# 余りと小数に対応した割り算の筆算計算機

> Long division calculator that shows every step: quotient and remainder, the decimal quotient with its repeating block marked, and the fraction.

操作できるページ：https://www.calcopenly.com/ja/math/long-division-calculator
分野：数学の計算機

Long division splits a division into one small division per digit. Take the leading digits of the dividend, divide by the divisor, write the digit above, multiply, subtract, then bring down the next digit and repeat. When the divisor has decimals, both numbers are first multiplied by the same power of 10 so the divisor is a whole number.

For 1234 ÷ 7 the working gives 176 with remainder 2, since 7 × 176 = 1232. Carrying on past the decimal point with appended zeros gives 176.285714…, and because the remainder 2 comes back after six more digits, the block 285714 repeats forever. The layout marks that block with a bar.

A fraction's decimal either ends or repeats. It ends when the reduced denominator has no prime factors other than 2 and 5, as with 487 ÷ 32 = 15.21875. Otherwise the length of the repeating block is the smallest k for which 10ᵏ − 1 is divisible by q once its factors of 2 and 5 are removed: 6 for q = 7 and 96 for q = 97.

## 入力

- **Dividend**: The number being divided. Whole numbers or decimals, e.g. 4.71.
- **Divisor**: The number you divide by. Decimals are shifted to a whole number first.
- **Answer as** (選択肢：Decimal, Remainder)
- **Decimal places**: How far to carry the division when the decimal doesn't end or repeat sooner. The rounded quotient uses the same number of places.

## 結果

- Decimal quotient — 主な結果
- Quotient and remainder
- Whole-number quotient
- Remainder
- Rounded quotient
- Repeating block length (digits)
- As a mixed number

## 計算式

$$
a = b \times q + r,\quad 0 \le r < b \qquad \frac{a}{b} = q + \frac{r}{b}
$$

## 計算例

### 1234 ÷ 7 (default)

- Dividend: 1234
- Divisor: 7
- Answer as: Decimal
- Decimal places: 10
- **Decimal quotient: 176.(285714)**
- **Quotient and remainder: 176 R 2**
- **Whole-number quotient: 176**
- **Remainder: 2**
- **Repeating block length: 6 digits**
- **As a mixed number: 176 2/7**
- **Rounded quotient: 176.2857142857**
- 照合元：Python 3.8: divmod(1234, 7) = (176, 2); decimal.Decimal(1234) / 7 = 176.28571428571428571…; Fraction(1234, 7) = 1234/7

### 487 ÷ 32 (Calculator Soup worked example)

- Dividend: 487
- Divisor: 32
- Answer as: Remainder
- **Quotient and remainder: 15 R 7**
- **Whole-number quotient: 15**
- **Remainder: 7**
- **Decimal quotient: 15.21875**
- **Repeating block length: 0 digits**
- 照合元：Calculator Soup long division tutorial: 487 ÷ 32 = 15 R 7; 487/32 = 15.21875 by Python Fraction

### 65,321 ÷ 31 (Omni Calculator worked example)

- Dividend: 65321
- Divisor: 31
- Answer as: Remainder
- **Quotient and remainder: 2,107 R 4**
- **Whole-number quotient: 2,107**
- **Remainder: 4**
- **As a mixed number: 2,107 4/31**
- **Repeating block length: 15 digits**
- 照合元：Omni Calculator long division page: 65321 / 31 = 2107 r 4; period of 1/31 is 15 (ord₃₁ 10 = 15, Python pow(10, 15, 31) == 1)

### 4.71 ÷ 3.2 (decimal divisor)

- Dividend: 4.71
- Divisor: 3.2
- Answer as: Decimal
- Decimal places: 10
- **Decimal quotient: 1.471875**
- **Whole-number quotient: 1**
- **Remainder: 1.51**
- **Repeating block length: 0 digits**
- **As a mixed number: 1 151/320**
- 照合元：Calculator Soup long division with decimals example (4.71 ÷ 3.2 = 1.471… to 3 places); exact 471/320 = 1.471875 and 4.71 − 3.2 = 1.51 by Python Fraction

