2.3: Significant Figures  Writing Numbers to Reflect Precision
 Page ID
 86012
The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the ruler illustration below, the bottom ruler gave a length with 2 significant figures, while the top ruler gave a length with 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. With either ruler, it would not be possible to report the length at \(2.553 \: \text{cm}\) as there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported.
Measurement Uncertainty
Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest \(0.1 \: \text{g}\), other highly sensitive balances are capable of measuring to the nearest \(0.001 \: \text{g}\) or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. Figure \(\PageIndex{1}\) shows two rulers making the same measurement of an object (indicated by the blue arrow).
With either ruler, it is clear that the length of the object is between \(2\) and \(3 \: \text{cm}\). The bottom ruler contains no millimeter markings. With that ruler, the tenths digit can be estimated and the length may be reported as \(2.5 \: \text{cm}\). However, another person may judge that the measurement is \(2.4 \: \text{cm}\) or perhaps \(2.6 \: \text{cm}\). While the 2 is known for certain, the value of the tenths digit is uncertain.
The top ruler contains marks for tenths of a centimeter (millimeters). Now the same object may be measured as \(2.55 \: \text{cm}\). The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be \(2.54 \: \text{cm}\) or \(2.56 \: \text{cm}\). In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.
When you look at a reported measurement, it is necessary to be able to count the number of significant figures. The table below details the rules for determining the number of significant figures in a reported measurement. For the examples in the table, assume that the quantities are correctly reported values of a measured quantity.
Rule  Examples 

1. All nonzero digits in a measurement are significant. 

2. Zeros that appear between other nonzero digits (middle zeros) are always significant. 

3. Zeros that appear in front of all of the nonzero digits are called leading zeros. Leading zeros are never significant. 

4. Zeros that appear after all nonzero digits are called trailing zeros. A number with trailing zeros that lacks a decimal point may or may not be significant. Use scientific notation to indicate the appropriate number of significant figures. 

5. Trailing zeros in a number with a decimal point are significant. This is true whether the zeros occur before or after the decimal point. 

Exact Numbers
Integers obtained either by counting objects or from definitions are exact numbers, which are considered to have infinitely many significant figures. If we have counted four objects, for example, then the number 4 has an infinite number of significant figures (i.e., it represents 4.000…). Similarly, 1 foot (ft) is defined to contain 12 inches (in), so the number 12 in the following equation has infinitely many significant figures:
Accuracy and Precision
Measurements may be accurate, meaning that the measured value is the same as the true value; they may be precise, meaning that multiple measurements give nearly identical values (i.e., reproducible results); they may be both accurate and precise; or they may be neither accurate nor precise. The goal of scientists is to obtain measured values that are both accurate and precise. The video below demonstrate the concepts accuracy and precision.
Video 2.3.1: Difference between precision and accuracy.
Summary
Uncertainty exists in all measurements. The degree of uncertainty is affected in part by the quality of the measuring tool. Significant figures give an indication of the certainty of a measurement. Rules allow decisions to be made about how many digits to use in any given situation.
Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
CK12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.
Henry Agnew (UC Davis)
 Sridhar Budhi