# Калькулятор округления и значащих цифр

> Округляйте число до заданных десятичных или значащих цифр, ближайшего десятка или сотни по девяти правилам, включая округление к чётному, вниз и вверх.

Интерактивная версия: https://www.calcopenly.com/ru/math/rounding-calculator
Тема: Математические калькуляторы

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.

## Входные данные

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

## Результаты

- Rounded — основной результат
- Without trailing zeros
- Rounding error (rounded − original)
- Научная запись
- Engineering notation

## Формула

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

## Примеры с решением

### 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**
- Источник проверки: 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**
- Источник проверки: 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**
- Источник проверки: 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**
- **Научная запись: 1.23 × 10⁻³**
- Источник проверки: 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**
- Источник проверки: 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**
- Источник проверки: Python Decimal('-2.5').quantize(Decimal('1'), ROUND_FLOOR) = -3

## Вопросы

### 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.

### Насколько точен «Калькулятор округления и значащих цифр»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 8. Например, «1234.5678 to 2 decimal places» проверяется по источнику Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP).

### Откуда взята методика?

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.

## Источники

- [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)