Rounding and Error Intervals
Approximating numbers through rounding and truncation, and expressing the range of original values (error intervals) that an approximated number could represent, is a fundamental skill for handling measurements and calculations in science and engineering.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Rounding to decimal places (d.p.) involves looking at the digit immediately to the right of the target decimal place. If it's 5 or more, you round the target digit up; otherwise, it stays the same.
- Significant figures (s.f.) are counted from the first non-zero digit. When rounding large numbers, trailing digits are replaced with placeholder zeros to maintain the number's magnitude.
- Truncation is simply cutting off digits after a certain point. No rounding up occurs, regardless of the value of the first digit removed.
- The error interval for a number rounded to a certain precision is typically half of that precision below the number (inclusive) and half above (exclusive).
- The error interval for a truncated number starts at the truncated value itself (inclusive) and extends up to the next value at that level of precision (exclusive).
Formulae
y - 0.5p ≤ x < y + 0.5p To find the error interval for a number x which has been rounded to a value y, where p is the place value of the last digit in y (e.g., for 3.6 (1 d.p.), p=0.1).
y ≤ x < y + p To find the error interval for a number x which has been truncated to a value y, where p is the place value of the last digit in y (e.g., for 3.6 (1 d.p.), p=0.1).
Definitions
- Significant Figures (s.f.)
- The digits in a number that are reliable and necessary to indicate its precision, starting from the first non-zero digit from left to right. For example, in 0.0450, the significant figures are 4, 5, and 0.
- Decimal Places (d.p.)
- The number of digits that appear after the decimal point. For example, 3.142 has three decimal places.
- Truncation
- The process of shortening a number by removing digits after a certain position without performing any rounding.
- Error Interval
- An inequality that specifies the range of possible values a number could have had before it was approximated by rounding or truncation.
Worked example
A number, N, is stated as 4300 correct to 2 significant figures. Find the error interval for N.
- 1
Identify the level of accuracy.
The number is 4300.
The second significant figure is the '3', which is in the hundreds place.
- 2
Determine the precision.
The number has been rounded to the nearest 100.
- 3
Calculate half of the precision unit.
Half of 100 is 50.
- 4
Determine the lower bound.
Subtract half the precision from the rounded number:
4300 - 50 = 4250The original number could be exactly 4250, so the inequality is '≤'.
- 5
Determine the upper bound.
Add half the precision to the rounded number:
4300 + 50 = 4350The original number must be less than 4350, as 4350 would round up to 4400.
So the inequality is '<'.
- 6
Combine the bounds into a single inequality.
Answer: 4250 ≤ N < 4350
Common mistakes
- ×Confusing significant figures with decimal places. For 0.0785, rounding to 2 s.f. gives 0.079, while rounding to 2 d.p. gives 0.08.
- ×Forgetting placeholder zeros for large numbers. 178,210 rounded to 2 s.f. is 180,000, not 18.
- ×Using '≤' for the upper bound of a rounding error interval. A value equal to the upper bound would round up, so it must be strictly less than '<'.
- ×Incorrectly identifying the place value for significant figures. In 4300 (to 2 s.f.), the rounding is to the nearest 100, not the nearest 10 or 1.
No-calculator tips
- ✓To quickly find error bounds for rounding, think 'half a unit up, half a unit down'. If rounded to the nearest 10, the bounds are ±5. If to the nearest 0.01, the bounds are ±0.005.
- ✓For numbers less than 1, converting to standard form can clarify significant figures. 0.00508 is 5.08 x 10-3, which clearly has 3 significant figures.
- ✓When asked for an interval for a calculated value (e.g., area), find the intervals for the inputs first, then calculate the minimum and maximum possible outcomes by combining the lower and upper bounds appropriately.