Long division

In arithmetic, long division is a method for division of two real numbers. It requires only the means to write the numbers down, and it simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

Another form of long division is used for dividing polynomials - this process can be simplified using synthetic division.

This is long division notation for 500 ÷ 4 = 125:

The method involves several steps:

1. Write the dividend and divisor in this form:

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Write the multiple divided by the divisor (4 divided by 4 = 1) above the line over the leftmost digit of the dividend.

3. Subtract the digit under the dividend from the digit used in the dividend. Write the result (remainder) (5 - 4 = 1) under the bottom digit, then drop the zero (the second digit) to the right of it.

4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit.

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

=See also=

• Polynomial long division