Significant figures play a crucial role in ensuring accuracy and precision in scientific calculations. When performing operations like addition, subtraction, multiplication, and division, specific rules must be f ollowed to maintain the correct number of significant digits in the final result. Understanding how to apply these rules helps avoid errors and ensures that measurements and calculations reflect true precision in physics and other scientific fields.
- Significant Figures Rules in Addition and Subtraction Operations
- Significant Figures in Multiplication and Division Operations
- Solved Problems
- Scientific Notation
- Counting Significant Figures in Scientific Notation
- Rounding Off to Significant Figures
- Operations in Scientific Notation
- Addition and Subtraction
- Multiplication and Division
- Very Short Frequently Asked Questions (FAQs) and Answers
- Buy Complete Study Material
Significant Figures Rules in Addition and Subtraction Operations
When adding or subtracting measured values, the final result should have the same number of decimal places as the measurement with the least decimal places.
The accuracy of a sum or a difference is limited to the accuracy of the least accurate observation in the addition and subtraction.
Rule. Do not retain a greater number of decimal places in a result computed from addition and/or subtraction than in the observation, which has the fewest decimal places.
Illustration 1. Add and subtract 428.5 and 17.23 with due regards to significant figures.
We have,
$$428.5 + 17.23 = 445.73$$
$$428.5 – 17.23 = 411.27$$
But in physics, the sum and difference taken in this manner are discouraged.
In fact, in the data 428.5, we have assumed zero to be in second place after decimal. The data 428.5 might have been written to first decimal only, because of the inability of the instrument to measure it to the further accuracy. Therefore, the choice of zero only in the second decimal place of the data is not justified.
To add or subtract in such a situation, there are two methods :
(i) By rounding off the answer. The data 428.5 is the weakest link as its value is known upto first decimal only. Therefore, the answer should also be retained only up to first decimal place i.e.
$428.5 + 17.23 = 445.73$ (before rounding)
$428.5 – 17.23 = 411.27$ (before rounding)
Rounding off the results of the above sum and difference to the first decimal,
correct sum $= 445.7$ and correct difference $= 411.3$
(ii) By rounding off the other data. The result can also be obtained by rounding off the other data in accordance with the data, which is the weakest link. The data 17.23 should be rounded off to 17.2 and then added to or subtracted from 428.5. Thus, we have
$$428.5 + 17.2 = 445.7$$
$$428.5 – 17.2 = 411.3$$
Hence, $$428.5 + 17.23 = 445.7$$ and $$428.5 – 17.23 = 411.3$$
Example: Find the sum of 420.42 m, 420.4 m, and 0.402 m.
Solution:
420.42 m
+ 420.4 m
+ 0.402 m
————
841.222 m
- The least precise measurement (420.4 m) has only one decimal place.
- Thus, the final answer must be rounded to one decimal place.
- Final Answer: 841.2 m
Let’s look at another example: 5,289 + 100?
The sum of 5,289 and 100 is 5,389 but we need to round the final answer. The first number (5,289) ends with the ones place, while the second number (100) is significant only to the hundreds place, and therefore, we round 5,389 to the hundreds place: 5,400 (no decimal at the end).
Significant Figures in Multiplication and Division Operations
When multiplying or dividing measured values, the final result should have the same number of significant figures as the measurement with the least significant figures.
When the values of different observations are multiplied or divided, the number of digits to be retained in the answer depends upon the number of significant figures in the weakest link.
Rule. Do not retain a greater number of significant figures in a result computed from multiplication and/or division than the least number of significant figures in the data from which the result is computed.
Illustration 2. Multiply 312.65 and 26.4 with due regards to significant figures.
We have,
$$312.65 \times 26.4 = 8253.960$$
But as the weakest link i.e. the data 26.4 has only three significant figures, the correct result of multiplication will be 8250. Hence,
$$312.65 \times 26.4 = 8250$$
Illustration 3. Divide $2.5 \times 10^5$ by $4.75 \times 10^3$ with due regards to significant figures.
We have,
$$\frac{2.5 \times 10^5}{4.75 \times 10^3} = 52.6316$$
Since the weakest link $2.5 \times 10^5$ has only two significant figures, the correct result of the division will be 53. Hence,
$$\frac{2.5 \times 10^5}{4.75 \times 10^3} = 53$$
Same for the division, 138 ÷ 11.9 = 11.596638, and we round up the 5 to 6 because of the 9 on the right. Therefore, the answer is 11.6.
Example 1: Calculate 1.2×36.72.
Solution:
1.2 × 36.72 = 44.064
- The least number of significant figures in the given values is 2.
- Hence, the result should be rounded to 2 significant figures.
- Final Answer: 44
Example 2: Calculate 1100 ÷ 10.2.
Solution:
1100 ÷ 10.2 = 107.8431373
- The least number of significant figures is 3 (10.2 has 3 significant figures).
- The result must be rounded to 3 significant figures.
- Final Answer: 108
Solved Problems
Problem.1.
A jeweller puts a diamond in a box weighing 1.2 kg. Find the total weight of the box and the diamond with due regard to significant figures, if the weight of diamond is 5.42 g.
