Upper and Lower Bounds
Give the upper and lower bounds of a rounded measurement and write the error interval, using the half-unit rule for the accuracy stated.
What a learner can do afterwards
- State the bounds of a length given as 12.4 cm to 1 decimal place
- Write an error interval with the correct strict and inclusive signs
- Say what a mass rounded to the nearest 10 g could really be
1 · Read
A rounded measurement hides a range, and half a unit each way finds it for you. 12.4 cm to 1 decimal place spans 12.35 cm up to 12.45 cm, since half of 0.1 is 0.05. Name the half step first, then add and subtract it.
A mass of 6.5 kg to 1 decimal place really means 6.45 kg <= mass < 6.55 kg. The lower bound joins in with <= and the upper bound stays out with <. Say the signs out loud as you write them.
Whole-number rounding scales the same idea up. A mass shown as 40 g to the nearest 10 g spans 35 g up to 45 g, because the half step is 5 g. To 2 decimal places, 2.35 spans 2.345 <= x < 2.355.
No calculation can promise more detail than its roughest input. Extra digits past the bounds are fiction, so keep the half step small and symmetric.
Half a unit each way, lower bound in, upper bound out.
More to read
4.3.6: Error Intervals
A measurement like 12.4 cm to 1 decimal place really means a range: from 12.35 up to but not including 12.45. Half the precision each way. The lower bound joins in, the upper bound stays out.
Write it as 12.35 ≤ x < 12.45, with the closed mark at the bottom and the open mark at the top. Mixing the marks up is the classic lose-a-mark slip. Say the signs out loud as you write them.
6.3: Rounding Decimals
Rounding to a place sets the bounds automatically: look at half a unit of that place. To 1 decimal place the half step is 0.05, so 7.2 spans 7.15 to 7.25. Keep the half step small and symmetric.
For whole number rounding the same idea scales up. Nearest 10 g means a half step of 5 g, so a 40 g reading spans 35 g to 45 g. Name the half step first, then add and subtract it.
1.3 Accuracy, Precision, and Significant Figures
Precision limits what you can claim. A length 8.6 cm to 1 decimal place tops out below 8.65 cm, and no arithmetic with it can promise more detail than that. Extra digits beyond the bounds are fiction.
Significant figures track this honesty through calculations. Count from the first nonzero digit, and round the answer to match the roughest input. Bounds first, figures second, and the answer stays believable.
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2 · Watch
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17 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.