Significant Figures


Why Significant Figures



Significant figures preserve the precision of measurements.  You record data and use numerical values in calculations. Further calculations can only be as accurate as the as the measurements from an experiment. Use of a specific set of rules prevents calculation results more precise than an original measurement.


Chemistry Has Measurements

Not Numbers!



When you write down a measurement, you record the most precise measurement, and then estimate the last digit. The value you or someone else recorded possesses a precision limited by its number of significant digits. 



This topic suffers from students viewing the subject as totally abstract. Keep in mind the physical real purpose of significant figures whenever you consider their importance.



Which Numbers Significant?



When you count significant digits, not all digits are created equal. 



Count all numbers between 1 and 9 as significant figures.



Zero has the cases where it is significant or not significant.



The Cases of Zero



zeros as significant figures
Figure 1: Examples of zero and significant figures


Figure 1 shows a break down of when a zero gets treated as a significant figure and when it is not treated as a significant figure.


Zero Significant



Treat zero as significant when it appears in a value between two numbers between 1 and 9. Count zero as a significant figure when you find it trailing significant figures if the value contains a decimal. 



Zero Not Significant



Non-significant zeros occur in two cases. If a zero occurs in a value after a 1 through 9 digit, but the value does not contain any decimals, the zero is not significant.



Rounding Values



When values exceed the allowed number of significant figures, shorten the value to conform to the level of precision. A digit beyond the last digit with a four (4) or lower rounds down. Otherwise, round up. 



Bear in mind when you use values in calculations, the uncertainty in any measurement propagates through the calculations. This prevents a calculation from becoming more precise only because you did a calculation.



You must consider addition and subtraction as a different case than multiplication and division.



Addition and Subtraction



simple addition significant figures
Figure 2: addition rounds to the least number of decimals


For addition and subtraction, limit  the result to the last tenth’s place. When the value has decimals smaller than one, the answer is rounded to the value which has the least number of decimal places, Figure 2.



Since 1.3 cm contains only one place after the decimal place, the rounded answer can possess no more than one decimal place.








subtraction of significant figures
Figure 3: subtraction follows the same rounding rules as addition.


Subtraction follows the same rounding rule as addition, Figure 3.  One decimal place in 25.0 mL sets the place to round the calculation result to one decimal place.








No Decimal

significant figures twns place
Figure 4: If there is no decimal place, use the last 10’s place

When no decimal place occurs in a value, some students get confused. In this situation you actually follow the same rule as before. 



Find the last tens place which has a significant figure, and then round to the last tens place, Figure 4.



The value 420 m has a 2 digit in the tens place. The result of addition gives 455 m. However, the 1’s place must be rounded, since the least precise measurement corresponds to a tens place.




Multiplication and Division





simple significant figure multiplied
Figure 5: Multiplication rounds the least number of significant digits.

The result of multiplication or division rounds by the number of significant digits, Figure 5.


The volume of a rectangular box is found by multiplying height times length times width. 2.5 cm and 3.0 cm both have 2 significant figures0.125 cm has 3 significant figures.



When the result of multiplying the three measurements is calculated using a standard calculator, the answer is 0.9325 cm. However two of the measurements have only two significant figures. The final result rounds to 0.94 cm3






significant digit multiplication with conversion factor
Figure 7: Division follows the same rounding rules as multiplication.


Division follows the same rule as multiplication, Figure 6.


When 733 mm Hg is multiplied by the conversion factor used to change mm Hg to atm, it plays out as a division when calculated. A calculator would give a resut of 0.9645 atm. 733 mm Hg has 3 significant digits


Though 760 mm Hg has 2 digits, it is not considered for the purpose of accounting for precsion of measurements. Exact conversion factors do not count as actual measurements. The calculation is rounded to 0.965 atm.



Scientific Notation

significant figures with scientific notation
Figure 7: Multiplication to a power rounded with scientific notation.



In a more involved example where a measurement is raised to a power, treat it as a type of multiplication, Figure 7.



Only one actual measurement contributes to a calculation of volume from the measured value of 6.3 cm for the radius. The ratio of (4/3) is an exact value and π has any desired number of significant figures. 



The measurement possesses 2 significant digits. The calculation results in a value of 1021.21 cm3. Rounding to the 2 significant digits would resultin an answer of 1000 cm3. However that loses part of the precision. Converting to scientific notation, (1.0 x 103 cm3), preserves the precision at 2 significant digits.



significant figures division with scientific notation
Figure 9: the only actual measurement is the atomic mass

A final example, Figure 9, shows a calculation that requires the calculation of the mass of a single phosphorus atom. Both (1 mole P/6.022 x 2023 atoms P) and 1 atom of P do not get considered for the purposes of precision. One is a conversion factor and the other a counted number.


The only experimental measurement is the atomic mass of phosphorus. The final answer rounds to 4 significant digits.