In the previous section, we saw the basics about significant figures. In this section, we will see the rules for finding the number of 'significant figures' in a given measurement.
• Before learning the rules, we must first find the answer to an important question:
Do 'changing of units' have any effect on significant figures?
• We will try to find the answer in steps:
1. Consider the measurement 2.308 cm
• Based on what we saw in the previous section, we get the following information:
(i) The last digit is uncertain
So 0.008 cm is the uncertain quantity
(ii) 2.30 cm is the reliable quantity
• The yellow arrow was stuck just beyond the 2.30 cm mark
(iii) Consider the two points:
♦ The 2.30 cm mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement. It is 0.008 cm
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
2. Now, let 2.308 cm be written as 0.02308 m
• Do we get the same 3 information that we wrote in (1)?
Let us try:
Consider the measurement 0.02308 m
(i) The last digit is uncertain
• So 0.00008 m is the uncertain quantity
• But 0.00008 m = 0.008 cm
• That means, 0.008 cm is the uncertain quantity
• This is the same result that we obtained in 1(i) above
(ii) 0.0230 m is a reliable quantity
♦ The yellow arrow was stuck beyond the 0.0230 m mark
• But 0.0230 m = 2.30 cm
• That means 2.30 cm is the reliable quantity
♦ The yellow arrow was stuck beyond the 2.30 cm mark
• This is the same result that we obtained in 1(ii) above
(iii) Consider the two points:
♦ The 0.0230 m mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement
♦ This length is 0.00008 m
• But 0.00008 m = 0.008 cm
♦ So 0.008 cm is obtained by judgement
• This is the same result that we obtained in 1(iii) above
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
3. Let 2.308 cm be written as 23.08 mm
• Do we get the same 3 information that we wrote in (1)?
Let us try:
Consider the measurement 23.08 mm
(i) The last digit is uncertain
• So 0.08 mm is an uncertain quantity
• But 0.08 mm = 0.008 cm
• That means, 0.008 cm is an uncertain quantity
• This is the same result that we obtained in 1(i) above
(ii) 23.0 mm is a reliable quantity
♦ The yellow arrow was stuck beyond the 23.0 mm mark
• But 23.0 mm = 2.30 cm
• That means 2.30 cm is a reliable quantity
♦ The yellow arrow was stuck beyond the 2.30 cm mark
• This is the same result that we obtained in 1(ii) above
(iii) Consider the two points:
♦ The 23.0 mm mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement
♦ This length is 0.08 mm
• But 0.08 mm = 0.008 cm
♦ So 0.008 cm is obtained by judgement
• This is the same result that we obtained in 1(iii) above
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
■ Let us write an inference based on (1), (2) and (3):
The units of 2.308 cm can be changed and written as 0.02308 m or 23.08 mm. But the reliable quantity and uncertain quantity will not change. So the number of significant figures do not change
Rule 1: All non-zero digits are significant
Explanation:
• In a measurement, we may see any of the 9 digits: 1, 2, 3, 4, 5, 6, 7, 8, and 9
• We must count them as significant
• The rest of the rules are all related to zeros
Rule 2: All captive zeros are significant
Explanation:
• Captive zeros are those zeroes which are in between two non-zero digits
• Such zeros are significant. We need not worry whether there is a decimal point or not
Examples:
♦ The under lined zero in 2037 is significant
♦ The under lined zeros in 5.70021 are significant
Rule 3: IF the measurement is less than 1 AND there is one or more zeros just to the right side of the decimal point, those zeros are NOT significant
Explanation:
• If the measurement is less than 1, there will not be any non-zero digits on the left side of the decimal point. There will be only zeros
• On the right side of the decimal point, there can be both zeros and non-zero digits
• Consider two items:
(i) the decimal point
(ii) the first non-zero digit coming after the decimal point
• If there are any zeros between the two items, none of those zeros are significant
