In the previous section, we saw accuracy and precision. In this section, we will learn about significant figures. In the later sections we will see errors.
We will learn about significant figures using three examples.
Example 1:
1. The length of a pencil is being measured using an ordinary ruler.
• The ruler is graduated in cm and mm.
• The situation is shown in fig.2.11 below:
2. The fig.2.11 is an enlarged view of the actual situation.
• The 'position of the tip of the pencil' is indicated by the yellow arrow.
3. Based on the fig., we can write the length.
• We will write it in 3 steps:
(i) The length of the pencil is clearly greater than 7 cm.
(ii) Consider the two points:
♦ The 7 cm mark.
♦ The tip of the yellow arrow.
(iii) Between those two points, there are three small divisions.
• We know that each of those divisions indicate 1 mm.
• 1 mm = 0.1 cm. So the length is clearly greater than 7.3 cm.
• At the same time, the length is less than 7.4 cm.
4. So the length is in between 7.3 and 7.4 cm.
• Here arises the problem.
• There is a ‘small length’ in between the 7.3 cm mark and the yellow arrow.
• We cannot ignore this length even if it is very small.
5. So how do we record that small length?
• Scientists have given clear rules regarding this situation. We can write those rules in 6 steps:
(i) we have to make a good judgment about the quantity of that small length.
• Assume that the '1 mm length' in between 7.3 and 7.4 is divided into 10 equal parts.
♦ So each of those parts will be 0.1 mm.
• How many of those parts fall between 7.3 and the yellow arrow?
• Make a good judgement.
• We can say: about 8 equal parts.
• So the extra quantity beyond 7.3 cm is (8 × 0.1 mm) = 0.8 mm.
• 0.8 mm = 0.08 cm.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 7.3 cm.
• So after addition, we get: (7.3 + 0.08) = 7.38 cm.
(iii) Consider the quantity 0.08 cm.
• It was not measured using any instrument. It was based on judgment. It is an uncertain quantity.
• We cannot be sure that it is exactly 0.08. It can be 0.09 or 0.07 .
• That is., (0.08 + 0.01) or (0.08 – 0.01).
• So we must record the final value as: 7.38 ± 0.01 cm.
(iv) The number 7.38 has three digits: 7, 3 and 8.
• The last digit ‘8’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 7, 3 and 8.
(vi) Remember that, the tip of the pencil was stuck in between two marks on the ruler.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get 'stuck with uncertainty' will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
6. Consider a person reading the following statement:
The length of the pencil = 7.38 ± 0.01 cm.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 8 is the uncertain digit.
• The rest of the digits are reliable.
• So the length of the pencil is certainly greater than 7.3 cm.
(ii) The quantity in excess of 7.3 cm may be any one of the following three:
0.07 cm, 0.08 cm, 0.09 cm.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.01 cm = least count⁄10
• So the least count of the instrument used for measurement = 0.01 cm × 10 = 0.1 cm. That is., 1 mm.
7. What if we do not write the digit '8'?
• Consider a person reading the following statement:
The length of the pencil = 7.3 cm.
• What all information does he get?
Answer: He gets the following information:
• The last digit 3 is the uncertain digit.
♦ So the length can be 7.2, 7.3 or 7.4 cm.
• Such a situation arises if we use the ruler shown in fig.2.12 below:
• We see no mm marks. So a judgement is made about the extra length beyond the 7 cm mark. Thus the '3' becomes the uncertain digit.
■ This is a misleading information because in our case, 7.3 cm is a certain quantity.
(i) If the tip of the pencil does not go beyond the 7.3 cm mark, we will get fig.2.13 below:
• The arrow coincides with 7.3 cm.
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '3'.
• We write: Length of the pencil = 7.30 cm.
• We must not write the length as 7.3 cm.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
The length of the pencil = 7.30 ± 0.01 cm.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 7.3 cm.
• The length is exactly 7.3 cm.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.01 cm = least count⁄10
• So the least count of the instrument used for measurement = 0.01 cm × 10 = 0.1 cm. That is., 1 mm.
Example 2:
1. The volume of a liquid is being measured using a measuring cylinder.
• The cylinder is graduated in mL and some smaller divisions.
♦ We will soon see 'how much' those smaller divisions indicate.
