What a learner can do afterwards
- Combines absolute uncertainties for a sum or difference and relative ones for a product or quotient
- Handles a power correctly and explains why the exponent multiplies the relative uncertainty
- Identifies the dominant contribution in a multi-step calculation and says what to improve first
- Explains why a small difference between two large numbers carries a large relative uncertainty
1 · Read
Some things cannot be measured directly. You cannot lay a ruler on the speed of a reaction or pour out the density of a coin, so you measure what you can reach, like mass and volume, and calculate the rest. Every range you feed into a formula moves through it into the answer, the way wobble in your angle readings follows along into the refraction index.
Combine with the matching rule. For a sum or a difference, add the absolute ranges. For a product or a quotient, add the percentage ranges. For a power, multiply the percentage range by the exponent, so a value that wobbles by 2 percent carries about 4 percent when squared.
Take 10.24 minus 10.11, which is 0.13, while each input wobbles by about 0.01. The answer is tiny next to the inputs, so its percentage wobble is huge. That is the pattern to watch: a small difference between two large numbers always carries a large relative uncertainty.
Ask which input moves your answer most, and improve that one first, the way you would ask how much the index shifts when your angles wobble by two degrees. Then round the final answer to stop where your least precise measurement stops.
Carry every range through the formula with its matching rule, and spend your effort on the input that moves the answer most.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.