Examples:
♦ The under lined zero in 0.035 is not significant
♦ The under lined zero in 0.0402 is not significant
♦ The under lined zeros in 0.00295 are not significant
Rule 4:
• This rule is related to trailing zeros. There can be two cases:
Case 1: The measurement has a decimal point
Case 2: The measurement does not have a decimal point
We will consider each case separately
Case 1:
• In the previous section, we saw this:
If length is exactly 2.3 cm (on a ruler which can measure both cm and mm), we must write it as 2.30 cm
• Another example:
A tape measure is graduated in m and cm. The yellow arrow is exactly at 4 m and 20 cm. Then the reliable measurement is 4.20 m. We must add one more zero to denote the uncertain part. Thus the measurement becomes: 4.200 m
■ Based on the two examples, we can write: All trailing zeros are significant
Case 2:
• In this case, the trailing zeros are not significant
Example:
♦ The underlined zeros in 87300 are not significant
Ambiguity in Rule 4 (case 2):
We will write about the ambiguity in steps:
1. Consider the measurement 4.700 m
• Based on Rule 4 (case 1), we know that the zeros after 7 are significant
• So the number of significant figures in 4.700 m is 4
2. Let us write this measurement in different units
• We have: 4.700 m = 470.0 cm = 4700 mm = 0.004700 km
3. Consider the third measurement: 4700 mm
♦ Based on Rule 4 (case 2), the number of significant figures will be 2
♦ When it is written as 4.700 m, the number of significant figures is 4
■ But the number of significant figures cannot change by mere 'changing of units'
4. To remove such ambiguities in determining the 'number of significant figures', the best way is to record every measurement in scientific notation
• The details about this notation can be written in 3 steps as follows:
(i) In this method, every number is expressed as a × 10b
• Where a has to satisfy two conditions:
♦ Condition 1: a should be greater than or equal to 1
♦ Example: 1, 2.5, 3.48, 7, 8.0569, 9 etc.,
♦ Condition 2: a should be less than 10
♦ Example: 7, 8, 9.21, 9.999 etc.,
• The two conditions can be written together mathematically as: 1 ≤ a <10
(ii) b is the positive or negative power of 10
• If b is positive and large, the number itself will be very large
♦ Example: 3.79 × 108
• If b is negative and large (numerically), the number itself will be very small
♦ Example: 2.341 × 10-15
(iii) It is often customary to put the decimal point after the first digit
■ If we follow the above 3 steps, the confusion can be avoided
5. Consider our example again. We have:
4.700 × 10-3 km = 4.700 m = 4.700 × 102 cm = 4.700 × 103 mm
• We see that, in all measurements, the base a = 4.700
• For the base, Rule 4 (case 1) can be easily applied
6. Scientific notation is ideal for reporting measurement. But if the measurement given to us is not in that notation, we can directly apply Rule 4 (case 1 OR case 2, which ever is applicable)
Rule 5:
• If a number is less than 1, we usually put a zero in front of the decimal point
• Such a zero is not significant
♦ Example: The under lined zero in 0.325 is not significant
Rule 6:
• We must not try to find the significant figures in 'multiplying or dividing factors'
• They are exact
• Their precision cannot be increased or decreased
• We can assume that, they have an infinite number of significant figures
Example:
♦ Radius is obtained by dividing diameter by the factor 2
♦ Circumference of a circle is obtained by multiplying 𝞹r by a factor 2
Solved example 2.12
Find the number of significant figures in the following measurements:
(a) 0.0002801 cm (b) 13.5 g (c) 4100 mL (d) 7.400 × 103 m
Solution:
(a) 0.0002801 cm
(i) Rule 1 says that all non-zero digits are significant
• So 2, 8 and 1 are significant
(ii) The zero between 8 and 1 is a captive zero
• Rule 2 says that it is significant
(iii) There are 3 zeros just after the decimal point
• Rule 3 says that they are not significant
(iv) There is just a zero in front of the decimal point
• Rule 5 says that it is not significant
