# 舍入与有效数字计算器

> 按指定小数位、有效数字或最接近的十或百的倍数进行舍入。提供九种规则，包括四舍六入五成双、向下取整和向上取整。

交互版本：https://www.calcopenly.com/zh/math/rounding-calculator
主题：数学计算器

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.

## 输入

- **Number**
- **Round to** (选项：Decimal places, Significant figures)
- **How many**: Negative decimal places round to tens (−1), hundreds (−2) and so on
- **Rounding rule** (选项：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)

## 结果

- Rounded — 主要结果
- Without trailing zeros
- Rounding error (rounded − original)
- 科学计数法
- Engineering notation

## 公式

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

## 计算示例

### 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**
- 核验来源：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**
- 核验来源：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**
- 核验来源：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**
- **科学计数法: 1.23 × 10⁻³**
- 核验来源：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**
- 核验来源：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**
- 核验来源：Python Decimal('-2.5').quantize(Decimal('1'), ROUND_FLOOR) = -3

## 常见问题

### 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.

### “舍入与有效数字计算器”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：8。 例如，“1234.5678 to 2 decimal places”根据Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP)进行核验。

### 这种方法出自哪里？

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.

## 来源

- [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)