Long division
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In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is 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.
A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).
In long division notation, 500 ÷ 4 = 125 is denoted as follows:
- <math>
\begin{matrix} \quad 125\\ 4\overline{)500}\\ \end{matrix} <math>
The method involves several steps:
1. Write the dividend and divisor in this form:
- <math>4\overline{)500}<math>
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.
- <math>
\begin{matrix} 1\\ 4\overline{)500}\\ 4 \end{matrix} <math>
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.
- <math>
\begin{matrix} 1\\ 4\overline{)500}\\ \underline{4}\\ \;\,10 \end{matrix} <math>
4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit.
- <math>
\begin{matrix} \,\,12\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20 \end{matrix} <math>
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.
- <math>
\begin{matrix} \quad 125\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20\\ \quad\;\,\underline{20}\\ \qquad0 \end{matrix} <math>