Rounding Accuracy

Why numbers are rounded

An exact value is not always necessary or sensible. Rounding replaces a number with a nearby value that is easier to read, report or use while keeping its approximate size. For example, a crowd count might be reported to the nearest hundred, whereas a measured length might be recorded to the nearest millimetre.

The instruction tells you the required degree of accuracy. This might be a place value, such as the nearest ten; a number of decimal places; or a number of significant figures.

The basic rounding decision

First identify the digit that must be kept as the final digit. Then inspect the single digit immediately to its right:

• if it is 0, 1, 2, 3 or 4, leave the final kept digit unchanged;
• if it is 5, 6, 7, 8 or 9, increase the final kept digit by 1.

Remove the remaining digits. For a whole-number place value, replace removed digits before the decimal point with zeros so that the number keeps the correct magnitude.

For example, to round 823.4 to the nearest ten, the tens digit is 2 and the next digit is 3. The 2 remains unchanged, so the answer is 820. By contrast, 816 rounds to 820 because the units digit is 6, so the tens digit increases from 1 to 2.

Rounding can involve carrying. In 2,394 to three significant figures, the final kept digit is the 9 in the tens column. The following digit is 4, so the result is 2,390. If the number were 2,398, the following digit would be 8: rounding the 9 upwards carries into the hundreds column, giving 2,400.

Nearest whole number, ten, hundred and beyond

For a named place value, locate that column and use the digit immediately to its right:

• nearest whole number: inspect the tenths digit;
• nearest ten: inspect the units digit;
• nearest hundred: inspect the tens digit;
• nearest thousand: inspect the hundreds digit.

Round in one step from the original number. For instance, 14,489 to the nearest 10,000 is 10,000 because the thousands digit is 4. Repeatedly rounding it to the nearest ten, then hundred, then thousand and finally 10,000 could incorrectly produce 20,000.

Rounding to decimal places

Decimal places are counted from the decimal point. The first digit after it is the tenths digit, the second is the hundredths digit and the third is the thousandths digit.

To round 58.1666 to 2 decimal places, keep the hundredths digit: $58.16\underline{6}6$. The next digit is 6, so the hundredths digit rounds up and the result is $58.17$.

Zeros may be needed to display the requested accuracy. For example, $6.084$ to 1 decimal place is $6.1$, while $6.02$ to 1 decimal place is $6.0$. Writing $6$ instead would not show that the value had been given to one decimal place.

Rounding to significant figures

Significant figures describe precision across numbers of very different sizes. Start counting at the first non-zero digit, wherever it occurs. Once counting has begun, zeros between significant digits count too.

For $0.05072$, the initial zeros only locate the decimal point and are not significant. The first two significant digits are 5 and 0. The following digit is 7, so $0.05072$ rounded to 2 significant figures is $0.051$.

For 234 rounded to 1 significant figure, keep the 2 in the hundreds column. The next digit is 3, so the answer is 200. The zeros preserve the number's magnitude.

Decimal places and significant figures therefore answer different questions. In $0.004073$:

• to 2 significant figures, begin at 4 and obtain $0.0041$;
• to 2 decimal places, keep the hundredths digit and obtain $0.00$.

A written zero can convey precision. For example, $3.50$ has three significant figures: the final zero shows that the value has been stated to the nearest hundredth. For whole numbers, trailing zeros can be ambiguous, so standard form can make the intended precision explicit: $1.20\times10^3$ clearly has three significant figures.

Choosing an appropriate accuracy

Sometimes the question specifies the accuracy. If it asks for 3 significant figures, that instruction must be followed even if the calculator displays many more digits.

At other times, choose an accuracy suited to the context:

• a number of people, cars or objects usually needs to be a whole number;
• money is normally written to the nearest penny, or 2 decimal places, unless the context asks for another unit such as the nearest pound;
• a measured answer should not suggest much greater precision than the measurements used to calculate it;
• a probability is commonly left as an exact fraction where possible, or rounded only when requested.

Carry unrounded values through a multi-step calculation and round the final answer. Early rounding can accumulate error and change the result. A sensible final answer will often have a similar degree of accuracy to the least precise measurements supplied, but the wording and context of the question take priority.

Finally, state the accuracy when it is not already clear: for example, “$7.3$ cm to 1 decimal place”. This distinguishes an intentional rounded result from an unexplained shortened decimal.

Rounding rule

Identify the final digit to keep. Look one digit to its right:

• 0–4: keep the final digit unchanged;
• 5–9: increase it by 1.

Round directly from the original value.

Place-value rounding

• Nearest whole number: inspect tenths.
• Nearest ten: inspect units.
• Nearest hundred: inspect tens.
• Nearest thousand: inspect hundreds.

Replace discarded whole-number digits with zeros to preserve magnitude.

Decimal places

Count digits after the decimal point. The last written digit must be in the requested decimal place, so retain trailing zeros where necessary.

Example: $4.5817=4.582$ to 3 dp.

Significant figures

Begin with the first non-zero digit, then count onwards. Zeros after counting has begun can be significant.

Example: $0.05072=0.051$ to 2 sf.

Do not confuse sf with dp: sf begins at the first non-zero digit; dp begins immediately after the decimal point.

Appropriate accuracy

Follow any stated instruction. Otherwise use the context: whole objects need whole numbers, money commonly needs 2 dp, and measured results should not claim unjustified precision. Keep full values in working and round the final answer.