• The situation is shown in fig.2.14(a) below:
2. The top surface of the liquid is called the meniscus. For most liquids, it will be curved downwards.
• If the meniscus is curved downwards, we must consider it's lower most portion for taking readings.
3. Based on the fig.2.14(a), we can write the volume.
• We will write it in 3 steps:
(i) First we want to know the quantity represented by the small divisions.
• We see that, between 9 mL and 10 mL, there are 5 equal divisions.
• So each mL is divided into 5 equal parts.
• So each small division is equal to 1⁄5 = 0.2 mL.
(ii) Now we can write the volume.
• The volume is clearly greater than 11 mL.
(iii) Consider the two points:
♦ The 11 mL mark.
♦ The lower meniscus.
(iv) Between those two points, there are two small divisions.
• We have seen that each of those divisions indicate 0.2 mL.
• So the volume is clearly greater than 11.4 mL.
• At the same time, the volume is less than 11.6 mL.
4. So the volume is between 11.4 and 11.6 mL.
• Here arises the problem.
• There is a ‘small volume’ in between the 11.4 mL mark and the lower meniscus.
• We cannot ignore this volume, even if it is very small.
5. So how do we record that small volume?
• Scientists have given clear rules regarding this situation. We have seen those rules in example 1. We will write them again:
(i) we have to make a good judgment about the quantity of that small volume.
• Assume that the '0.2 mL volume' in between 11.4 and 11.6 is divided into 10 equal parts.
♦ So each of those parts will be 0.2⁄10 = 0.02 mL.
• How many of those parts fall between 11.4 and the lower meniscus?
• Make a good judgement.
• We can say: about 3 equal parts.
• So the extra quantity beyond 11.4 mL is (3 × 0.02 mL) = 0.06 mL.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 11.4 mL.
• So after addition, we get: (11.4 + 0.06) = 11.46 mL.
(iii) Consider the uncertain quantity 0.06 mL.
• It was not measured using any instrument. It was based on judgment.
• We cannot be sure that it is exactly 0.06. It can be 0.08 or 0.04.
• That is., (0.06 + 0.02) or (0.06 – 0.02).
(This is because, each of the 10 divisions that we assumed, is 0.02 mL).
• So we must record the final value as: 11.46 ± 0.02 mL.
(iv) The number 11.46 has four digits: 1, 1, 4 and 6.
• The last digit ‘6’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 1, 1, 4 and 6.
(vi) Remember that, the lower meniscus was stuck in between two marks.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get stuck with uncertainty will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
6. Consider a person reading the following statement:
The volume of liquid = 11.46 ± 0.02 mL.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 6 is the uncertain digit.
• The rest of the digits are reliable.
• So the volume is certainly greater than 11.4 mL.
(ii) The quantity in excess of 11.4 mL may be any one of the following three:
0.04 mL, 0.06 mL, 0.08 mL.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.02 = least count⁄10
• So the least count of the instrument used for measurement = 0.02 × 10 = 0.2 mL.
7. What if we do not write the digit '6'?
• Consider a person reading the following statement:
The volume of liquid = 11.4 mL.
• What all information does he get?
Answer: He gets the following information:
• The last digit 4 is the uncertain digit.
♦ So the volume can be 11.3, 11.4 or 11.5 mL.
• Such a situation arises if we use the cylinder shown in fig.2.14(b) above.
• We see no small divisions. So a judgement is made about the extra volume beyond the 11 mL mark.
• Thus the '4' becomes the uncertain digit.
■ This is a misleading information because in our case, 11.4 mL is a certain quantity.
(i) If the lower meniscus does not rise above the 11.4 mL mark, we will get fig.2.14(c)
• The lower meniscus coincides with 11.4 mL.
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '4' .
• We write: Volume of liquid = 11.40 mL.
• We must not write the volume as 11.4 mL.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
Volume of liquid = 11.40 ± 0.02 mL.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 11.4 mL.
• The volume is exactly 11.4 mL.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.02 = least count⁄10
• So the least count of the instrument used for measurement = 0.02 × 10 = 0.2 mL.
Example 3:
1. The length of a wall is being measured using a measuring tape.
• The tape is graduated in m and cm.
• The situation is shown in fig.2.15 below:
2. The fig.2.15 is an enlarged view of the actual situation.
(There will be a large number of 1 cm marks between 4 m and 5 m. A 100 to be exact. In order to avoid showing all those 100 marks, cut lines are given in between 4 m and 5 m)
• The 'position of the end of the wall' is indicated by the yellow arrow.
