Rounding Numbers and Error Intervals

Rounding to Decimal Places

Rounding to a specified number of decimal places (d.p.) involves identifying the digit at the target position and looking at the digit immediately to its right. To round to 3 d.p., for example, you look at the 4th decimal place. If that 4th digit is 5 or more, you add 1 to the digit in the 3rd decimal place. If it is less than 5, you leave the 3rd digit unchanged.

Worked Example: Rounding Decimal Places

Consider the number $31.56387$.

1. Round to 4 d.p.: Count four places to the right of the decimal point, which is the digit 8. The digit to its right is 7. Since 7 is greater than or equal to 5, we add 1 to the 8. The result is $31.5639$.

2. Round to 2 d.p.: Count two places to the right of the decimal point, which is the digit 6. The digit to its right is 3. Since 3 is less than 5, we make no correction and leave the 6 as it is. The result is $31.56$.

Rounding to Significant Figures

Significant figures (sig. figs.) indicate the precision of a number regardless of the decimal point location. To round to 3 sig. figs., you must count three digits from left to right, starting with the first non-zero digit. Digits to the right of the third significant figure are removed. If they are to the left of the decimal point, they are replaced with zeros as placeholders. If the 4th significant figure is 5 or more, add 1 to the 3rd significant figure.

Worked Example: Significant Figures with Decimals

Consider the number $0.0023824$.

1. Round to 4 sig. figs.: The first non-zero digit is 2. Counting four digits (2, 3, 8, 2), the next digit is 4. Since 4 is less than 5, no correction is needed. The result is $0.002382$.

2. Round to 2 sig. figs.: The first two significant figures are 2 and 3. The next digit is 8. Since 8 is greater than or equal to 5, the 3 is corrected to a 4. The result is $0.0024$.

Worked Example: Significant Figures with Large Numbers

Consider the number $365,892$. To round this to 2 sig. figs., identify the first two digits (3 and 6). The next digit is 5. Since it is 5 or more, the 6 becomes a 7. We then fill in zeros for all positions to the left of the decimal point. The result is $370,000$.

Rounding Measures and Unit Conversions

When rounding measures to a specific accuracy, you must ensure the units match the target before rounding. For example, to round a value in centimetres to the nearest metre, convert to metres first by dividing by 100.

Worked Example: Rounding Measures

Write $36,548$ cm in metres, correct to the nearest metre.

1. Convert: $36,548 \div 100 = 365.48$ m.

2. Round: To round to the nearest whole metre, we look at the first digit after the decimal point, which is 4. Since 4 is less than 5, we do not round up. The result is $365$ m.

Truncation of Decimals

Truncation is different from rounding. Truncating a decimal to a specified number of places means simply cutting off all digits to the right of that position without any adjustment to the final digit.

Worked Example: Truncation

Truncate $3.45699$ after 3 decimal places. We look at the first three digits after the decimal point (4, 5, 6) and remove the rest. No correction is made despite the next digit being 9. The result is $3.456$.

Error Intervals and Inequality Notation

Error intervals define the range of possible values that a number could have been before it was rounded or truncated. We use inequality notation to express these intervals. Usually, the lower bound is inclusive (uses $\leq$) and the upper bound is exclusive (uses $<$).

Rounding Intervals

If a number $x$ is written as 3.6 correct to 1 d.p., the interval is $3.55 \leq x < 3.65$. The smallest value that rounds to 3.6 is 3.55. Any value exactly at 3.65 would round up to 3.7, which is why the upper bound is strictly less than 3.65.

Truncation Intervals

If a number $x$ is truncated to 1 d.p. as 3.6, the interval is $3.6 \leq x < 3.7$. The smallest possible value is 3.6 itself. Any value up to but not including 3.7 (such as 3.699...) would truncate to 3.6. Once the value reaches 3.7, the truncated version becomes 3.7.