Long division

From CryptoDox, The Online Encyclopedia on Cryptography and Information Security

For the album by Rustic Overtones, see Long Division.

In arithmetic, long division is a procedure which breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. Long division requires no advanced technology nor mental gymnastics, merely paper and pencil (or any similar means for writing). It is very powerful, enabling computations involving arbitrarily large numbers to be performed by following a series of simple steps.

Contents 1 Notation 2 Example 3 Division algorithm 4 Generalizations 4.1 Rational numbers 4.2 Polynomials 5 Standards-based mathematics reform 6 See also 7 External links

Notation

Long division does not use the / (slash) or ÷ (obelus) signs, instead displaying the dividend, divisor, and (once it is found) quotient in a tableau. An example is shown below, representing the division of 500 by 4 (with a result of 125).

Example

The procedure involves several steps. As an example, consider the problem of 950 divided by 4:

1. The dividend and divisor are written in the long division tableau:

Now instead of dividing the whole dividend (950) by the divisor (4) we will simply divide each digit of the dividend by the divisor, one at a time, starting from the most significant (leftmost) digit:

2.The first number to be divided by the divisor (4) is the leftmost digit (9) of the dividend. Ignoring any remainder, we write the integer part of the result (2) above the division bar over the leftmost digit of the dividend.

Since we ignored the remainder, though, we have not accounted for the leftmost place entirely. That is to say: 4•2 is merely 8, and the relevant digit of the dividend was 9. Thus we subtract 8 from 9, yielding 1, to tell us how much of the leftmost place remains unaccounted for.

3. We "bring down" this unaccounted-for remainder from the leftmost place (1) and copy the next digit of the dividend (5) to the right.

4. Next we repeat steps 2 and 3, using the newly created bottom number (15) as the active part of the dividend, dividing it by the divisor (4) and writing the results as before above and under the next digit of the dividend.

5. We repeat step 4 until there are no digits remaining in the dividend. The number written above the bar (237) is the quotient, and the result of the last subtraction is the remainder for the entire problem (2).

The answer to the above example is expressed as 237 with remainder 2. Alternatively, one can continue the above procedure to produce a decimal answer. We continue the process by adding a decimal and zeroes as necessary to the right of the dividend, treating each zero as another digit of the dividend. Thus the next step in such a calculation would give the following:

Division algorithm

The above procedure relies on the division algorithm, which states that given any two integers a and d, with d ≠ 0, there exist unique integers q and r such that a = qd + r and 0 ≤ r < |d |, where |d | denotes the absolute value of d.

Generalizations

Rational numbers

Long division of integers can easily be extended to include non-integer dividends, as long as they are rational. This is because every rational number has a recurring decimal expansion. The procedure can also be extended to include divisors which have a finite or terminating decimal expansion (i.e. decimal fractions). In this case the procedure involves multiplying the divisor and dividend by the appropriate power of ten so that the new divisor is an integer — taking advantage of the fact that a/b = (ca)/(cb) — and then proceeding as above.

Polynomials

A generalized version of this method called polynomial long division is also used for dividing polynomials (sometimes using a shorthand version called synthetic division).

Standards-based mathematics reform

Many mathematics text series were created in response to the recommendations of the NCTM. Some of these, such as TERC omit any instruction in long division. In fact the fifth grade teachers manual states that mathematicans no longer use the notation of long division, students should be discouraged from using the method if they were taught outside the classroom. It also states that the letter "R" should not be used to signify a remainder. Parents who dislike such methods of teaching division have protested the adoption of such texts on websites such as Mathematically Correct.

See also

• Elementary arithmetic
• Arbitrary-precision arithmetic
• Polynomial long division

External links

• Alternative Division Algorithms: Double Division, Partial Quotients & Column Division, Partial Quotients Movie

• Step By Step Polynomial Long Division: WebGraphing.com