1.5.2.2. Multiplication or Division

If Q = x × y or Q = x ÷ y, then relative (percent) uncertainties add in quadrature:

$\frac{\delta Q}{Q} = \sqrt{\left( \frac{\delta x}{x} \right)^{2} + \left( \frac{\delta y}{y} \right)^{2}}$

Or, equivalently:

$\text{absolute } \delta Q = Q \times \sqrt{\left( \frac{\delta x}{x} \right)^{2} + \left( \frac{\delta y}{y} \right)^{2}}$

When you carry out multiplication or division of values that have uncertainties, it's essential to determine how those uncertainties combine. For example, if you're measuring two quantities, x and y, each with their uncertainties (denoted as $\delta x$ and $\delta y$), the uncertainty in the result (Q) from multiplying or dividing those values is not simply added together. Instead, you express it in terms of relative uncertainty and combine these using a method called 'adding in quadrature.' This process implies that the overall uncertainty in Q depends on the proportional uncertainties of x and y. The formula given clarifies that you need to take each quantity's percentage uncertainty, square it, sum these squares, and then take the square root of the result to find the total relative uncertainty in Q.

Imagine you are baking cookies, and the recipe requires 2 cups of flour (with a possible measurement error of 0.1 cups) and 1 cup of sugar (with a possible measurement error of 0.05 cups). The total amount of ingredients (Q) you need is the product of flour and sugar. To find out how uncertain your total ingredient amount is, you would express the uncertainty in the flour and sugar relative to their amounts and use the formula to combine these uncertainties to understand how much dough you're actually making.

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, $\delta Q/Q = 2 \times (\delta x/x)$. Absolute uncertainty ( \delta Q = Q \times 2 \times (\delta x/x) ].

When working with powers, the way we handle uncertainty changes slightly. If Q is defined as x raised to the power of n, the uncertainty in Q is expressed as a product of the absolute value of n and the percentage uncertainty of x. In other words, if x is squared, as in Q = x², the uncertainty gets multiplied by two. Thus, if the x increases, not only does Q increase but the uncertainty in Q reflects this exponential change.

Think of this like balloon inflation. If you have a balloon that doubles in size every second (squared relationship), any small error in tracking how much air you put in (uncertainty) will also multiply by the rate at which the balloon grows. So any slight inaccuracy becomes substantially larger as the balloon grows bigger, just like the uncertainty magnifies as you square the figure.

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

When you're combining multiple measurements with uncertainties, particularly when they involve more than two variables, you use a more generalized formula. This formula requires that you calculate how each variable's uncertainty contributes to the overall uncertainty in your function by using partial derivatives. Essentially, each term corresponds to a different measurement, and you will need to sum all these terms to get the complete uncertainty for the calculated result.

Imagine you're planning a road trip and calculating your total driving time. You consider how far each leg of the trip is and the average speed for that leg. Each distance and speed has some uncertainty (detours, speed limit changes). By applying partial derivatives, you can figure out how uncertainties in distance from each leg and speeds compound together to affect your total estimated time. Each separate leg's impact becomes clearer through this precise calculation.

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