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The significant figures include the digit for which we are certain and one digit, the smallest, which is uncertain.
When quantitative data is obtained or manipulated, it may be numbers with several decimal places. This is when significant figures are useful, as they are directly related to the accuracy of the instrument used.
The number of figures to be written when a measurement is taken depends on the accuracy of the instrument used.
The more accurate the device, the greater the number of digits obtained by the measurement, and the greater the number of significant digits. For example, a beam scale measures with an accuracy of one hundredth of a gram, whereas an electronic scale can measure with an accuracy of one thousandth of a gram. This means that the mass measured by an electronic scale will have a greater number of significant figures than the mass measured by a beam scale.
If a beam scale is used to weigh a pencil, it will be accurate to one hundredth of a gram. The mass could be |\small 3.40 \:\text{g}|, and not |\small 3.4 \:\text{g}| or |\small 3.400 \:\text{g}|, because these levels of accuracy do not correspond to the accuracy that can actually be achieved using the beam scale.
When a value is determined by counting, such as by counting the number of cars passing in a street during an hour, this value has an infinite number of significant figures. The same applies to numbers obtained by definition, such as the number of moles, or the value 1 sometimes present in a formula.
The following rules are useful for identifying the number of significant figures.
|
Rule |
Precision |
Examples |
| All numbers other than zero 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. The number |456.5687| has 7 significant figures. |
| All zeros between non-zero digits are significant. | The number |4.507| has 4 significant figures. The number |40.56| has 4 significant figures. |
|
| Zeros at the beginning of a number are not significant. | To determine the number of significant figures, find the first digit other than 0 and count the number of digits to the right of this 0. | The number |0.0056| has 2 significant figures. The number |9.56| has 3 significant figures. |
| The zeros at the end of a number are significant. | To determine the number of significant figures, count the number of digits in the number. | The number |23.700| has 5 significant figures. The number |0.560| has 3 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. The number |2.5\times10^{-2}| has 2 significant figures. |
When adding or subtracting values, the result must always be expressed with the same accuracy as the least accurate value, i.e. the value with the fewest decimal places.
The answer obtained by the sum or difference between two values cannot be more accurate than the least accurate of the values. So, when performing this type of operation, we need to determine which value is the least accurate, i.e. the one with the fewest decimal places.
What is the total distance of a wall if it is made up of two sections measuring |\small 3.75 \:\text{km}| and |\small 6.1 \:\text{km}| respectively?
First of all, we need to determine the sum of the two sections.
||3.75 \: \text{km} + 6.1 \: \text{km}= 9.85 \: \text{km}||
This sum must be expressed to the same number of decimal places as the least accurate value, i.e. |\small 6.1|. This number is accurate to the nearest tenth. In this case, the value must be rounded to 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 |\small 100.67 \:\text{N}| is applied to the right and a force of |\small 3.768 \:\text{N}| is applied to the left?
First of all, we need to determine the difference between the two forces.
||100.67 \: \text{N} - 3.768 \: \text{N}= 96.902 \: \text{N}||
This difference must be expressed to the same number of decimal places as the least accurate value, i.e. |\small 100.67|. This number is accurate to one hundredth. The answer must therefore also be accurate to one 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 have the same number of significant figures of the value with the smallest number of significant figures.
The answer obtained by the product or quotient between two values cannot contain more significant figures than the value which contains the fewest at the start. Therefore, when performing this type of operation, it is necessary to determine which value contains the lowest number of significant digits.
What mass of alcohol is present in the |\small 0.225 \:\text{L}| of blood of a person with |\small 0.2 \:\text{g}| of alcohol per litre of blood?
First of all, we need to determine the product of these two values.
||0.225 \: \text{L} \times \displaystyle \frac{ 0.2 \: \text {g}}{\text {L}} = 0.045 \: \text {g}||
The result gives two significant figures. The value with the fewest significant figures is the concentration, which contains just one. So, the final answer must contain as many significant figures as the number present in the concentration value.
||0.225 \: \text {L} \times \displaystyle \frac{ 0.2 \: \text {g}}{\text {L}} = 0.05 \: \text {g}||
What is the speed of an animal travelling a distance of |\small 12.776 \:\text{m}| in |\small 3.1| seconds?
||12.776 \: \text{m} \div 3.1 \: \text{s} = 4.121290322... \: \text {m/s}||
This quotient must be expressed with the same number of significant digits as the value which has the fewest. In this situation, the entry with the fewest significant figures is |\small 3.1|, as it only has two. We therefore need to round off the speed so that the answer also has two significant figures.
||12.776 \: \text {m} \div 3.1 \: \text {s} = 4.1 \: \text {m/s}||