### 100 ÷ 7 (calculator.net example)

- Dividend: 100
- Divisor: 7
- Answer as: Decimal
- Decimal places: 10
- **Quotient and remainder: 14 R 2**
- **Decimal quotient: 14.(285714)**
- **As a mixed number: 14 2/7**
- 照合元：calculator.net long division page: 100 ÷ 7 = 14 R 2 = 14.285714…

### 1 ÷ 97: 96-digit repeating block (edge case)

- Dividend: 1
- Divisor: 97
- Answer as: Decimal
- Decimal places: 10
- **Whole-number quotient: 0**
- **Remainder: 1**
- **Repeating block length: 96 digits**
- **Rounded quotient: 0.0103092784**
- 照合元：Python: smallest k with pow(10, k, 97) == 1 is 96; Decimal(1)/97 = 0.01030927835051546…

## よくある質問

### How do you do long division step by step?

Repeat four moves: divide, multiply, subtract, bring down. For 487 ÷ 32: 48 ÷ 32 = 1, write 1, 1 × 32 = 32, 48 − 32 = 16; bring down 7 to make 167; 167 ÷ 32 = 5, 5 × 32 = 160, 167 − 160 = 7. No digits are left, so 487 ÷ 32 = 15 remainder 7. Check it: 32 × 15 + 7 = 487.

### How do you divide by a decimal?

Move the decimal point in the divisor to the right until it is a whole number, and move the point in the dividend the same number of places. For 4.71 ÷ 3.2, move both one place: 47.1 ÷ 32. The quotient stays the same because both numbers were multiplied by 10. Then divide as usual, putting the quotient's point directly above the dividend's: 47.1 ÷ 32 = 1.471875.

### How do you write a remainder as a decimal or fraction?

As a fraction, put the remainder over the divisor: 1234 ÷ 7 = 176 R 2 = 176 2/7. As a decimal, keep dividing: write a decimal point and zeros after the dividend and bring them down one at a time. 2/7 = 0.285714285714…, so 1234 ÷ 7 = 176.285714…, where the six digits 285714 repeat.

### How do you know when a decimal repeats?

Watch the remainders. Once only appended zeros are being brought down, each step depends only on the current remainder, and there are fewer possible remainders than the divisor. As soon as a remainder appears a second time, the digits between the two appearances repeat forever. Dividing by 7 always repeats within 6 digits; 1 ÷ 97 has a 96-digit repeating block.

### What is the remainder when dividing negative numbers?

It depends on the convention. This page divides the absolute values and gives the remainder the dividend's sign, as C and Java do: −17 ÷ 5 = −3 R −2, since 5 × (−3) − 2 = −17; Excel's QUOTIENT truncates to −3 the same way. The floored convention used by Python's // and % and by Excel's MOD gives −4 R 3 instead, since 5 × (−4) + 3 = −17.

### 「余りと小数に対応した割り算の筆算計算機」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：8。 例えば、「1234 ÷ 7 (default)」はPython 3.8: divmod(1234, 7) = (176, 2); decimal.Decimal(1234) / 7 = 176.28571428571428571…; Fraction(1234, 7) = 1234/7と照合しています。

### この計算方法の出典は何ですか？

Common Core State Standards for Mathematics, 6.NS.B.2 and 6.NS.B.3: divide multi-digit numbers and decimals using the standard algorithm; Hardy & Wright, An Introduction to the Theory of Numbers, §9.6 (period of a recurring decimal is the order of 10 modulo q); Wolfram MathWorld: Long division; Wolfram MathWorld: Repeating decimal.

## 出典

- [Common Core State Standards for Mathematics, 6.NS.B.2 and 6.NS.B.3: divide multi-digit numbers and decimals using the standard algorithm](https://www.thecorestandards.org/Math/Content/6/NS/)
- Hardy & Wright, An Introduction to the Theory of Numbers, §9.6 (period of a recurring decimal is the order of 10 modulo q)
- [Wolfram MathWorld: Long division](https://mathworld.wolfram.com/LongDivision.html)
- [Wolfram MathWorld: Repeating decimal](https://mathworld.wolfram.com/RepeatingDecimal.html)