Sol. Weight of box, $a = 1.2$ kg
Weight of diamond, $b = 5.42$ g $= 0.00542$ kg
$\therefore\ a + b = 1.2 + 0.00542 = 1.20542$ kg
Of all the weight measurements, weight of the box (1.2 kg) has the least number of decimal places i.e. one. Therefore, rounding off the above result to the first decimal place, we have
the total weight of the box, $a + b = 1.2$ kg
Problem.2.
Subtract with due regard to significant figures:
$$3.9 \times 10^5 – 2.5 \times 10^4$$
Sol. Now, $$3.9 \times 10^5 – 2.5 \times 10^4 = 390000 – 25000$$
$$= 365000 = 3.65 \times 10^5$$
Since each of $3.9 \times 10^5$ and $2.5 \times 10^4$ has got two significant figures, the result should also have two significant figures. Rounding off to two significant figures, we have
$$3.9 \times 10^5 – 2.5 \times 10^4 = 3.7 \times 10^5$$
Problem.3.
Solve with due regard to significant figures:
$$4.0 \times 10^{-4} – 2.5 \times 10^{-6}$$
Sol. Now, $$4.0 \times 10^{-4} – 2.5 \times 10^{-6}$$
$$= 0.0004 – 0.0000025 = 0.0003975 = 3.975 \times 10^{-4}$$
Since each of $4.0 \times 10^{-4}$ and $2.5 \times 10^{-6}$ has got two significant figures, the result should also have two significant figures. Rounding off to two significant figures, we have
$$4.0 \times 10^{-4} – 2.5 \times 10^{-6} = 4.0 \times 10^{-4}$$
Problem.4.
Solve the following with due regard to significant figures:
$$\frac{0.9996 \times 3.52}{1.758}$$
Sol. $$\frac{0.9996 \times 3.52}{1.758} = 2.00147$$ (by actual calculations)
3.52 has least number of significant figures i.e. three. Therefore, rounding off the result to three significant figures, we have
$$\frac{0.9996 \times 3.52}{1.758} = 2.00$$
Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It is especially useful in chemistry and physics, where precision is essential. The general form of scientific notation is:
N×10n
Where:
- N is a number between 1 and 10.
- n is an integer (positive or negative) that represents the exponent.
For example:
- 1.34×105 = 134,000
- 8.6×10−4 = 0.00086
Counting Significant Figures in Scientific Notation
The number of significant figures is determined by N, not by the exponent.
- 1.34×105 has 3 significant figures.
- 8.6×10−4 has 2 significant figures.
To express a number with a certain number of significant figures, zeros may be added after the decimal point:
- 1.34×105 with 6 significant figures = 1.34000×105
Rounding Off to Significant Figures
If a number has more digits than required, it is rounded appropriately.
- Example: Express 46897 in two significant figures.
- The first two digits are 46.
- Since the next digit is 8 (greater than 5), we round up.
- Final result: 4.7×104.
Operations in Scientific Notation
Addition and Subtraction
- If the exponents are the same, simply add or subtract the coefficients.
- Example: (2.65×103)+(6.4×103)
- (2.65+6.4)×103=9.05×103
- Rounded to one decimal place: 9.1 × 103
- If the exponents are different, adjust one number to match the other.
- Example: (9.578×103)−(5.326×102)
- Convert to decimal: 9578 – 532.6 = 9045.4
- Convert back to scientific notation: 9.045 × 103
Multiplication and Division
- Multiplication: Multiply the coefficients and add the exponents.
- Example: (3.4×10−6)×(2.5×104)
- (3.4×2.5)×10−6+4 = 8.5×10−2
- Division: Divide the coefficients and subtract the exponents.
- Example: (6.0×105)÷(2.0×102)
- (6.0÷2.0)×105−2 = 3.0×103
Very Short Frequently Asked Questions (FAQs) and Answers
Why do we use significant figures in calculations?
Significant figures indicate the precision of measured values and help maintain accuracy while performing mathematical operations.
What happens if we do not follow the rules of significant figures in calculations?
Ignoring significant figures may lead to overestimation or underestimation of results, causing errors in scientific and engineering calculations.
Why do we use scientific notation?
To simplify the representation of very large or small numbers and to avoid misplacing zeros.
What determines the number of significant figures in scientific notation?
The coefficient N (not the exponent).
Summary
To strengthen your understanding of significant figures in addition, subtraction, multiplication, and division, it is important to first build a clear foundation of basic measurement concepts. You should review topics like significant figures rules, rounding off rules, and errors in measurement, as these are directly connected to how precision is maintained in calculations. For deeper practice, explore solved examples and numerical problems from units and measurements, especially JEE-level questions, which help you apply these rules in real exam scenarios. This interconnected approach will improve both conceptual clarity and problem-solving accuracy.
Strengthen your understanding with Class 11 Physics Notes for complete theory and concepts.
Revise the Units and Measurements chapter to build a clear foundation in systems of units.
Practice JEE Units and Measurements PYQs to improve accuracy and understand exam-level questions.
For IMU CET Previous Years Questions, click the link.
Buy Complete Study Material
Get comprehensive study material for JEE, NEET, and CBSE Board Class 11 Exams, including detailed notes, MCQs, worksheets, and explanations at Anand Technical Publishers.
Published by: Anand Technical Publishers
Anand Classes Jalandhar
Author: Neeraj Anand
Proprietor: NIRMAL ANAND Educations
Contact: +91-9463138669, Physics Anand Classes
Academic Building : Physics Anand Classes