Rules for Determining the Uncertainty in the Results of Arithmetic Calculations

Section 1.3.3 – Rules for Determining the Uncertainty in the Results of Arithmetic Calculations from CBSE Class 11 Physics Chapter 1: Units and Measurements.

📏 Section 1.3.3: Rules for Determining the Uncertainty in the Results of Arithmetic Calculations

In experimental physics, every measured quantity carries a certain degree of uncertainty. When we perform calculations using these quantities, it's essential to propagate or combine the uncertainties in a systematic way to ensure that the final result reflects the limitations of the measurements.

This section outlines the rules and reasoning for how uncertainties (or errors) should be managed during addition, subtraction, multiplication, division, and multi-step calculations.

🔢 1. Error Propagation in Multiplication and Division

When quantities are multiplied or divided, the relative errors (percentage errors) are added to estimate the total uncertainty in the result.

📌 Example:

Let’s say:

• Length $l = 16.2 \pm 0.1\ \text{cm} \Rightarrow \text{Relative error} = \frac{0.1}{16.2} \times 100 \approx 0.6\%$

• Breadth $b = 10.1 \pm 0.1\ \text{cm} \Rightarrow \text{Relative error} = \frac{0.1}{10.1} \times 100 \approx 1.0\%$

Then:

$$\text{Area} = l \times b = 16.2 \times 10.1 = 163.62\ \text{cm}^2$$ $$\text{Total relative error} = 0.6\% + 1.0\% = 1.6\%$$ $$\text{Uncertainty in area} = \frac{1.6}{100} \times 163.62 \approx 2.6\ \text{cm}^2$$

✅ Final answer:

$$A = (164 \pm 3)\ \text{cm}^2$$

➕ 2. Error Propagation in Addition and Subtraction

For addition or subtraction, the absolute uncertainties (not percentage) are combined. The result must be reported with the least number of decimal places among the given quantities.

📌 Example:

$$12.9\ \text{g} - 7.06\ \text{g} = 5.84\ \text{g}$$

But:

• 12.9 has 1 decimal place

• 7.06 has 2 decimal places

Thus, the result must be rounded to 1 decimal place:

$$\boxed{5.8\ \text{g}}$$

⚖️ 3. Dependence of Relative Error on Magnitude

The same absolute error has different impacts on measurements of different magnitudes.

📌 Example:

Measurement	Absolute Error	Relative Error
$1.02\ \text{g}$	$\pm 0.01 \text{g}$	$\frac{0.01}{1.02} \times 100 \approx 1\%$
$9.89\ \text{g}$	$\pm 0.01 \text{g}$	$\frac{0.01}{9.89} \times 100 \approx 0.1\%$

🔎 Inference: Smaller measurements suffer more from the same level of absolute uncertainty.

🧮 4. Multi-Step Calculations and Intermediate Rounding

In multi-step calculations, if we round off after every step, it can lead to compounding of errors and loss of accuracy.

🔑 Rule: Retain one extra significant figure in intermediate results than the least precise measurement. Round off only in the final step.

📌 Example:

• Reciprocal of $9.58$:

$$1/9.58 = 0.1044 \ (\text{Retain one extra digit})$$

• If we round off this as 0.104 (3 sig. figs),
  Then:

$$1/0.104 = 9.62 \ne 9.58$$

🧠 This small discrepancy emphasizes why we should avoid premature rounding and retain extra precision temporarily.

✅ Final Summary (Quick Rules):

Operation	Uncertainty Rule	Result Rule
Multiplication / Division	Add relative errors	Keep result to least sig. figs
Addition / Subtraction	Add absolute errors	Keep result to least decimal places
Multi-step Calculation	Retain extra digit in steps	Round at the end

📝 Important Note:

Exact numbers (like 2, π, etc.) in physical formulae are considered to have infinite significant figures and do not contribute to uncertainty.

Formulas for Combination of Errors (Uncertainty Propagation), which is an essential part of experimental physics and measurement analysis in CBSE Class 11 Physics (Chapter 1: Units and Measurements).

📘 Combination of Errors: A Comprehensive Note

In experimental sciences, measured quantities are often used to calculate other physical quantities through mathematical relationships (e.g., products, quotients, powers). Since each measurement comes with its own uncertainty, we need rules to combine these uncertainties to find the final uncertainty in the result.

These rules are known as error propagation formulas or combinations of errors.

⚙️ Let’s Define Some Terms First:

• $\Delta x$: Absolute error in measurement $x$

• $\Delta y$: Absolute error in measurement $y$

• $\delta x = \frac{\Delta x}{x}$: Relative error in $x$

• $\% \text{error} = \delta x \times 100$: Percentage error in $x$

🧮 Rules for Combination of Errors

📌 1. Addition and Subtraction

If:

$$Q = A + B \quad \text{or} \quad Q = A - B$$

Then:

$$\Delta Q = \Delta A + \Delta B$$

• Absolute errors are added directly.

• This is a conservative estimate to account for the worst-case uncertainty.

✅ Example:

$$A = 12.3 \pm 0.1,\quad B = 5.6 \pm 0.2 \Rightarrow Q = A + B = 17.9 \pm 0.3$$

📌 2. Multiplication and Division

If:

$$Q = A \times B \quad \text{or} \quad Q = \frac{A}{B}$$

Then:

$$\frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \quad \text{or} \quad \delta Q = \delta A + \delta B$$

• Relative errors are added.

✅ Example:

$$A = 3.4 \pm 0.1,\quad B = 2.0 \pm 0.1$$ $$Q = A \times B = 6.8,\quad \frac{\Delta Q}{Q} = \frac{0.1}{3.4} + \frac{0.1}{2.0} \approx 0.029 + 0.05 = 0.079 \Rightarrow \Delta Q \approx 6.8 \times 0.079 \approx 0.537 \Rightarrow Q = 6.8 \pm 0.54$$

📌 3. Error in Powers (Exponents)

If:

$$Q = A^n$$

Then:

$$\frac{\Delta Q}{Q} = |n| \cdot \frac{\Delta A}{A} \quad \text{or} \quad \delta Q = |n| \cdot \delta A$$

• The relative error is multiplied by the absolute value of the exponent.

✅ Example:

$$A = 2.0 \pm 0.1,\quad Q = A^3 \Rightarrow Q = 8.0$$ $$\frac{\Delta Q}{Q} = 3 \cdot \frac{0.1}{2.0} = 0.15 \Rightarrow \Delta Q = 8.0 \times 0.15 = 1.2 \Rightarrow Q = 8.0 \pm 1.2$$

📌 4. Error in a Function of Multiple Powers

If:

$$Q = A^p \cdot B^q / C^r$$

Then:

$$\frac{\Delta Q}{Q} = |p| \cdot \frac{\Delta A}{A} + |q| \cdot \frac{\Delta B}{B} + |r| \cdot \frac{\Delta C}{C}$$

• Relative errors of all variables are added, each multiplied by the magnitude of its exponent.

✅ Example:

$$Q = \frac{A^2 \cdot B}{C^3} \Rightarrow \frac{\Delta Q}{Q} = 2 \cdot \frac{\Delta A}{A} + 1 \cdot \frac{\Delta B}{B} + 3 \cdot \frac{\Delta C}{C}$$

🧠 Best Practices:

1. Always carry extra digits during intermediate calculations. Round off only in the final result.

2. Use appropriate significant figures based on the least precise measured value.

3. Remember that errors should reflect the precision of the instruments used and should not be overstated or understated.

4. Exact numbers (like π, 2, etc.) are error-free and are not included in error propagation.