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Rounding Calculator
Round a number to a chosen number of decimal places, or to the nearest ten or hundred, and compare how half-up, banker's rounding, floor, ceiling and truncation each treat the same input.
The rounding methods compared
Rounding replaces a number with a nearby one that has fewer digits. Nearly every method agrees unless the digits after the cut are exactly a half, or you ask for a directional method.
- Half up (half away from zero): the classroom rule. 2.5 becomes 3 and −2.5 becomes −3.
- Banker's rounding (half to even): a half goes to whichever neighbour ends in an even digit, so 2.5 becomes 2 and 3.5 becomes 4. This is the default mode in the IEEE 754 standard and avoids a steady upward bias when many halves are added.
- Half down: halves move toward zero, so 2.5 becomes 2.
- Floor and ceiling: always go to the lower or the higher whole step, so floor of −1.2 is −2 and ceiling of 1.2 is 2.
- Truncate: drops the extra digits, moving toward zero, so −1.9 becomes −1.
Decimal places, tens and hundreds
A positive number of decimal places keeps that many digits after the point: 3.14159 to two places is 3.14. Zero places gives a whole number. Negative values round to the left of the point, so −1 rounds to the nearest ten, −2 to the nearest hundred and −3 to the nearest thousand. For example 1,234.5678 rounded to −2 places is 1,200. The result keeps trailing zeros after the decimal point, so 2.5 to two places reads 2.50, which signals the precision you chose. If you want the number of meaningful digits instead of decimal places, use the significant figures calculator.
Why 2.675 is a special case
Many software tools round 2.675 to 2.67, which surprises people who expect 2.68. The reason is that computers store most decimals in binary, and 2.675 cannot be stored exactly. The nearest stored value is slightly below 2.675, so a tool that rounds that stored value goes down. This calculator reads the digits you type as decimal text and rounds them with integer arithmetic, so 2.675 is exactly 2.675 and rounds half up to 2.68. Be aware of this difference when you compare results with a spreadsheet or a program, which may apply the stored binary value instead.
Pitfalls and good practice
- Rounding twice. Rounding 1.4999 to 1.5 and then to 2 gives a different answer from rounding 1.4999 to 1 directly. Round once, from the original value.
- Rounding too early. Keep full precision through a calculation and round only the final answer.
- Assuming one rule everywhere. Accounting, tax, statistics and programming languages follow different conventions for halves, and some rules for money are set by regulation that varies by country. Check the rule your task requires.
- Confusing floor with truncate. They agree for positive numbers but differ for negatives.
Worked example
Round 2.675 to two decimal places. The digit after the cut is exactly 5, so the method chosen decides the answer.
Inputs
| Number | 2.675 |
|---|---|
| Round to this many decimal places | 2 |
| Method | Half up (halves round away from zero) |
Result
Half up, 2 decimal places.
| Original | 2.675 |
|---|---|
| Rounded | 2.68 |
| Method | Half up |
| Rounding error (rounded − original) | +0.005 |
The same input under every method
| Method | Result |
|---|---|
| Half up | 2.68 |
| Banker's (half even) | 2.68 |
| Half down | 2.67 |
| Floor | 2.67 |
| Ceiling | 2.68 |
| Truncate | 2.67 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
What is banker's rounding?
Banker's rounding, also called round half to even, sends an exact half to the neighbour with an even last digit. So 2.5 rounds to 2 and 3.5 rounds to 4, which balances out over many values.
How do I round to the nearest ten or hundred?
Use a negative number of decimal places: −1 for the nearest ten and −2 for the nearest hundred. For example, 1,234 rounded to −2 places is 1,200.
What is the difference between floor and truncate?
Floor moves toward negative infinity, while truncate moves toward zero. They give the same answer for positive numbers, but −1.9 floors to −2 and truncates to −1.
Why does my spreadsheet round 2.675 differently?
Software that rounds the stored binary value of 2.675 sees a number slightly below the half and rounds down. This page uses the decimal digits you type, so the half is exact.
Does rounding change the value?
Yes. The rounding error row shows the exact difference between the rounded and original values, so you can see how much precision was given up.
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References and sources
- IEEE 754-2019, Standard for Floating-Point Arithmetic: defines the roundTiesToEven default and the directed rounding modes.
- Knuth, D. E., The Art of Computer Programming, Volume 2, section 4.2.2, on rounding.