(v) So the total number of significant figures is: 4
(b) 13.5 g
• Here we apply Rule 1
• All digits are non-zero digits
• So the number of significant figures = 3
(c) 4100 mL
• This measurement should have been given in scientific notation
• But the person who recorded it, does not give importance for presenting it in that notation
• So we can apply Rule 4 (case 2)
• We get: Number of significant figures = 2
(d) 7.400 × 103 m
• This measurement is in scientific notation
• We need to consider the base only
• The base is 7.400
• So the number of significant figures = 4
1. We have seen the details about scientific notation
• If a number is given to us in that notation, we can get an approximate idea about the 'size' of that number by rounding off a
2. We know that a will be greater than or equal to 1
• If a is less than or equal to 5, it will become 1
3. We know that a will also be less than 10
• If a is greater than 5, it will become 10
4. When a is rounded off in this manner, b gets a new name: order of magnitude
5. Let us see some examples:
Example 1:
• The diameter of the earth is 1.28 × 107 m
• To get an approximate idea about the diameter, we round of 1.28
• Based on (2) above, when 1.28 is rounded off, we get 1
• After rounding off 1.28 to 1, we cannot say this:
The diameter of the earth is 107 m
• We must say this:
Diameter of the earth is of the order of 107 m with the order of magnitude 7
• 107 is very large: 1 followed by seven zeroes
Example 2:
• The actual diameter of hydrogen atom is 1.06 × 10-10 m
• To get an approximate idea about the diameter, we round off 1.06
• Based on (2) above, when 1.06 is rounded off, we get 1
• After rounding off 1.06 to 1, we cannot say this:
The diameter of the hydrogen atom is 10-10 m
• We must say this:
Diameter of the hydrogen atom is of the order of 10-10 m with the order of magnitude -10
• 10-10 is very small:
♦ There is only a zero in front of the decimal point
♦ Just after the decimal point, there will be 9 zeroes
♦ After those 9 zeroes, there will be just a '1'
6. What is the difference between the following two items:
(i) Order of magnitude of diameter of hydrogen atom
(ii) Order of magnitude of diameter of earth
• Obviously, the difference is: [7 - (-10)] = 17
• Thus we can write:
Diameter of earth is 17 orders of magnitude larger than the diameter of hydrogen atom
• Before learning the rules, we must first find the answer to an important question:
Do 'changing of units' have any effect on significant figures?
• We will try to find the answer in steps:
1. Consider the measurement 2.308 cm
• Based on what we saw in the previous section, we get the following information:
(i) The last digit is uncertain
So 0.008 cm is the uncertain quantity
(ii) 2.30 cm is the reliable quantity
• The yellow arrow was stuck just beyond the 2.30 cm mark
(iii) Consider the two points:
♦ The 2.30 cm mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement. It is 0.008 cm
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
2. Now, let 2.308 cm be written as 0.02308 m
• Do we get the same 3 information that we wrote in (1)?
Let us try:
Consider the measurement 0.02308 m
(i) The last digit is uncertain
• So 0.00008 m is the uncertain quantity
• But 0.00008 m = 0.008 cm
• That means, 0.008 cm is the uncertain quantity
• This is the same result that we obtained in 1(i) above
(ii) 0.0230 m is a reliable quantity
♦ The yellow arrow was stuck beyond the 0.0230 m mark
• But 0.0230 m = 2.30 cm
• That means 2.30 cm is the reliable quantity
♦ The yellow arrow was stuck beyond the 2.30 cm mark
• This is the same result that we obtained in 1(ii) above
(iii) Consider the two points:
♦ The 0.0230 m mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement
♦ This length is 0.00008 m
• But 0.00008 m = 0.008 cm
♦ So 0.008 cm is obtained by judgement
• This is the same result that we obtained in 1(iii) above
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
3. Let 2.308 cm be written as 23.08 mm
• Do we get the same 3 information that we wrote in (1)?
