Long Division Calculator

Long division splits a big division into small ones you can do in your head. You work from left to right, one digit at a time, and repeat the same four steps: divide, multiply, subtract and bring down. The number you divide is the dividend, the number you divide by is the divisor, the answer is the quotient and what is left over is the remainder.

The formula

dividend = quotient × divisor + remainder, where 0 ≤ remainder < divisor
each round: digit = (current number) ÷ divisor, rounded down
new number = (current number − digit × divisor) × 10 + next digit
mixed number = quotient and remainder/divisor
decimal: keep bringing down zeros after the point

Worked example

437 ÷ 12. 12 doesn’t go into 4, so start with 43. 43 ÷ 12 = 3 (because 3 × 12 = 36 and 4 × 12 = 48 is too big). Write 3 on top, then 43 − 36 = 7. Bring down the 7 to make 77. 77 ÷ 12 = 6, 6 × 12 = 72 and 77 − 72 = 5. There are no digits left, so the answer is 36 remainder 5. Check: 36 × 12 = 432 and 432 + 5 = 437. As a mixed number it is 36 5/12.

To carry on as a decimal, put a point after 36 and bring down zeros: 50 ÷ 12 = 4 (50 − 48 = 2), 20 ÷ 12 = 1 (20 − 12 = 8), 80 ÷ 12 = 6 (80 − 72 = 8). The remainder 8 has come back, so the 6 repeats for ever: 36.41666…, written 36.416. Rounded to 2 places it is 36.42, because the next digit (6) is 5 or more.

Decimals in the numbers

You can’t easily divide by a decimal, so make the divisor whole first. Move its decimal point to the right until it is a whole number, and move the point in the dividend the same number of places (this multiplies both by the same 10, 100 or 1,000, so the answer doesn’t change). For 12.5 ÷ 0.25, move both points 2 places: 1250 ÷ 25 = 50. If the dividend still has a point, like 127.5 ÷ 5, put the point straight above it in the answer and carry on: 25.5.

For a whole-number answer with a remainder, move the point in both numbers until both are whole. Then move the remainder’s point back the same number of places: 7.5 ÷ 2 becomes 75 ÷ 20 = 3 R 15, so the real remainder is 1.5 (check: 3 × 2 + 1.5 = 7.5).

Negative numbers

Divide the numbers without their signs, then fix the sign: if exactly one number is negative, the answer is negative; if both are, it is positive. For the remainder there are two common rules. Most calculators and computer languages give the remainder the same sign as the number divided: −437 ÷ 12 = −36 R −5. In number theory the remainder is never negative: −437 ÷ 12 = −37 R 7. Both pass the check: −36 × 12 − 5 = −437 and −37 × 12 + 7 = −437.

Short division

Short division is the same method with the multiplying and subtracting done in your head. Write the small remainder in front of the next digit instead of below. For 437 ÷ 12: 12 into 4 won’t go, so carry 4 to make 43; 12 into 43 is 3 remainder 7, so carry 7 to make 77; 12 into 77 is 6 remainder 5. Answer: 36 r 5. It is quickest when you know the divisor’s times table.

Polynomial long division

Polynomials divide the same way. Write both with the highest power first and fill any missing powers with 0. Divide the first term by the divisor’s first term, multiply the whole divisor by that, subtract and bring down the next term. For (x³ − 2x² − 4) ÷ (x − 3): x³ ÷ x = x², and x²(x − 3) = x³ − 3x² leaves x² + 0x. Then x² ÷ x = x, and x(x − 3) = x² − 3x leaves 3x − 4. Then 3x ÷ x = 3, and 3(x − 3) = 3x − 9 leaves 5. So the answer is x² + x + 3 with remainder 5.

Synthetic division is a shortcut when you divide by x − c. Write c (here 3) and the coefficients 1, −2, 0, −4. Bring down the 1, multiply by 3 and add to the next coefficient, and repeat: 1, 1, 3, 5. The last number is the remainder (5) and the others are the quotient’s coefficients (x² + x + 3). By the remainder theorem, the remainder is also the value of the polynomial at x = 3: 27 − 18 − 4 = 5.

Divisibility shortcuts

All answers are exact: the calculator works with whole numbers of any size up to 120 digits, so nothing is rounded until you choose a number of decimal places. Repeating decimals are found by watching for a remainder that comes back (up to 120 decimal places in the written layout).