Rounding off the Uncertain Digits

 Section 1.3.2: Rounding off the Uncertain Digits, from CBSE Class 11 Physics Chapter 1: Units and Measurements:

📘 Rounding Off Uncertain Digits: Precision and Convention

In experimental physics and calculations involving measured quantities, significant figures express the precision of measurements. Often, results derived from measurements include uncertain digits—values that go beyond the precision of the original data. To preserve meaningful accuracy and avoid misleading results, it is essential to round off such numbers correctly.

🔢 General Rule of Rounding

Given a number, when rounding to a required number of significant figures:

• If the digit to be dropped (i.e., the first insignificant digit) is less than 5, the preceding digit remains unchanged.

  • Example: $1.743 \rightarrow 1.74$ (dropping 3)

• If the digit to be dropped is more than 5, the preceding digit is increased by 1.

  • Example: $2.746 \rightarrow 2.75$ (dropping 6)

🔁 Special Case: Digit to Be Dropped = 5

When the digit to be dropped is exactly 5, the rule depends on the digit preceding the 5:

• If the preceding digit is even, leave it unchanged.

  • Example: $2.745 \rightarrow 2.74$

• If the preceding digit is odd, raise it by 1.

  • Example: $2.735 \rightarrow 2.74$

This approach is called "round to even" or "bankers' rounding", and it helps reduce rounding bias over large datasets.

🧮 Multi-step Calculations: Rule of Thumb

In complex or multi-step calculations:

• Do not round off at intermediate steps.

• Instead, carry one extra digit than required.

• Perform rounding only at the final step to maintain overall accuracy.

Example:
If you need 3 significant figures in the final result, use 4 in the intermediate steps.

⚖️ Examples

🔷 Example 1.1: Surface Area and Volume of a Cube

Given: Side of cube = $7.203\ \text{m}$ (4 significant figures)

• Surface Area =

  $$6 \times (7.203)^2 = 6 \times 51.849 \approx 311.299\ \text{m}^2 \Rightarrow \boxed{311.3\ \text{m}^2}$$

• Volume =

  $$(7.203)^3 = 373.7147\ \text{m}^3 \Rightarrow \boxed{373.7\ \text{m}^3}$$

Result is rounded to 4 significant figures, matching the original measurement.

🔷 Example 1.2: Density of a Substance

Given:

• Mass = $5.74\ \text{g}$ (3 significant figures)

• Volume = $1.2\ \text{cm}^3$ (2 significant figures)

$$\text{Density} = \frac{5.74}{1.2} = 4.783\ \text{g/cm}^3 \Rightarrow \boxed{4.8\ \text{g/cm}^3}$$

Since volume has only 2 significant figures, the final answer must be limited to 2 significant figures.

🔁 Use of Constants and Infinite Precision

• Constants like $2$, $\pi$, or $10$ (used in formulas) are considered exact numbers, having infinite significant figures.

• For practical calculations:

  • Use $\pi \approx 3.14$ (2 sig. figs) or $\pi \approx 3.142$ (4 sig. figs) as required.

  • Speed of light $c = 2.99792458 \times 10^8\ \text{m/s} \approx 3.00 \times 10^8\ \text{m/s}$ for simplicity.

✅ Conclusion

• Rounding ensures consistency with the precision of measured quantities.

• Always round the final answer, not intermediates.

• Apply standard rounding rules, especially for digits like 5, using the even–odd rule to prevent cumulative bias.

This approach ensures that results are scientifically meaningful and not overstated in precision.

Worksheet on Rounding Off Uncertain Digits based on CBSE Class 11 Physics, Chapter 1 – Units and Measurement (Section 1.3.2):

🔬 Worksheet: Rounding Off Uncertain Digits

Subject: Physics | Class: 11 | Chapter: Units and Measurements
Topic: Rounding Off Uncertain Digits | Time: 25 mins
Name: _____________________   Date: ____________
 

🧠 Part A: Conceptual Understanding

Q1. State the rule for rounding off when the digit to be dropped is:
a) Less than 5 → __________________________
b) Greater than 5 → _______________________
c) Equal to 5, and the preceding digit is even → _____________________
d) Equal to 5, and the preceding digit is odd → ______________________

Q2. Why should rounding be done only at the final step in a multi-step calculation?

