Division, one digit at a time
Long division breaks a big division into small steps you can do in your head: divide, multiply, subtract, bring down the next digit, and repeat. It is taught in school because it works for any size of number and shows exactly where the remainder comes from.
This long division calculator writes out each of those steps for whole numbers, then gives the quotient and remainder, the answer as a mixed fraction in its lowest terms, and the exact decimal with any repeating digits marked in brackets.
How to use it
- Type the dividend, the number being divided.
- Type the divisor, the number you are dividing by. It cannot be 0.
- Read the quotient and remainder, written as 1,305 R 4.
- Open the working to follow each step, then check the decimal and the fraction underneath.
The rule behind it
dividend = divisor × quotient + remainder
0 ≤ remainder < divisor
decimal: keep dividing the remainder, adding a 0 each time, until it is 0 or repeatsEvery division of whole numbers ends in one of two ways. Either the remainder reaches 0 and the decimal stops, or a remainder comes round again and the digits repeat for ever. Brackets mark the repeating part, so 0.41(6) means 0.41666…
Worked example: 7,834 ÷ 6
| Step | Divide | Multiply | Subtract |
|---|---|---|---|
| Start with 7 | 7 ÷ 6 = 1 | 1 × 6 = 6 | 7 − 6 = 1 |
| Bring down 8: 18 | 18 ÷ 6 = 3 | 3 × 6 = 18 | 18 − 18 = 0 |
| Bring down 3: 3 | 3 ÷ 6 = 0 | 0 × 6 = 0 | 3 − 0 = 3 |
| Bring down 4: 34 | 34 ÷ 6 = 5 | 5 × 6 = 30 | 34 − 30 = 4 |
Reading the digits down gives 1,305 R 4. As a fraction that is 1,305 4/6, which reduces to 1,305 2/3. As a decimal, the remainder of 4 keeps producing 6s, so the answer is 1,305.(6), or 1,305.666…
Check it: 1,305 × 6 = 7,830, and 7,830 + 4 = 7,834.
Which fractions repeat?
A fraction in its lowest terms ends neatly as a decimal only when its bottom number has no prime factors other than 2 and 5. So 3/8 = 0.375 and 7/20 = 0.35 stop, while 1/3, 1/6 and 1/7 repeat for ever. The length of the repeating part is always shorter than the divisor: 1/7 repeats six digits, the most it can.
Tips for doing it by hand
- Remember the cycle: divide, multiply, subtract, bring down.
- If the divisor does not go into a step at all, write 0 in the answer and bring down the next digit.
- Each subtraction must leave less than the divisor; if not, the division step was too small.
- For fractions such as 3/8, the fraction calculator works with the parts directly.
Frequently asked questions
How do you do long division?
Work from left to right. Divide the first part of the dividend by the divisor, write the result above, multiply it back, subtract, and bring down the next digit. Repeat until every digit is used. Whatever is left at the end is the remainder. The calculator writes these steps out for you.
What does R mean in a division answer?
R stands for remainder: what is left over when the divisor does not go in exactly. In 7,834 ÷ 6 = 1,305 R 4, six goes into 7,834 a whole 1,305 times with 4 left over. The remainder is always smaller than the divisor.
How do I turn a remainder into a decimal?
Put a decimal point after the quotient, add a zero to the remainder, and keep dividing. A remainder of 4 out of 6 becomes 40 ÷ 6 = 6 remainder 4, which repeats, so the decimal part is 0.666…. The calculator marks repeating digits in brackets.
What is a repeating decimal?
It is a decimal whose digits settle into a pattern that goes on for ever, such as 1 ÷ 3 = 0.333… or 1 ÷ 7 = 0.142857142857…. It happens when a remainder comes round a second time. The calculator writes these as 0.(3) and 0.(142857).
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