Let us try:
Consider the measurement 23.08 mm
(i) The last digit is uncertain
• So 0.08 mm is an uncertain quantity
• But 0.08 mm = 0.008 cm
• That means, 0.008 cm is an uncertain quantity
• This is the same result that we obtained in 1(i) above
(ii) 23.0 mm is a reliable quantity
♦ The yellow arrow was stuck beyond the 23.0 mm mark
• But 23.0 mm = 2.30 cm
• That means 2.30 cm is a reliable quantity
♦ The yellow arrow was stuck beyond the 2.30 cm mark
• This is the same result that we obtained in 1(ii) above
(iii) Consider the two points:
♦ The 23.0 mm mark
♦ The tip of the yellow arrow
• The length between those two points is found by judgement
♦ This length is 0.08 mm
• But 0.08 mm = 0.008 cm
♦ So 0.008 cm is obtained by judgement
• This is the same result that we obtained in 1(iii) above
(iv) So 8 is the last significant figure
• 2, 3 and 0 are the other significant figures
• In total, there are 4 significant figures
■ Let us write an inference based on (1), (2) and (3):
The units of 2.308 cm can be changed and written as 0.02308 m or 23.08 mm. But the reliable quantity and uncertain quantity will not change. So the number of significant figures do not change
Now we can write the rules for 'finding the number of significant figures'
Explanation:
• In a measurement, we may see any of the 9 digits: 1, 2, 3, 4, 5, 6, 7, 8, and 9
• We must count them as significant
• Once we understand Rule 1, there should be no doubts regarding non-zero digits
Rule 2: All captive zeros are significant
• Captive zeros are those zeroes which are in between two non-zero digits
• Such zeros are significant. We need not worry whether there is a decimal point or not
Examples:
♦ The under lined zero in 2037 is significant
♦ The under lined zeros in 5.70021 are significant
Rule 3: IF the measurement is less than 1 AND there is one or more zeros just to the right side of the decimal point, those zeros are NOT significant
Explanation:
• If the measurement is less than 1, there will not be any non-zero digits on the left side of the decimal point. There will be only zeros
• On the right side of the decimal point, there can be both zeros and non-zero digits
• Consider two items:
(i) the decimal point
(ii) the first non-zero digit coming after the decimal point
• If there are any zeros between the two items, none of those zeros are significant
Examples:
♦ The under lined zero in 0.035 is not significant
♦ The under lined zero in 0.0402 is not significant
♦ The under lined zeros in 0.00295 are not significant
Rule 4:
• This rule is related to trailing zeros. There can be two cases:
Case 1: The measurement has a decimal point
Case 2: The measurement does not have a decimal point
We will consider each case separately
Case 1:
• In the previous section, we saw this:
If length is exactly 2.3 cm (on a ruler which can measure both cm and mm), we must write it as 2.30 cm
• Another example:
A tape measure is graduated in m and cm. The yellow arrow is exactly at 4 m and 20 cm. Then the reliable measurement is 4.20 m. We must add one more zero to denote the uncertain part. Thus the measurement becomes: 4.200 m
■ Based on the two examples, we can write: All trailing zeros are significant
Case 2:
• In this case, the trailing zeros are not significant
Example:
♦ The underlined zeros in 87300 are not significant
Ambiguity in Rule 4 (case 2):
We will write about the ambiguity in steps:
1. Consider the measurement 4.700 m
• Based on Rule 4 (case 1), we know that the zeros after 7 are significant
• So the number of significant figures in 4.700 m is 4
2. Let us write this measurement in different units
• We have: 4.700 m = 470.0 cm = 4700 mm = 0.004700 km
3. Consider the third measurement: 4700 mm
♦ Based on Rule 4 (case 2), the number of significant figures will be 2
♦ When it is written as 4.700 m, the number of significant figures is 4
■ But the number of significant figures cannot change by mere 'changing of units'
4. To remove such ambiguities in determining the 'number of significant figures', the best way is to record every measurement in scientific notation
• The details about this notation can be written in 3 steps as follows:
(i) In this method, every number is expressed as a × 10b
• Where a has to satisfy two conditions:
♦ Condition 1: a should be greater than or equal to 1
♦ Example: 1, 2.5, 3.48, 7, 8.0569, 9 etc.,
♦ Condition 2: a should be less than 10
♦ Example: 7, 8, 9.21, 9.999 etc.,
• The two conditions can be written together mathematically as: 1 ≤ a <10