🧮 Part B: Rounding Practice

Round the following to 3 significant figures:

Q3. $2.748$ → __________
Q4. $5.646$ → __________
Q5. $0.03567$ → __________
Q6. $19.4505$ → __________
Q7. $7.500$ → __________
Q8. $4.735$ → __________

🔢 Part C: Application-Based Problems

Q9. The radius of a sphere is measured as $6.378\ \text{m}$. Calculate:
a) Surface Area $A = 4\pi r^2$ → _______________
b) Volume $V = \frac{4}{3}\pi r^3$ → _______________
(Use $\pi = 3.14$; give your answers to 4 significant figures.)

Q10. A student measured the mass of a metal block to be $8.527\ \text{g}$ and its volume as $2.1\ \text{cm}^3$. Calculate the density, expressing the final answer with correct significant figures.

$$\text{Density} = \frac{\text{Mass}}{\text{Volume}} = ?$$

Answer: ___________________________

Q11. If the speed of light is $2.99792458 \times 10^8\ \text{m/s}$, express it rounded off to:
a) 3 significant figures → ________________
b) 2 significant figures → ________________

✍️ Part D: True or False

Q12. ( ) Trailing zeros in a number like 4.500 are not significant.
Q13. ( ) The digit zero placed before the decimal point in 0.0487 is not significant.
Q14. ( ) The number 5.745 rounded to 3 significant figures is 5.74.
Q15. ( ) $\pi$ is a constant with infinite significant figures.

Worksheet on Rounding Off Uncertain Digits (Class 11 Physics – CBSE Chapter 1: Units and Measurement, Section 1.3.2):

✅ Answer Key

🧠 Part A: Conceptual Understanding

Q1.
a) Preceding digit remains unchanged
b) Preceding digit is raised by 1
c) Preceding digit remains unchanged
d) Preceding digit is raised by 1

Q2.
Because rounding at intermediate steps may lead to accumulated rounding errors. Keeping one extra digit during intermediate steps maintains precision, and final rounding ensures the correct number of significant figures in the final result.

🧮 Part B: Rounding Practice

To 3 significant figures:

Q3. $2.748$ → 2.75
Q4. $5.646$ → 5.65
Q5. $0.03567$ → 0.0357
Q6. $19.4505$ → 19.5
Q7. $7.500$ → 7.50
Q8. $4.735$ → 4.74

🔢 Part C: Application-Based Problems

Q9. Given:
$r = 6.378\ \text{m}$, $\pi = 3.14$

a)

$$A = 4\pi r^2 = 4 \times 3.14 \times (6.378)^2 \approx 4 \times 3.14 \times 40.684 \approx 510.99\ \text{m}^2 \Rightarrow \boxed{511.0\ \text{m}^2}$$

(because 4 significant figures)

b)

$$V = \frac{4}{3}\pi r^3 \approx \frac{4}{3} \times 3.14 \times (6.378)^3 \approx \frac{4}{3} \times 3.14 \times 259.2 \approx 1085.35 \Rightarrow \boxed{1085\ \text{m}^3}$$

(Rounded to 4 significant figures)

Q10.

$$\text{Density} = \frac{8.527\ \text{g}}{2.1\ \text{cm}^3} = 4.060\ \text{g/cm}^3$$

Volume has 2 significant figures, so final answer = $\boxed{4.1\ \text{g/cm}^3}$

Q11.
a) $2.99792458 \times 10^8$ → $\boxed{3.00 \times 10^8\ \text{m/s}}$
b) → $\boxed{3.0 \times 10^8\ \text{m/s}}$

✍️ Part D: True or False

Q12. ❌ False (Trailing zeros in 4.500 are significant because of the decimal point)
Q13. ✅ True
Q14. ✅ True (5.745 → 5.74; digit before 5 is even)
Q15. ✅ True (π is an irrational number with infinite significant digits)