3. Based on the fig., we can write the length.
• We will write it in 3 steps:
(i) The length of the wall is clearly greater than 4 m.
(ii) Consider the two points:
♦ The 4 m mark.
♦ The tip of the yellow arrow.
(iii) Between those two points, there are two small divisions.
• Each of those divisions indicate 1 cm.
• 1 cm = 0.01 m. So the length is clearly greater than 4.02 m.
• At the same time, the length is less than 4.03 cm.
4. So the length is in between 4.02 and 4.03 m.
• Here arises the problem.
• There is a ‘small length’ in between the 4.02 m mark and the yellow arrow.
• We cannot ignore this length even if it is very small.
5. So how do we record that small length?
• Scientists have given clear rules regarding this situation. We have seen those rules in examples 1 and 2. We will write them again:
(i) we have to make a good judgment about the quantity of that small length.
• Assume that the '1 cm length' in between 4.02 and 4.03 is divided into 10 equal parts.
♦ So each of those parts will be 1 mm.
• How many of those parts fall between 4.02 and the yellow arrow?
• Make a good judgement.
• We can say: about 7 equal parts.
• So the extra quantity beyond 4.02 m is (7 × 1 mm) = 7 mm.
• 7 mm = 0.007 m.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 4.02 m.
• So after addition, we get: (4.02 + 0.007) = 4.027 m.
(iii) Consider the quantity 0.007 m.
• It was not measured using any instrument. It was based on judgment. It is an uncertain quantity.
• We cannot be sure that it is exactly 0.007. It can be 0.008 or 0.006 m .
• That is., (0.007 + 0.001) or (0.007 – 0.001).
• So we must record the final value as: 4.027 ± 0.001 m.
(iv) The number 4.027 has four digits: 4, 0, 2 and 7.
• The last digit ‘7’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 4, 0, 2 and 7.
(vi) Remember that, the end of the wall was stuck in between two marks on the tape.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get 'stuck with uncertainty' will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
6. Consider a person reading the following statement:
The length of the wall = 4.027 ± 0.001 m.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 7 is the uncertain digit.
• The rest of the digits are reliable.
• So the length of the wall is certainly greater than 4.02 m.
(ii) The quantity in excess of 4.02 m may be any one of the following three:
0.006 m, 0.007 m, 0.008 m.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.001 m = least count⁄10
• So the least count of the instrument used for measurement = 0.001 m × 10 = 0.01 m. That is., 1 cm.
7. What if we do not write the digit '7'?
• Consider a person reading the following statement:
The length of the wall = 4.02 m.
• What all information does he get?
Answer: He gets the following information:
• The last digit 2 is the uncertain digit.
♦ So the length can be 4.01, 4.02 or 4.03 m.
• Such a situation arises if we use the tape shown in fig.2.16 below:
• We see no cm marks. So a judgement is made about the extra length beyond the 4 m mark. Thus the '2' becomes the uncertain digit .
■ This is a misleading information because in our case, 4.02 m is a certain quantity.
(i) If the end of the wall does not go beyond the 4.02 m mark, we will get fig.2.17 below:
• The arrow coincides with 4.02 m.
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '2'.
• We write: Length of the wall = 4.020 m.
• We must not write the length as 4.02 m.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
The length of the wall = 4.020 ± 0.001 m.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 4.02 m.
• The length is exactly 4.02 m.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.001 m = least count⁄10
• So the least count of the instrument used for measurement = 0.001 m × 10 = 0.01 m. That is., 1 cm.
We will learn about significant figures using three examples.
Example 1:
1. The length of a pencil is being measured using an ordinary ruler.
• The ruler is graduated in cm and mm.
• The situation is shown in fig.2.11 below:
 |
| Fig.2.11 |
• The 'position of the tip of the pencil' is indicated by the yellow arrow.
3. Based on the fig., we can write the length.
• We will write it in 3 steps:
(i) The length of the pencil is clearly greater than 7 cm.
(ii) Consider the two points:
♦ The 7 cm mark.
♦ The tip of the yellow arrow.
(iii) Between those two points, there are three small divisions.
• We know that each of those divisions indicate 1 mm.
