# Rounding and significant figures calculator

> Round a number to decimal places, significant figures or the nearest ten or hundred under nine rules, including banker's rounding, floor and ceiling.

Interactive version: https://www.calcopenly.com/math/rounding-calculator
Subject: Math calculators

Rounding replaces a number with the nearest value that has a chosen number of decimal places or significant figures. The calculator finds the two candidates either side of the number at that precision and applies one of nine rules: the five “half” rules (up, to even, down, toward +∞, toward −∞) differ only when the number sits exactly halfway, while floor, ceiling, truncate and away-from-zero always move in one direction.

Lab reports round measurements to significant figures, prices are rounded to two decimal places, and programmers check which rule their language uses. The default, 1234.5678 to 2 decimal places, gives 1234.57, a change of +0.0022; the table lists the result under all nine rules at once. Negative decimal places round to tens (−1), hundreds (−2) and so on.

The rounded result keeps significant trailing zeros, so 999.96 to one place shows 1000.0, not 1000. Scientific notation shows exactly the significant figures kept, which removes the ambiguity of trailing zeros in a number like 1200.

## Inputs

- **Number**
- **Round to** (options: Decimal places, Significant figures)
- **How many**: Negative decimal places round to tens (−1), hundreds (−2) and so on
- **Rounding rule** (options: Half up (school rounding), Half to even (banker's), Half down, Half toward +∞, Half toward −∞, Floor (always down), Ceiling (always up), Truncate (toward zero), Away from zero)

## Results

- Rounded — main result
- Without trailing zeros
- Rounding error (rounded − original)
- Scientific notation
- Engineering notation

## Formula

$$
\operatorname{round}_d(x) = \frac{\operatorname{rule}(x \cdot 10^{d})}{10^{d}}
$$

## Worked examples

### 1234.5678 to 2 decimal places

- Number: 1234.5678
- Round to: Decimal places
- How many: 2
- Rounding rule: Half up (school rounding)
- **Rounded: 1234.57**
- **Rounding error (rounded − original): 0.0022**
- Checked against: Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP)

### Banker's rounding of a tie, 2.345

- Number: 2.345
- Round to: Decimal places
- How many: 2
- Rounding rule: Half to even (banker's)
- **Rounded: 2.34**
- Checked against: Python Decimal('2.345').quantize(Decimal('0.01'), ROUND_HALF_EVEN) = 2.34

### Same tie rounded half up

- Number: 2.345
- Round to: Decimal places
- How many: 2
- Rounding rule: Half up (school rounding)
- **Rounded: 2.35**
- Checked against: Python Decimal quantize with ROUND_HALF_UP = 2.35

### 0.0012345 to 3 significant figures

- Number: 0.0012345
- Round to: Significant figures
- How many: 3
- Rounding rule: Half up (school rounding)
- **Rounded: 0.00123**
- **Scientific notation: 1.23 × 10⁻³**
- Checked against: Python Context(prec=3, rounding=ROUND_HALF_UP).plus(Decimal('0.0012345')) = 0.00123

### 1234.5 to 2 significant figures

- Number: 1234.5
- Round to: Significant figures
- How many: 2
- Rounding rule: Half up (school rounding)
- **Rounded: 1200**
- **Engineering notation: 1.2 × 10³**
- Checked against: Python Context(prec=2).plus(Decimal('1234.5')) = 1.2E+3

### Floor of −2.5

- Number: -2.5
- Round to: Decimal places
- How many: 0
- Rounding rule: Floor (always down)
- **Rounded: −3**
- Checked against: Python Decimal('-2.5').quantize(Decimal('1'), ROUND_FLOOR) = -3

## Questions

### How do you round to significant figures?

Count from the first non-zero digit, keep the number of digits you need, and round on the next one. For 0.0012345 to 3 significant figures, skip the leading zeros, keep 1, 2 and 3, and look at the 4: it is below 5, so the answer is 0.00123, or 1.23 × 10⁻³. Rounding 1234.5 to 2 significant figures gives 1200, where the two zeros only hold place value.

### What is banker's rounding?

Banker's rounding, or round half to even, sends an exact tie to whichever candidate ends in an even digit: 2.345 to two places becomes 2.34, and 2.355 becomes 2.36. Ties then go up and down equally often, so long sums do not drift upward. NIST SP 811 (B.7.1) gives this as its rule for discarding a 5 followed by zeros, and it is the default in IEEE 754 arithmetic and Python's round(), where round(2.5) is 2.

### Does 5 round up or down?

Under school rounding (half up) an exact 5 rounds up, away from zero: 2.345 to two places is 2.35, and −2.5 to a whole number is −3. Under banker's rounding the same ties go to the even digit, giving 2.34 and −2. The rule matters only for exact ties; 2.3451 becomes 2.35 under both because it lies above halfway.

### What is the difference between floor, ceiling and truncate?

Floor always moves down toward −∞, ceiling always up toward +∞, and truncation drops the extra digits, which moves toward zero. Floor and truncate agree for positive numbers but split for negatives: −2.5 to a whole number floors to −3, truncates to −2, and has a ceiling of −2. None of them looks at which candidate is nearer, so 1.99 floors to 1.

### How do you round to the nearest ten, hundred or thousand?

Enter negative decimal places: −1 rounds to tens, −2 to hundreds and −3 to thousands. 1234.5678 to −2 places is 1200 under every half rule. 1250 sits exactly halfway between 1200 and 1300, so it becomes 1300 under half up but 1200 under banker's rounding, because 2 is the even digit.

### How accurate is the rounding and significant figures calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “1234.5678 to 2 decimal places” is checked against Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP).

### Where does the method come from?

IEEE 754-2019 — rounding-direction attributes (roundTiesToEven, roundTiesToAway, toward ±∞, toward 0); NIST SP 811, Appendix B.7 — rounding numerical values; Python decimal module — rounding modes.

## Sources

- [IEEE 754-2019 — rounding-direction attributes (roundTiesToEven, roundTiesToAway, toward ±∞, toward 0)](https://standards.ieee.org/ieee/754/6210/)
- [NIST SP 811, Appendix B.7 — rounding numerical values](https://www.nist.gov/pml/special-publication-811)
- [Python decimal module — rounding modes](https://docs.python.org/3/library/decimal.html#rounding-modes)