Rounding Numbers: Rules, Ties and How Errors Build Up

The basic rule

Decide which place you are keeping, then look only at the digit immediately to its right. If that digit is 0 to 4, leave the kept digit alone (round down). If it is 5 to 9, add 1 to the kept digit (round up). Digits beyond that one do not matter, except to tell you whether a 5 is exactly a tie.

• 3.14159 to 2 decimal places: the next digit is 1, so the result is 3.14.
• 7.865 to 1 decimal place: the next digit is 6, so round up to 7.9.
• 2.995 to 2 decimal places: the next digit is 5, so round up. The 9 carries over, giving 3.00. Keep the trailing zeros to show the precision.
• 2,847 to the nearest hundred: the tens digit is 4, so the result is 2,800. To the nearest ten it is 2,850.

Decimal places versus significant figures

Decimal places count digits after the decimal point. Significant figures count digits starting from the first non-zero digit, wherever the point is. Significant figures suit very large and very small numbers, because leading zeros do not count.

Worked example: round 0.004572 to 2 significant figures. The first non-zero digit is 4, so the two significant digits are 4 and 5. The next digit is 7, so round up: 0.0046. Now round 4,572 to 2 significant figures: the digits 4 and 5 stay, the 7 rounds the 5 up, and zeros fill the rest, giving 4,600.

A measurement should not be reported with more precision than the inputs justify. If a room is measured as 4.3 m by 2.1 m (two significant figures each), the product 4.3 × 2.1 = 9.03 should be reported as about 9.0 m².

What happens at an exact tie

A tie occurs when the discarded part is exactly half, as in 2.5 rounded to a whole number. The basic rule says to round up, and that is the round half up convention taught in most schools. A second method, round half to even (sometimes called banker's rounding), sends a tie to the nearest even digit.

Always rounding ties upward pushes totals slightly high over many values. Half to even is as likely to round down as up, so these small biases tend to cancel. That is why it is the default rounding mode in the IEEE 754 standard for floating-point arithmetic, and why Python's built-in round() gives round(2.5) = 2 and round(3.5) = 4.

Software does not agree on ties or on negative numbers, and many decimals such as 2.675 are stored as binary approximations that sit just below the tie. If it matters, check how your spreadsheet or programming language rounds before relying on it.

How rounding errors build up

Each rounding step discards a little information, and in a chain of operations those small losses add up and can be magnified.

Worked example: compute 1/7 × 49, whose exact answer is 7. If you round 1/7 to 0.14 first, the product is 0.14 × 49 = 6.86, which is off by 2%. With 0.1429 the product is 7.0021, much closer. Keeping a few extra digits throughout costs nothing and protects the final answer.

The same thing happens when splitting. Divide 100 into three equal shares and each is 33.33 when rounded to cents, but 3 × 33.33 = 99.99, so one cent has vanished. Percentages show it too: three shares of 33.3% total 99.9%. In reports, a note such as "figures may not sum due to rounding" is honest. For money, the leftover cent is usually assigned to one share.

A sensible policy

• Round at the end, not in the middle. Carry at least two extra digits through intermediate steps.
• Round in the direction that suits the purpose. If a job needs 12.2 boxes of tile, you buy 13, because rounding down leaves you short. If you can afford 7.8 items, you buy 7.
• Match the precision to the input. Do not report seven digits from a figure measured to two.
• State what you did. "Rounded to the nearest cent" or "to 3 significant figures" avoids confusion.

To check a rounded value, use the significant figures calculator, and for very large or small values, the scientific notation calculator shows the same number in compact form.