• 1 mm = 0.1 cm. So the length is clearly greater than 7.3 cm.
• At the same time, the length is less than 7.4 cm.
4. So the length is in between 7.3 and 7.4 cm.
• Here arises the problem.
• There is a ‘small length’ in between the 7.3 cm mark and the yellow arrow.
• We cannot ignore this length even if it is very small.
5. So how do we record that small length?
• Scientists have given clear rules regarding this situation. We can write those rules in 6 steps:
(i) we have to make a good judgment about the quantity of that small length.
• Assume that the '1 mm length' in between 7.3 and 7.4 is divided into 10 equal parts.
♦ So each of those parts will be 0.1 mm.
• How many of those parts fall between 7.3 and the yellow arrow?
• Make a good judgement.
• We can say: about 8 equal parts.
• So the extra quantity beyond 7.3 cm is (8 × 0.1 mm) = 0.8 mm.
• 0.8 mm = 0.08 cm.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 7.3 cm.
• So after addition, we get: (7.3 + 0.08) = 7.38 cm.
(iii) Consider the quantity 0.08 cm.
• It was not measured using any instrument. It was based on judgment. It is an uncertain quantity.
• We cannot be sure that it is exactly 0.08. It can be 0.09 or 0.07 .
• That is., (0.08 + 0.01) or (0.08 – 0.01).
• So we must record the final value as: 7.38 ± 0.01 cm.
(iv) The number 7.38 has three digits: 7, 3 and 8.
• The last digit ‘8’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 7, 3 and 8.
(vi) Remember that, the tip of the pencil was stuck in between two marks on the ruler.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get 'stuck with uncertainty' will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
• The above rules are accepted world wide. So any third person seeing the recorded value at a later stage will get the same information.
The length of the pencil = 7.38 ± 0.01 cm.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 8 is the uncertain digit.
• The rest of the digits are reliable.
• So the length of the pencil is certainly greater than 7.3 cm.
(ii) The quantity in excess of 7.3 cm may be any one of the following three:
0.07 cm, 0.08 cm, 0.09 cm.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.01 cm = least count⁄10
• So the least count of the instrument used for measurement = 0.01 cm × 10 = 0.1 cm. That is., 1 mm.
7. What if we do not write the digit '8'?
• Consider a person reading the following statement:
The length of the pencil = 7.3 cm.
• What all information does he get?
Answer: He gets the following information:
• The last digit 3 is the uncertain digit.
♦ So the length can be 7.2, 7.3 or 7.4 cm.
• Such a situation arises if we use the ruler shown in fig.2.12 below:
 |
| Fig.2.12 |
■ This is a misleading information because in our case, 7.3 cm is a certain quantity.
8. So we see that, it is necessary to record the uncertain quantity also.
• Here we must discuss a possible scenario. It can be written in 2 steps:
• Here we must discuss a possible scenario. It can be written in 2 steps:
 |
| Fig.2.13 |
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '3'.
• We write: Length of the pencil = 7.30 cm.
• We must not write the length as 7.3 cm.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
The length of the pencil = 7.30 ± 0.01 cm.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 7.3 cm.
• The length is exactly 7.3 cm.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.01 cm = least count⁄10
• So the least count of the instrument used for measurement = 0.01 cm × 10 = 0.1 cm. That is., 1 mm.
Example 2:
1. The volume of a liquid is being measured using a measuring cylinder.
• The cylinder is graduated in mL and some smaller divisions.
♦ We will soon see 'how much' those smaller divisions indicate.
• The situation is shown in fig.2.14(a) below:
 |
| Fig.2.14 |
• If the meniscus is curved downwards, we must consider it's lower most portion for taking readings.
3. Based on the fig.2.14(a), we can write the volume.
• We will write it in 3 steps:
(i) First we want to know the quantity represented by the small divisions.
• We see that, between 9 mL and 10 mL, there are 5 equal divisions.
• So each mL is divided into 5 equal parts.
• So each small division is equal to 1⁄5 = 0.2 mL.
(ii) Now we can write the volume.
• The volume is clearly greater than 11 mL.
(iii) Consider the two points:
♦ The 11 mL mark.
♦ The lower meniscus.
(iv) Between those two points, there are two small divisions.
• We have seen that each of those divisions indicate 0.2 mL.
