How do uncertainties combine for multiplication/division?

When you multiply or divide measured quantities, add their relative uncertainties in quadrature to get the relative uncertainty of the result, then multiply that relative uncertainty by the result to obtain the absolute uncertainty.

Think of relative uncertainty as the size of the error compared to the measurement, like saying fifty percent of ten is five; to find the uncertainty when you multiply or divide, first turn each absolute uncertainty into a fraction of its value by dividing the uncertainty by the measurement, then square each of those fractions, add the squares together, take the square root of that sum to get the combined relative uncertainty, and finally multiply that relative uncertainty by the calculated product or quotient to find the absolute uncertainty. For example, if you multiply two numbers, each with its own percent error, you do the math with the relative errors rather than the absolute errors so the final uncertainty scales correctly with the result. This method assumes the errors are independent and random, which is common in laboratory measurements. If you are working with powers, multiply the relative uncertainty by the absolute value of the power before squaring and summing.

A common mistake is adding absolute uncertainties directly when quantities are multiplied, which overestimates the error, so always convert to relative uncertainties first. Another error is forgetting to square the relative uncertainties before adding, which underestimates the combined uncertainty, so remember the squaring and square root steps. People also sometimes forget to multiply the final relative uncertainty by the result to get the absolute uncertainty, so always convert the final relative value back to an absolute uncertainty by multiplying by the calculated measurement.

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Uncertainty Propagation for × and ÷

1. Use relative uncertainties
2. Divide uncertainty by measurement
3. Square each relative uncertainty
4. Add squared relative uncertainties
5. Take square root of sum
6. Multiply relative uncertainty by result

Common Mistakes in Uncertainty Propagation

1. Adding absolute uncertainties directly
2. Forgetting to square relative uncertainties
3. Not converting relative to absolute

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