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The significant figures include the digits of which we are certain and one number, the smallest, which is uncertain.
When quantitative data is obtained or manipulated, there are times when the numbers involved have several decimal places. This is where significant figures are useful, because they are directly related to the precision of the instrument used to determine the value.
The number of digits to be written when a measurement is taken depends on the precision of the device used.
The more precise the device, the greater the number of digits obtained by the measurement, and the greater the number of significant digits. For example, a balance scale can measure to one hundredth of a gram, whereas an electronic scale can measure to one thousandth of a gram. This means that the mass measured by an electronic scale will have a greater number of significant digits than the mass measured by the balance scale.
If a beam balance is used to weigh a pencil, it will be accurate to one hundredth. The mass might be , not or , as these degrees of accuracy do not correspond to the accuracy that can actually be achieved using the beam balance.
If a balance scale is used to weigh a pencil, the accuracy will be to the hundredth. The mass could be |3.40 \:\text{g},| but not |3.4 \:\text{g}| or |3.400 \:\text{g},| since these degrees of precision do not correspond to the accuracy that can be achieved in reality when using a balance scale.
When a value is determined by counting, such as the number of cars on a street in an hour, the 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 that is sometimes found in a formula.
The following rules are useful for identifying the number of significant figures.
|
Rule |
More Precisely |
Examples |
| All numbers other than zero are significant. | To determine the number of significant digits, simply count the number of digits in the number. | The number |9.56| has 3 significant figures. The number |456.568 7| has 7 significant figures. |
| All the zeros between non-zero digits are significant. | The number |4.507| has 4 significant figures. The number |40.56| has 4 significant figures. |
|
| Zeros found at the beginning of a number are not significant. | To determine the number of significant digits, find the first digit that isn't 0 and count the number of digits to the right of this 0. | The number |0.005 6| has 2 significant figures. The number |9.56| has 3 significant figures. |
| The zeros found at the end of a number are significant. | To determine the number of significant digits, 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 digits, 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 data, or values, the result must always be expressed using the same precision as the least precise value, or the value with the fewest decimal places.
The answer of the sum or difference between two values should not be more precise than the least precise value. So, when performing this type of operation, we need to determine which value is the least precise, that is, the one with the fewest decimal places.
What is the total distance of a wall if it is composed of two sections that measure |3.75 \:\text{km}| and |6.1 \:\text{km}|?
First off, 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 precise data, which is |6.1.| This value is expressed to the nearest tenth. In this case, the value must be rounded to the same precision.
||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 of all, we need to find 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 data, which is |100.67.|
This data is rounded to the one hundredth position. The answer must therefore also be expressed to the one hundredth.
||100.\color{red}{67} \: \text{N} - 3,.68 \: \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 digits as the lowest number.
The product or quotient between two values cannot contain more significant digits than the value that contains the least significant figures. Therefore, when performing this type of operation, we must determine which value has the fewest significant figures.
What mass of alcohol is present in |0.225 \:\text{L}| of blood of a person with |0.2 \:\text{g}| of alcohol per litre of blood?
First off, we need to find 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 has two significant figures. The data value with the least number of significant figures is the concentration, which contains only one. The final answer must therefore contain the same amount of significant figures as 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 that travels |12\,776 \:\text{m}| in |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 figures as the data with the least. In this situation, the data with the fewest significant figures is |3.1,| since it only has two. Therefore we must round the speed up so that the answer also has two significant figures.
||12.776 \: \text {m} \div 3.1 \: \text {s} = 4.1 \: \text {m/s}||