1.5.2.4. More Complex Functions

Common Special Cases

1. Addition or Subtraction If Q = x + y or Q = x – y, and x has uncertainty δx and y has δy, then $$\delta Q = \sqrt{(\delta x)² + (\delta y)²}$$
   1. Note: One never adds absolute uncertainties linearly; instead, add in quadrature (square root of sum of squares).

When adding or subtracting two measurements (for example, two distances), the total uncertainty is not simply the sum of their uncertainties. Instead, we combine them using the square root of the sum of their squares. This approach is called adding uncertainties in quadrature.

For instance, if you measure length A with an uncertainty of ±0.1 cm and length B with an uncertainty of ±0.2 cm, the total length C = A + B would have an uncertainty calculated as follows:

• Calculate the square of each uncertainty: (0.1 cm)² + (0.2 cm)² = 0.01 + 0.04 = 0.05
• Then take the square root: √0.05 = 0.2236 cm. Thus, C = A + B ± 0.22 cm.

Imagine you are to measure ingredients for a recipe, such as flour and sugar. If you measure flour and find it weighs 200 grams with an accuracy of ±1 gram, and sugar weighs 150 grams with an accuracy of ±2 grams, when you combine these two ingredients, the total weight will not just be 200 + 150 grams ± (1 + 2) grams. Instead, it will be calculated using the method described. This ensures a more accurate total which reflects the precision of both measurements.

2. Multiplication or Division If Q = x × y or Q = x ÷ y, then relative (percent) uncertainties add in quadrature: $$\frac{\delta Q}{Q} = \sqrt{(\frac{\delta x}{x})² + (\frac{\delta y}{y})²}$$ Or, equivalently: $$\text{absolute } \delta Q = Q \times \sqrt{(\frac{\delta x}{x})² + (\frac{\delta y}{y})²}$$

When multiplying or dividing measurements, the way we handle their uncertainties is a bit different. Instead of adding absolute uncertainties, we combine their relative uncertainties.

For example, if you have two measurements: a length of 5 cm measured with an uncertainty of ±0.1 cm and a width of 10 cm with an uncertainty of ±0.2 cm, the area (A) can be calculated as:

• A = length × width. To calculate the uncertainty in area, you first find the relative uncertainties:
• Relative uncertainty in length = (0.1 cm / 5 cm) = 0.02.
• Relative uncertainty in width = (0.2 cm / 10 cm) = 0.02. Next, you add these relative uncertainties in quadrature:
• Total relative uncertainty in area = √(0.02² + 0.02²) = √0.0008 = 0.0283. Thus, area A = 50 cm² ± (50 cm² * 0.0283) ≈ 50 cm² ± 1.415 cm².

Consider the task of creating a new garden bed. If you calculate the area based on measurements of length and width as above, you wouldn’t want to just add the uncertainties when you are looking for the total area. Just like calculating the amount of paint needed for a fence; if you know the height and width of the fence, combining those dimensions while considering their uncertainties will give you a better estimate of how much paint you need. This method provides a more reliable and accurate estimation.

3. Powers or Exponentials If Q = x^n (x raised to power n), then $$\frac{\delta Q}{Q} = |n| \times \frac{\delta x}{x}$$ For example, if Q = x², then: $$\delta Q = Q \times 2 \times \frac{\delta x}{x}.$$

When dealing with functions where a variable is raised to a power, the uncertainty in the result can be calculated using the power's absolute value multiplied by the relative uncertainty of that variable.

For instance, if you are measuring the volume of a cube where the edge length (x) is measured as 3 cm with an uncertainty of ±0.1 cm, the volume (Q = x³) is calculated as:

• Q = 3³ = 27 cm³. The relative uncertainty is calculated as:
• Relative uncertainty = (0.1 cm / 3 cm) = 0.03333. Then: δQ = |3| × (27 cm³ × 0.03333) = 2.9998 cm³, meaning the volume could be reported as 27 cm³ ± 3 cm³.

Think about filling a rectangular box with sand. If you are unsure about the exact dimensions (each length), the total volume calculation gets influenced by both the measurements of length and their associated uncertainties. Each measurement’s uncertainty can significantly affect the estimated amount of sand required to fill the box.

4. More Complex Functions Apply the general formula using partial derivatives. For functions of more than two variables, sum all corresponding terms.

No detailed explanation available.

No real-life example available.