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Error in product or quotient
Intermediate
Relative errors add when quantities are multiplied or divided.
Δz/z = ΔA/A + ΔB/BFor Z = AB or A/B the relative errors add: ΔZ/Z = ΔA/A + ΔB/B. Express each error as a fraction first, then convert the total back to an absolute error if needed.
A = 5.0 ± 0.1 (2%) and B = 2.0 ± 0.1 (5%): AB = 10.0 with 7% error, i.e. 10.0 ± 0.7.
Do not add absolute errors when multiplying; the units differ.
In more detail
When you multiply or divide quantities, their relative uncertainties add. A rectangle measured as 2.0 ± 0.1 cm by 3.0 ± 0.1 cm has a relative error of about 8% in its area.
Real-world uses
- Density is mass divided by volume, so uncertainties in both combine to give the density uncertainty.
- Estimating the area of a room from two measured sides uses this rule.
- Density is mass divided by volume, so its relative error is the sum of the relative errors in mass and volume.
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Frequently asked questions
Are relative errors added directly?
For products and quotients, yes, as a first estimate. The formal rule uses root-sum-square for independent errors.
What about multiplying by an exact number?
An exact number such as 2 adds no uncertainty, so the relative error of the product stays the same.
What happens to the relative error when I multiply by an exact number like 2?
An exact number has no uncertainty, so the relative error stays the same, while the absolute error doubles.
Related topics in Measurement & Math Tools
Absolute error · Relative and percent error · Error in a sum or difference · Error in powers · Mean and standard deviation · Dimensions of a quantity