(ii) b is the positive or negative power of 10
• If b is positive and large, the number itself will be very large
♦ Example: 3.79 × 108
• If b is negative and large (numerically), the number itself will be very small
♦ Example: 2.341 × 10-15
(iii) It is often customary to put the decimal point after the first digit
■ If we follow the above 3 steps, the confusion can be avoided
5. Consider our example again. We have:
4.700 × 10-3 km = 4.700 m = 4.700 × 102 cm = 4.700 × 103 mm
• We see that, in all measurements, the base a = 4.700
• For the base, Rule 4 (case 1) can be easily applied
6. Scientific notation is ideal for reporting measurement. But if the measurement given to us is not in that notation, we can directly apply Rule 4 (case 1 OR case 2, which ever is applicable)
Rule 5:
• If a number is less than 1, we usually put a zero in front of the decimal point
• Such a zero is not significant
♦ Example: The under lined zero in 0.325 is not significant
Rule 6:
• We must not try to find the significant figures in 'multiplying or dividing factors'
• They are exact
• Their precision cannot be increased or decreased
• We can assume that, they have an infinite number of significant figures
Example:
♦ Radius is obtained by dividing diameter by the factor 2
♦ Circumference of a circle is obtained by multiplying 𝞹r by a factor 2
Now we will see some solved examples:
Find the number of significant figures in the following measurements:
(a) 0.0002801 cm (b) 13.5 g (c) 4100 mL (d) 7.400 × 103 m
Solution:
(a) 0.0002801 cm
(i) Rule 1 says that all non-zero digits are significant
• So 2, 8 and 1 are significant
(ii) The zero between 8 and 1 is a captive zero
• Rule 2 says that it is significant
(iii) There are 3 zeros just after the decimal point
• Rule 3 says that they are not significant
(iv) There is just a zero in front of the decimal point
• Rule 5 says that it is not significant
(v) So the total number of significant figures is: 4
(b) 13.5 g
• Here we apply Rule 1
• All digits are non-zero digits
• So the number of significant figures = 3
(c) 4100 mL
• This measurement should have been given in scientific notation
• But the person who recorded it, does not give importance for presenting it in that notation
• So we can apply Rule 4 (case 2)
• We get: Number of significant figures = 2
(d) 7.400 × 103 m
• This measurement is in scientific notation
• We need to consider the base only
• The base is 7.400
• So the number of significant figures = 4
Order of magnitude
We will explain this in steps:1. We have seen the details about scientific notation
• If a number is given to us in that notation, we can get an approximate idea about the 'size' of that number by rounding off a
2. We know that a will be greater than or equal to 1
• If a is less than or equal to 5, it will become 1
3. We know that a will also be less than 10
• If a is greater than 5, it will become 10
4. When a is rounded off in this manner, b gets a new name: order of magnitude
5. Let us see some examples:
Example 1:
• The diameter of the earth is 1.28 × 107 m
• To get an approximate idea about the diameter, we round of 1.28
• Based on (2) above, when 1.28 is rounded off, we get 1
• After rounding off 1.28 to 1, we cannot say this:
The diameter of the earth is 107 m
• We must say this:
Diameter of the earth is of the order of 107 m with the order of magnitude 7
• 107 is very large: 1 followed by seven zeroes
Example 2:
• The actual diameter of hydrogen atom is 1.06 × 10-10 m
• To get an approximate idea about the diameter, we round off 1.06
• Based on (2) above, when 1.06 is rounded off, we get 1
• After rounding off 1.06 to 1, we cannot say this:
The diameter of the hydrogen atom is 10-10 m
• We must say this:
Diameter of the hydrogen atom is of the order of 10-10 m with the order of magnitude -10
• 10-10 is very small:
♦ There is only a zero in front of the decimal point
♦ Just after the decimal point, there will be 9 zeroes
♦ After those 9 zeroes, there will be just a '1'
6. What is the difference between the following two items:
(i) Order of magnitude of diameter of hydrogen atom
(ii) Order of magnitude of diameter of earth
• Obviously, the difference is: [7 - (-10)] = 17
• Thus we can write:
Diameter of earth is 17 orders of magnitude larger than the diameter of hydrogen atom
We have seen the basics about significant figures. In the next section, we will see the rules for arithmetic operations with significant figures
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