are significant.
- Trailing zeros in the decimal portion only are significant.
Examples:
- \(4308\) has four significant figures.
- \(40.05\) has four significant figures.
- \(470,000\) has two significant figures.
- \(4.00\) has three significant figures.
- \(0.0004\) has one significant figure.
Counting, and using, significant figures correctly becomes very easy if one uses scientific notation. Then, the numbers before the “\(\times 10^{\rm{n}}\)” are all significant. This also eliminates the vagueness of whether, for example, \(400\) has one, two, or three significant figures.
Examples with Scientific Notation:
- \(4\times 10^{2}\) has one significant figure.
- \(4.0\times 10^{2}\) has two significant figures.
- \(4.00\times 10^{2}\) has three significant figures, etc..
The uncertainty \(\delta x\) in a measurement \(x\) will be presented with the measured value as \(x\pm\delta x\). Three steps for getting significant figures right in calculations involving uncertainty:
- Calculate your answer, \(x\), ignoring uncertainty, keeping a couple more significant figures than you think you will ultimately need.
- Calculate the uncertainty, \(\delta x\), of the answer using the appropriate rule(s) from the section to follow on propagating errors in calculations. Round it to one significant figure. One exception: Keep two significant figures if the calculated uncertainty value would start with a one if written in scientific notation, as in \(0.014\), \(1.7\), \(11\), etc..
- Use the decimal position of the only/last digit in the uncertainty (whose number of sig. figs. is now only one, or maybe two) to determine how many decimal places to keep in your answer.
The number of decimal places (not significant figures) of the answer and its uncertainty must be the same.
Examples with Uncertainty:
If \(\delta x\) turns out to be \(0.007\) (from step two), and your answer \(x\) was initially \(1.46129\) (from step one), then make your final statement of the result $$ x = 1.461\pm 0.007.$$ If you use scientific notation – definitely recommended for very large or very small numbers – it is easier to present the answer and uncertainty to the same magnitude, as in $$ x = (2.52\pm 0.04)\times 10^{14}$$ instead of $$ x = 2.52\times 10^{14}\pm 4\times 10^{12}.$$
