Less common M2.13

Rounding and Error Intervals

This topic covers how to correctly approximate numbers through rounding and truncation, and how to express the range of original values (error intervals) that an approximated number could represent. This 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. 1

    Identify the level of accuracy.

    The number is 4300.

    The second significant figure is the '3', which is in the hundreds place.

  2. 2

    Determine the precision.

    The number has been rounded to the nearest 100.

  3. 3

    Calculate half of the precision unit.

    Half of 100 is 50.

  4. 4

    Determine the lower bound.

    Subtract half the precision from the rounded number:

    4300 - 50 = 4250

    The original number could be exactly 4250, so the inequality is '≤'.

  5. 5

    Determine the upper bound.

    Add half the precision to the rounded number:

    4300 + 50 = 4350

    The original number must be less than 4350, as 4350 would round up to 4400.

    So the inequality is '<'.

  6. 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.

Read this topic in the official UAT-UK ESAT guide →

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