Uncertainty in products, quotients and powers

5 minAS Level1.3.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find the uncertainty in a quantity calculated by multiplying, dividing or raising to a power, and turn it back into an absolute uncertainty.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\rho$	density
$\Delta$	the uncertainty in

Rule 2: multiplying or dividing? Add the percentage uncertainties

If $y = ab$ or $y = \dfrac{a}{b}$ , then

$\%\,y = \%\,a + \%\,b$

(Fractional uncertainties add in exactly the same way.) As with sums, you add for a quotient too, never subtract.

Rule 3: powers multiply the percentage

If $y = a^n$ , then $\%\,y = n \times \%\,a$ .

$a^2$ is just $a \times a$ , so its percentage uncertainty is $\%a + \%a = 2\%a$ .
$\sqrt{a} = a^{1/2}$ has half the percentage uncertainty of $a$ .
A negative power works the same way: $\dfrac{1}{a^2}$ still has $2\%a$ .
Pure numbers such as $\pi$ , $2$ or $\tfrac{1}{2}$ are exact and add no uncertainty.

Why percentages?

If $a$ is 2% too big and $b$ is 3% too big, then $ab$ is $1.02 \times 1.03 = 1.0506$ times too big: about 5% out. For a quotient, the worst case is the top too big and the bottom too small: $\dfrac{1.02}{0.97} = 1.052$ , again about 5%. Absolute uncertainties can't combine like this in a product, because $a$ and $b$ may not even have the same unit.

Quick check

Multiple choiceQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.