Significant figures include digits that are known with certainty and one digit, the smallest, that is uncertain.
When quantitative data is obtained or manipulated, the resulting numbers may have several decimal places. When this happens, significant figures come into play, because they are directly related to the accuracy of the device used to measure the value.
The number of digits in a measurement depends on the accuracy of the device used.
The more accurate the device, the greater the number of digits included in the measurement and the greater the number of significant figures. For example, a triple beam balance has a measurement accuracy of a hundredth of a gram, while an electronic balance has a measurement accuracy of a thousandth of a gram. The mass measured by the electronic balance will have a greater number of significant figures than the mass measured by the triple beam balance.
When using a triple beam balance to weigh a pencil, the accuracy will be to the hundredth. The mass could be |3.40\ \text{g}| and not |3.4\ \text{g}| or |3\ 400\ \text{g},| because these degrees of accuracy do not correspond to what can actually be obtained using the triple beam balance.
When a value is determined by counting, such as the number of cars travelling on a street in one hour, this value has an infinite number of significant figures. The same is true for defined numbers, such as the number of moles or the value |1| that may be in a formula.
The following rules help identify the number of significant figures.
|
Rule |
Accuracy |
Examples |
|
All non-zero digits are significant. |
To determine the number of significant figures, simply count the number of digits in the number. |
The number |9.56| has 3 significant figures. |
|
All zeros between non-zero digits are significant. |
The number |4\ 507| has 4 significant figures. |
|
|
Zeros at the beginning of a number are not significant. |
To determine the number of significant figures, locate the first digit other than 0 and count the number of digits to the right of that 0. |
The number |0.0056| has 2 significant figures. |
|
All zeros at the end of a decimal number are significant. |
To determine the number of significant figures, count the digits in the number (excluding the zeros at the beginning of the number). |
The number |0.50600| has 5 significant figures. |
|
Depending on the context, the zeros at the end of an integer may or may not be significant. |
Determining the number of significant figures requires context, such as the precision of the instrument. |
If the value |23\ 700| is measured with a device accurate to the nearest unit, then |23\ 700| has 5 significant figures. |
|
In scientific notation, the digits before the power of |10| are significant. |
To determine the number of significant figures, count the number of digits to the left of the power of 10. |
The number |9.568\times10^{3}| has 4 significant figures. |
When adding or subtracting data, the result must always be expressed with the same accuracy as the least accurate value, meaning the one with the fewest digits after the decimal point.
The sum or difference of two values cannot be more accurate than the least accurate of the values. When performing this type of operation, the least accurate value must be determined, meaning the one with the fewest decimal places.
What is the total length of a wall if it is composed of two sections measuring |3.75 \ \text{km}| and |6.1\ \text{km}| respectively?
First, the sum of the two sections must be calculated.
||3.75\ \text{km} + 6.1\ \text{km}= 9.85\ \text{km}||
The answer must be expressed with the same number of decimal places as the least accurate given value |(6.1).| This value is accurate to the tenth place. Therefore, the answer must be rounded to obtain the same accuracy.
||3.75 \: \text{km}+ 6.\color{red}{1} \: \text{km}= 9.85 \rightarrow 9.\color{red}{9} \: \text{km}||
What force is applied to an object if a force of |100.67\ \text{N}| is applied to the right and a force of |3.768\ \text{N}| is applied to the left?
First, the difference between the two forces must be calculated.
||100.67\ \text{N} – 3.768\ \text{N}= 96.902\ \text{N}||
This difference must be expressed with the same number of decimal places as the least accurate given value |(100.67).| This value is accurate to the hundredth place. Therefore, the answer must also be accurate to the nearest hundredth.
||100.\color{red}{67}\ \text{N} – 3.768\ \text{N}= 96.902 \rightarrow 96.\color{red}{90}\ \text{N}||
When multiplying or dividing numbers, the result must always be expressed with the same number of significant figures as the least accurate value.
The product or quotient of two values cannot contain more significant figures than the value containing the fewest significant figures initially. When performing this type of operation, it is necessary to determine which value contains the fewest significant figures.
How much alcohol is present in |0.225\ \text{L}| of blood of a person with |0.2\ \text{g}| of alcohol per litre of blood?
The first step is to determine the product of the two values.
||0.225\ \cancel{\text {L}} \times\dfrac{0.2\ \text {g}}{\cancel{\text {L}}} = 0.045\ \text {g}||
The result has two significant figures. Of the given values, it is the concentration that is the least accurate, with only one significant figure. Therefore, the final answer must contain the same number of significant figures.
||0.225\ \cancel{\text {L}} \times\dfrac{0.2\ \text {g}}{\cancel{\text {L}}} = 0.045\ \text {g} \rightarrow 0.05\ \text {g}||
What is the speed of an animal travelling a distance of |12.776\ \text{m}| in |3.1\ \text{s}|?
The first step is to determine the quotient of the two values.
||12.776\ \text {m} \div 3.1\ \text {s} = 4.121290322...\ \text {m/s}||
The result must be expressed with the same number of significant figures as the value that has the fewest. In this situation, the time interval is the least accurate, with only two significant figures |3.1.| Therefore, the speed must be rounded to two significant figures.
||12.776\ \text {m} \div 3.1\ \text {s} = 4.121290322...\ \text {m/s} \rightarrow 4.1\ \text {m/s}||