Rounding Calculator

About the Rounding Calculator

Round a number — or a whole list — to N decimal places, N significant figures or the nearest multiple of any step: 10, 100, 0.05, 0.25 or 1/8. Every common rounding mode is shown side by side, with what it does and where it is used: half away from zero (the school rule), half up, half down, half to even (banker’s rounding), half toward zero, ceiling, floor, truncate and away from zero. The modes only disagree on ties and in the direction they push, and the explanation says which case your number is.

The arithmetic is exact decimal arithmetic, so 2.675 is exactly 2.675. In ordinary binary floating point it is stored as 2.67499999…, which is why Python’s round(2.675, 2) gives 2.67. For a list, the tool also adds up the rounded values in each mode, so you can see how much the rounding shifts the total.

Common uses

• Rounding prices, measurements or survey results consistently, and choosing the rule a standard or teacher asks for.
• Checking how Excel, JavaScript or Python will round a value.
• Rounding inch measurements to the nearest 1/8 or 1/16, or minutes to the nearest 15.
• Seeing how much rounding every item in a list changes the total.

The rounding modes

• Half away from zero: ties go away from zero — 2.5 → 3, −2.5 → −3. This is the rule usually taught at school, and Excel’s ROUND follows it.
• Half up: ties go toward +∞ — 2.5 → 3, −2.5 → −2. JavaScript’s Math.round works this way (MDN).
• Half down: ties go toward −∞ — 2.5 → 2, −2.5 → −3.
• Half to even (banker’s rounding): ties go to the even neighbour — 2.5 → 2, 3.5 → 4, −2.5 → −2. Python’s round() uses it (Python docs).
• Half toward zero: 2.5 → 2, −2.5 → −2.
• Ceiling, floor, truncate and away from zero always move one way, tie or not: ceiling 2.1 → 3, floor −2.1 → −3, truncate −2.9 → −2, away from zero 2.1 → 3. Excel’s ROUNDUP rounds away from zero and ROUNDDOWN toward zero.

The names are not used consistently: Java’s RoundingMode and Python’s decimal module call half away from zero “half up”.

Why half to even?

Rounding every tie up pushes totals and averages up a little. Half to even sends half of the ties up and half down, so on average there is no bias. It is the default rounding of IEEE 754 binary floating point (roundTiesToEven, IEEE 754-2019 §4.3) and the rule for measurement data in ASTM E29 and in NIST’s guide to the SI, SP 811, appendix B.7. The list mode shows the effect: 1.25 + 2.35 + 3.45 + 4.55 = 11.6, but rounding each to one place half away from zero gives a total of 11.8, while half to even gives 11.6.

Round once, from the original value

Rounding in stages can give a different result from rounding once. 9.46 to one decimal place is 9.5, and 9.5 to a whole number (half to even) is 10 — but 9.46 rounded straight to a whole number is 9. Keep full precision while you calculate and round only the final answer.

Significant figures and multiples

Significant figures are counted from the first non-zero digit: 0.004567 to 2 significant figures is 0.0046. When a rounded whole number ends in zeros, the zeros may or may not be significant — 1499.5 to 3 figures is 1500 — so the calculator also writes it as 1.50 × 10³ (and 150. with a decimal point when the last zero counts). For a multiple, any positive step works, including fractions: 1/8 or 1/16 for inches, 0.25 for quarters, 15 for minutes.

Limitations

• Up to 1,000 numbers at a time, each with up to 100 digits. Decimal places can be from −20 (round to the left of the point) to 100; up to 100 significant figures.
• A step that is not a terminating decimal, such as 1/3, gives fractions (10 1/3) with their decimal expansion.
• The table lists where a mode is used only where that is documented; software can also offer other options, so check the documentation of the program you use.

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