• So the volume is clearly greater than 11.4 mL.
• At the same time, the volume is less than 11.6 mL.
4. So the volume is between 11.4 and 11.6 mL.
• Here arises the problem.
• There is a ‘small volume’ in between the 11.4 mL mark and the lower meniscus.
• We cannot ignore this volume, even if it is very small.
5. So how do we record that small volume?
• Scientists have given clear rules regarding this situation. We have seen those rules in example 1. We will write them again:
(i) we have to make a good judgment about the quantity of that small volume.
• Assume that the '0.2 mL volume' in between 11.4 and 11.6 is divided into 10 equal parts.
♦ So each of those parts will be 0.2⁄10 = 0.02 mL.
• How many of those parts fall between 11.4 and the lower meniscus?
• Make a good judgement.
• We can say: about 3 equal parts.
• So the extra quantity beyond 11.4 mL is (3 × 0.02 mL) = 0.06 mL.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 11.4 mL.
• So after addition, we get: (11.4 + 0.06) = 11.46 mL.
(iii) Consider the uncertain quantity 0.06 mL.
• It was not measured using any instrument. It was based on judgment.
• We cannot be sure that it is exactly 0.06. It can be 0.08 or 0.04.
• That is., (0.06 + 0.02) or (0.06 – 0.02).
(This is because, each of the 10 divisions that we assumed, is 0.02 mL).
• So we must record the final value as: 11.46 ± 0.02 mL.
(iv) The number 11.46 has four digits: 1, 1, 4 and 6.
• The last digit ‘6’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 1, 1, 4 and 6.
(vi) Remember that, the lower meniscus was stuck in between two marks.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get stuck with uncertainty will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
• The above rules are accepted world wide. So any third person seeing the recorded value at a later stage will get the same information.
The volume of liquid = 11.46 ± 0.02 mL.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 6 is the uncertain digit.
• The rest of the digits are reliable.
• So the volume is certainly greater than 11.4 mL.
(ii) The quantity in excess of 11.4 mL may be any one of the following three:
0.04 mL, 0.06 mL, 0.08 mL.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.02 = least count⁄10
• So the least count of the instrument used for measurement = 0.02 × 10 = 0.2 mL.
7. What if we do not write the digit '6'?
• Consider a person reading the following statement:
The volume of liquid = 11.4 mL.
• What all information does he get?
Answer: He gets the following information:
• The last digit 4 is the uncertain digit.
♦ So the volume can be 11.3, 11.4 or 11.5 mL.
• Such a situation arises if we use the cylinder shown in fig.2.14(b) above.
• We see no small divisions. So a judgement is made about the extra volume beyond the 11 mL mark.
• Thus the '4' becomes the uncertain digit.
■ This is a misleading information because in our case, 11.4 mL is a certain quantity.
8. So we see that, it is necessary to include the uncertain quantity also.
• Here we must discuss a possible scenario. It can be written in 2 steps:
• Here we must discuss a possible scenario. It can be written in 2 steps:
• The lower meniscus coincides with 11.4 mL.
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '4' .
• We write: Volume of liquid = 11.40 mL.
• We must not write the volume as 11.4 mL.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
Volume of liquid = 11.40 ± 0.02 mL.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 11.4 mL.
• The volume is exactly 11.4 mL.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.02 = least count⁄10
• So the least count of the instrument used for measurement = 0.02 × 10 = 0.2 mL.
Example 3:
1. The length of a wall is being measured using a measuring tape.
• The tape is graduated in m and cm.
• The situation is shown in fig.2.15 below:
 |
| Fig.2.15 |
(There will be a large number of 1 cm marks between 4 m and 5 m. A 100 to be exact. In order to avoid showing all those 100 marks, cut lines are given in between 4 m and 5 m)
• The 'position of the end of the wall' is indicated by the yellow arrow.
3. Based on the fig., we can write the length.
• We will write it in 3 steps:
(i) The length of the wall is clearly greater than 4 m.
(ii) Consider the two points:
♦ The 4 m mark.
♦ The tip of the yellow arrow.
(iii) Between those two points, there are two small divisions.
• Each of those divisions indicate 1 cm.
• 1 cm = 0.01 m. So the length is clearly greater than 4.02 m.
• At the same time, the length is less than 4.03 cm.
