Rounding Calculator
Round any decimal value to a chosen precision, from thousandths to thousands, with the mathematical floor and ceiling shown alongside the rounded result.
How It's Calculated
Formula
\text{rounded} = \mathrm{round}\!\left(\frac{v}{10^{p}}\right) \times 10^{p}Rounding a value to a given precision means finding the nearest multiple of that precision — rounding 123.456 to the nearest hundredth finds the nearest multiple of 0.01, which is 123.46. This calculator supports precision targets from thousandths up through thousands: positive targets (tenths, hundredths, thousandths) round to decimal places, and negative-magnitude targets (tens, hundreds, thousands) round to whole-number place values. Three related operations are shown side by side: rounding finds the nearest value at the chosen precision (ties round away from zero, so 2.5 rounds to 3 and -1.5 rounds to -2); floor always rounds down toward negative infinity, regardless of sign (the floor of -1.2 at the nearest integer is -2, not -1); and ceiling always rounds up toward positive infinity (the ceiling of -1.2 is -1). This calculator uses exact decimal arithmetic rather than native floating-point math, which matters for edge cases like 1.005 — floating point can't represent 1.005 exactly, so naive rounding silently produces 1.00 instead of the mathematically correct 1.01.
Worked Examples
Round 123.456 to the nearest hundredth
- Nearest hundredth means the nearest multiple of 0.01
- 123.456 sits between 123.45 and 123.46
- 123.456 is closer to 123.46
- Result: 123.46
Round 123.456 to the nearest ten
- Nearest ten means the nearest multiple of 10
- 123.456 sits between 120 and 130
- 123.456 is closer to 120
- Result: 120
Frequently Asked Questions
What's the difference between rounding, floor, and ceiling?
Rounding finds the nearest value at the chosen precision, going up or down depending on which is closer (with ties breaking away from zero). Floor always moves toward negative infinity — it never rounds up, even if the value is very close to the next multiple. Ceiling always moves toward positive infinity — it never rounds down. For a positive number like 4.7 rounded to the nearest integer, floor gives 4 and ceiling gives 5. For a negative number like -4.7, floor gives -5 (further from zero) and ceiling gives -4 (closer to zero) — floor and ceiling are defined by direction on the number line, not by distance from zero.
Why does 1.005 round to 1.01 here, when some calculators show 1.00?
The number 1.005 cannot be represented exactly in standard binary floating-point (the format behind JavaScript's native `Number` type) — it's actually stored as a value fractionally less than 1.005. A calculator that rounds using native floating-point math inherits that tiny error and can round down to 1.00 instead of the mathematically correct 1.01. This calculator avoids that by using exact decimal arithmetic instead of native floating point, so precision-sensitive edge cases like this round the way a human doing the math by hand would expect.
Which way do ties round — up or down?
This calculator rounds ties (exactly halfway between two values) away from zero: 2.5 rounds to 3, and -2.5 rounds to -3. This matches everyday rounding convention and how the rest of this project's calculators round currency values, rather than 'round half to even' (banker's rounding), which some other tools use.