Rounding Calculator

Rounding is a comparison, not a rule

To round a number you only need two candidates: the multiple of your chosen place just below it, and the one just above. 3.472 to one decimal place sits between 3.4 and 3.5, and it is 0.072 above the first and 0.028 below the second - so 3.5 wins. "Look at the next digit and go up if it is 5 or more" is a shortcut for exactly that comparison, and it is a very good one.

The shortcut only becomes a real decision at the halfway point, where neither neighbour is nearer. There the answer genuinely depends on which convention you use, and there are six in common use: school rounding sends a tie away from zero, a bank sends it to the even neighbour, a spreadsheet's FLOOR always goes down, and truncation just drops the rest. This page shows all six side by side on your number, because that is the question a rounding page ought to answer.

Seeing a tie at all needs exact arithmetic. In binary floating point 2.675 is really 2.67499999999999982…, which is why nearly every calculator, spreadsheet and programming language rounds it to 2.67 and insists it was never a tie. Here the number is held as an exact fraction, so 2.675 is a tie, the page says so, and the article explains why your other calculator disagreed.

What to notice

• The two neighbours are always multiples of the place you are rounding to. To the nearest ten they are 1270 and 1280; to two decimal places, 2.67 and 2.68.
• Only the distance matters, not the digits. The digit rule works because a digit of 5 or more IS the far half of the interval - it is a shorthand for the comparison, not a separate law.
• Significant figures count from the first non-zero digit, not from the decimal point. 0.004567 to two significant figures is 0.0046 - the leading zeros are placeholders, not figures.
• The conventions only ever disagree on a tie. Off a tie, half-up, half-even and half-down all give the same answer, which is why most people never notice there is more than one rule.
• The school rule is "away from zero", not "towards positive infinity". That is why -2.5 rounds to -3 under it, while a ceiling function would give -2.
• Rounding twice is not the same as rounding once. 2.446 to two decimals is 2.45, and rounding THAT to one gives 2.5 - but rounding the original straight to one decimal gives 2.4.

How to round a number

1. 1

   Decide what you are rounding to

   A number of decimal places, a place value (tens, hundreds), or a number of significant figures. Everything after this is the same work whichever you chose.

2. 2

   Find the two neighbours

   The multiple of that place just below your number, and the one just above. For 3.472 to one decimal place: 3.4 and 3.5.

3. 3

   See which is nearer

   Either compare the distances (0.072 against 0.028) or use the shortcut: look at the first digit past the place, and go up if it is 5 or more. Both answer the same question.

4. 4

   If it is exactly halfway, apply a convention

   The school rule rounds away from zero: 2.5 goes to 3 and -2.5 goes to -3. Statisticians and banks often round to the even neighbour instead, so 2.5 goes to 2 and 3.5 goes to 4.

5. 5

   Keep the place you rounded to

   Rounding 1274 to the nearest hundred gives 1300, not 13. The zeros are load-bearing: they say which place the answer is accurate to.

The same numbers, rounded four ways

School rounding (5 or more goes up) applied to each place. Note the numbers that do not move: they are already exact multiples of that place.

Worked examples

3.472

2.675

-2.5

0.004567

9.96

149.5

Common mistakes

• Rounding in stages. Round from the ORIGINAL number every time: 2.446 taken to two decimals and then to one gives 2.5, but rounded straight to one decimal it is 2.4.
• Dropping the place-holding zeros. 1274 to the nearest hundred is 1300; writing 13 changes the number by a factor of a hundred.
• Counting significant figures from the decimal point. In 0.004567 the first significant figure is the 4, so two significant figures is 0.0046 and not 0.00.
• Assuming the calculator's tie behaviour is the maths. A spreadsheet's ROUND rounds a tie away from zero, Python's round() rounds it to even, and both are defensible - the disagreement is about convention, not correctness.
• Reading "round up" as "towards positive infinity" for negatives. Under the school rule -2.5 goes to -3, because the digit rule works on size and ignores sign.
• Forgetting that a rounded number is an approximation. Once you have written 3.5, the 0.028 is gone for good - so round at the END of a calculation, never in the middle.

Rounding FAQ