Rounding and Error Intervals
Rounding and Error Intervals
Rounding expresses a value to a specified level of precision. Any rounded value carries an implicit error interval — a range of values that could have produced the rounded result. Expressing this range using inequality notation is both precise and compact, and is required throughout examination questions involving measurement and approximation.
Rounding to Decimal Places
To round to \( n \) decimal places: look at the \( (n+1) \)-th decimal digit. If it is 5 or more, round up the \( n \)-th digit; if it is less than 5, leave the \( n \)-th digit unchanged (round down).
\[ 3.748 \text{ to 2 dp} = 3.75 quad (\text{since the 3rd dp is 8} \geq 5) \] \[ 3.742 \text{ to 2 dp} = 3.74 quad (\text{since the 3rd dp is 2} < 5) \]Rounding to Significant Figures
Count significant figures from the first non-zero digit. To round to \( n \) sf, look at the \( (n+1) \)-th sf and apply the same rule as decimal places. Fill remaining places with zeros to preserve magnitude.
\[ 47,820 \text{ to 3 sf} = 48,000 quad (\text{4th sf is 2} < 5\text{, but wait — 3rd sf is 8, 4th is 2, round down}) \] \[ 0.004,587 \text{ to 2 sf} = 0.0046 quad (\text{3rd sf is 8} \geq 5\text{, round up}) \]For numbers like 47 820 to 3 sf: the three sfs are 4, 7, 8; the next digit is 2 < 5, so: \( 47,800 \).
Error Intervals
When a value \( x \) is given as a rounded number, the true value lies within an error interval. The bounds are: lower bound = rounded value minus half a unit; upper bound = rounded value plus half a unit. The lower bound is included (\( \leq \)); the upper bound is excluded (\( < \)), because a value at exactly the upper bound would round up to the next value.
For \( x = 3.4 \) (rounded to 1 dp): the half-unit is 0.05 (half of 0.1). Lower bound: \( 3.4 - 0.05 = 3.35 \). Upper bound: \( 3.4 + 0.05 = 3.45 \). Error interval: \( 3.35 \leq x < 3.45 \).
Truncation Higher
Truncation means cutting off digits at a specified position without rounding. The digits are simply removed, regardless of whether the next digit is ≥ 5 or < 5.
\[ 4.87 \text{ truncated to 1 dp} = 4.8 \quad (\text{not 4.9 — the 7 is just removed}) \]The error interval for a truncated value is different from a rounded one. If \( x = 4.8 \) after truncation to 1 dp, the true value could be anywhere in \( [4.8, 4.9) \) — the lower bound is the truncated value itself (included), and the upper bound is one unit of precision above it (excluded):
\[ 4.8 \leq x < 4.9 \]This is asymmetric — the error is always positive (the true value is always ≥ the truncated value), unlike rounding where the error can be in either direction.
Worked Examples
Round 0.005 87 to 2 sf and to 3 dp. Round 482 000 to 2 sf.
0.005 87 to 2 sf: first sf is 5, second is 8, third is 7 ≥ 5 → round up: \( 0.0059 \)
0.005 87 to 3 dp: the 3rd decimal place is 5, the 4th is 8 ≥ 5 → \( 0.006 \)
482 000 to 2 sf: sfs are 4 and 8; the next digit is 2 < 5 → \( 480,000 \)
Write error intervals for: \( m = 7.3 \) cm (1 dp) and \( n = 4700 \) (nearest 100).
Half a unit at 1 dp = 0.05. Error interval for \( m \):
\[ 7.25 \leq m < 7.35 \text{ cm} \]Half a unit at nearest 100 = 50. Error interval for \( n \):
\[ 4650 \leq n < 4750 \]Truncate 8.637 to 2 dp. Write its error interval and compare it to the error interval if 8.637 had been rounded to 2 dp.
Truncated to 2 dp: remove the 7 → \( 8.63 \)
Error interval (truncated): \( 8.63 \leq x < 8.64 \)
Rounded to 2 dp: 3rd dp is 7 ≥ 5, so round up → \( 8.64 \)
Error interval (rounded): \( 8.635 \leq x < 8.645 \)
Key difference: truncation always produces a lower-biased error (true value ≥ truncated value). Rounding is symmetric around the rounded value (error in either direction).
Key Takeaways
- Rounding to \( n \) dp: look at the \( (n+1) \)-th dp — if ≥ 5 round up; if < 5 round down.
- Rounding to \( n \) sf: count sfs from the first non-zero digit; apply the same rule; preserve magnitude with zeros.
- Error interval for a value rounded to precision \( u \): \( ( ext{value} - u/2) \leq x < ( ext{value} + u/2) \). The lower bound is included; the upper bound is excluded.
- Truncation removes digits without rounding up. The error interval is one-sided: \( ext{value} \leq x < ext{value} + u \). Higher
- For measurements rounded to the nearest unit, the error interval spans from (value − half a unit) to (value + half a unit), exclusive of the upper bound.