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# Long Division

• define a long division;
• identify the parts of a long division;
• properly use the method of long division; and
• solve problems involving long division.

## What is a long division?

Division is one of the four fundamental mathematical operations, with addition, subtraction, and multiplication being the other three. Long Division is a technique for dividing large numbers and algebraic expressions into multiple steps following a predetermined sequence. It is the most frequently used method for dividing problems.

The illustration below shows how long division works.

## What are the parts of long division?

If we are asked to divide two numbers, we usually see the notation 35 ÷6, and it is the same thing in long division. We have a dividend, a divisor, a quotient, and for some, a remainder.

Let us look deep into it!

• The dividend is the number being divided to, which in this case is 100.
• The divisor is the number that divides the dividend; in this case, 3 is the divisor.
• The quotient is the result of dividing the dividend and the divisor. In this case, it is 33.
• The remainder is the leftover part of a number after doing the process of division. In this case, the remainder is 1.

In English-speaking countries, the long division does not use the division slash or division sign symbols but instead creates a tableau. A right parenthesis “)” or vertical bar “|” separates the divisor from the dividend; a vinculum separates the dividend from the quotient (i.e., an overbar). These two symbols are sometimes referred to as a long division symbol or division bracket. It evolved in the 18th century from a single-line notation in which the dividend and quotient were separated by a left parenthesis.

## How to do long division?

The long division method can help us divide two whole numbers, two numbers with decimals, and even dividing polynomials. The process of long division begins by dividing the dividend’s left-most digit by the divisor. The result’s first digit is the quotient (rounded to an integer), and the remainder is calculated (this step is notated as a subtraction). This remainder is carried forward when the process is repeated on the next digit of the dividend (referred to as ‘bringing the next digit down’ to the remainder). The process is complete when all digits have been processed and no remainders remain.

In simple terms, long division involves only 6 steps.

1. Divide.
2. Multiply.
3. Subtract.
4. Bring down.
5. Repeat or Remainder.
6. Check.

### Dividing two whole numbers

To divide two whole numbers:

Step 1: Compare the dividend and the first digit of the divisor.

• If the divisor is less than or equal to the first digit of the dividend, then divide the two numbers.
• If the divisor is greater than the first digit of the dividend, then divide the divisor to the first and second digit of the dividend.

Step 2: Write the quotient on the top of the division bracket.

Step 3: Multiply the quotient and the divisor and write it below the dividend.

Step 4: Subtract the product of the quotient and the divisor from the dividend.

Step 5: Bring down the next digit that you weren’t able to use.

Step 6: Repeat the same process until there is no digit to bring down.

Step 7: If a remainder exists, ensure that it is less than the divisor. Otherwise, there must be a process in the multiplication or subtraction process.

Step 8: Lastly, the remainder must be added to the quotient and must be written in the form of rm , where r is the quotient and m is the divisor.

Example #1

What is the quotient if 424 is divided by 4?

Solution

Example #2

What is the quotient if 583 is divided by 7?

Solution

Example #3

What is the quotient when 2448 is divided by 19?

Solution

### Dividing two numbers with decimal points

1. Locate all decimal points within the dividend n and divisor m.
2. If necessary, simplify the long division problem by shifting the decimals of the divisor and dividend to the right (or to the left) by the same number of decimal places, so that the decimal of the divisor is to the right of the final digit.
3. When performing long division, maintain a straight line from top to bottom beneath the tableau.
4. Ensure that the remainder for each step is less than the divisor. If it is not, one of three things could be wrong: the multiplication is incorrect, the subtraction is incorrect, or a larger quotient is required.
5. Finally, the remainder, r, is added as a fraction, rm, to the quotient.

Example

Divide 506.25 by 5.

When the quotient is not an integer and the division operation is extended beyond the decimal point, one of two things may occur:

The process may terminate when a remainder of 0 is reached; or it may continue.

A remainder that is identical to a previous remainder that occurred after the decimal points were written could be obtained. In the latter case, continuing the process would be pointless, as the same sequence of digits would appear in the quotient repeatedly from that point on. As a result, a bar is drawn over the repeating sequence to indicate that it will continue to repeat indefinitely (i.e., every rational number is either a terminating or repeating decimal).

## What is the significance of long division?

Long division enables the division problem to be broken down into a series of simpler steps. Long division of integers can easily be extended to include rational non-integer dividends. This is due to the fact that every rational number has a recursive decimal expansion. Additionally, the procedure can be extended to encompass divisors with a finite or terminating decimal expansion (i.e. decimal fractions).

The technique used in long division also made way in other areas of division such as:

• Division of polynomials – when dividing polynomials we usually use synthetic division or polynomial long division.
• Binary division – calculations using the binary number system are simplified because each digit in the course can only be either 1 or 0 – no multiplication is required because multiplying by either results in the same number or zero.

Moreover, it enables the execution of computations involving arbitrarily large numbers via a series of simple steps. Short division is the abbreviation for long division, and it is almost always used instead of long division when the divisor has only one digit. Chunking (affectionately referred to as the partial quotients method or the hangman method) is a less mechanical method of long division popular in the United Kingdom that contributes to a more holistic understanding of the division process.

## Is long division still necessary to learn?

Calculators and computers have become the most frequently used tools for solving division problems, obviating the need for a traditional mathematical exercise and reducing educational opportunities to demonstrate how to do so using paper and pencil techniques. (Internally, those devices employ a variety of division algorithms, the quickest of which rely on approximations and multiplications to accomplish the task.) Long division has been singled out for de-emphasis or even omission from the school curriculum in the United States, despite its traditional introduction in the fourth or fifth grades.