4. So the length is in between 4.02 and 4.03 m.
• Here arises the problem.
• There is a ‘small length’ in between the 4.02 m mark and the yellow arrow.
• We cannot ignore this length even if it is very small.
5. So how do we record that small length?
• Scientists have given clear rules regarding this situation. We have seen those rules in examples 1 and 2. We will write them again:
(i) we have to make a good judgment about the quantity of that small length.
• Assume that the '1 cm length' in between 4.02 and 4.03 is divided into 10 equal parts.
♦ So each of those parts will be 1 mm.
• How many of those parts fall between 4.02 and the yellow arrow?
• Make a good judgement.
• We can say: about 7 equal parts.
• So the extra quantity beyond 4.02 m is (7 × 1 mm) = 7 mm.
• 7 mm = 0.007 m.
■ This quantity obtained by judgment is called the uncertain quantity.
(ii) This uncertain quantity should be added to the certain quantity.
• In our present case, the certain quantity is 4.02 m.
• So after addition, we get: (4.02 + 0.007) = 4.027 m.
(iii) Consider the quantity 0.007 m.
• It was not measured using any instrument. It was based on judgment. It is an uncertain quantity.
• We cannot be sure that it is exactly 0.007. It can be 0.008 or 0.006 m .
• That is., (0.007 + 0.001) or (0.007 – 0.001).
• So we must record the final value as: 4.027 ± 0.001 m.
(iv) The number 4.027 has four digits: 4, 0, 2 and 7.
• The last digit ‘7’ is the uncertain digit.
• The other digits are reliable digits.
(v) The reliable digits and the uncertain digit taken together is called significant digits or significant figures.
• In our present example, the significant figures are: 4, 0, 2 and 7.
(vi) Remember that, the end of the wall was stuck in between two marks on the tape.
• The distance between those two marks will be the least count of the measuring instrument.
(The point where we get 'stuck with uncertainty' will be always between 'two consecutive smallest divisions'. The distance between 'two consecutive smallest divisions' is the least count)
• We divided that least count into 10 equal parts.
• Then we made a judgement on the question:
How many of those equal parts will constitute the uncertain portion?
• The above rules are accepted world wide. So any third person seeing the recorded measurement at a later stage will get the same information.
The length of the wall = 4.027 ± 0.001 m.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 7 is the uncertain digit.
• The rest of the digits are reliable.
• So the length of the wall is certainly greater than 4.02 m.
(ii) The quantity in excess of 4.02 m may be any one of the following three:
0.006 m, 0.007 m, 0.008 m.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.001 m = least count⁄10
• So the least count of the instrument used for measurement = 0.001 m × 10 = 0.01 m. That is., 1 cm.
7. What if we do not write the digit '7'?
• Consider a person reading the following statement:
The length of the wall = 4.02 m.
• What all information does he get?
Answer: He gets the following information:
• The last digit 2 is the uncertain digit.
♦ So the length can be 4.01, 4.02 or 4.03 m.
• Such a situation arises if we use the tape shown in fig.2.16 below:
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| Fig.2.16 |
■ This is a misleading information because in our case, 4.02 m is a certain quantity.
8. So we see that, it is necessary to record the uncertain quantity also.
• Here we must discuss a possible scenario. It can be written in 2 steps:
• Here we must discuss a possible scenario. It can be written in 2 steps:
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| Fig.2.17 |
• There is no quantity to record as uncertain.
(ii) In such a situation, it is compulsory to put a zero after the '2'.
• We write: Length of the wall = 4.020 m.
• We must not write the length as 4.02 m.
♦ If we do, we will be in the situation described in (7).
9. Consider a person reading the following statement:
The length of the wall = 4.020 ± 0.001 m.
• What all information does he get?
Answer: He gets the following 3 information.
(i) The last digit 0 is the uncertain digit.
• The rest of the digits are reliable.
• So there is no uncertain quantity.
(ii) There is no quantity in excess of 4.02 m.
• The length is exactly 4.02 m.
(iii) The portion after the '± ' sign is obtained by 'dividing the least count by 10'.
• That is., 0.001 m = least count⁄10
• So the least count of the instrument used for measurement = 0.001 m × 10 = 0.01 m. That is., 1 cm.
We have seen the basics about significant figures. In the next section, we will see how to find the 'number of significant figures' in a given measurement.
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