# Division

Division is the mathematical operation that is the inverse of **multiplication**. If one multiplies 47 by 92 then divides by 92, the result is the original 47. In general, (ab)/b = a. Likewise, if one divides first then multiplies, the two operations nullify each other: (a/b)b = a. This latter relationship can be taken as the definition of division: a/b is a number which, when multiplied by b, yields a.

In the real world using ordinary **arithmetic**, division is used in two basic ways. The first is to partition a quantity of something into parts of a known size, in which case the quotient represents the number of parts, for example, finding how many three-egg omelets can be made from a dozen eggs. The second is to share a quantity among a known number of shares, as in finding how many eggs will be available per omelet for each of five people. In the latter case the quotient represents the size of each share. For omelets, if made individually you could use two eggs each for the five people and still have two left over, or you could put all the eggs in one bowl, so each person would get 2^{2}⁄_{3} eggs.

The three components of a division situation can represent three distinct categories of things. While it would not make sense to add dollars to earnings-per-share, one can divide dollars by earnings-per-share and have a meaningful result (in this case, shares). This is true, too, in the familiar rate-distance-time relationship R = D/T. Here the categories are even more distinct. **Distance** is measured with a tape; **time** by a clock; and **rate** is the result of these two quantities.

Another example would be in preparing a quarterly report for share holders; a company treasurer would divide the total earnings for the quarter by the number of shares in order to compute the earnings-per-share. On the other hand, if the company wanted to raise $6,000,000 in new capital by issuing new shares, and if shares were currently selling for $18^{1}⁄_{8}, the treasurer would use division to figure out how many new shares would be needed, i. e., about 330,000 shares.

Division is symbolized in two ways, with the symbol ÷ and with a bar, horizontal or slanted. In a/b or a ÷ b, a is called the dividend; b, the divisor; and the entire expression, the quotient.

Division is not commutative; 6/4 is not the same as 4/6. It is not associative; (8 ÷ 4) ÷ 2 is not the same as 8 ÷ (4 ÷ 2). For this reason care must be used when writing expressions involving division, or interpreting them. An expression such as

is meaningless. It can be given meaning by making one bar noticably longer than the other

to indicate that 3/4 is to be divided by 7. The horizontal bar also acts as a grouping symbol. In the expressions

The division indicated by the horizontal bar is the last operation to be performed.

In computing a quotient one uses an **algorithm**, which finds an unknown multiplier digit by digit or term by term.

In the algorithm on the left, one starts with the digit 7 (actually 0.7) because it is the biggest digit one can use so that 4 × 7 is 30 or less. That is followed by 5 (actually 0.05) because it is the biggest digit whose product with the divisor equals what remains of the dividend, or less. Thus one has found (.7 + 0.05) which, multiplied by 4 equals 3. In the algorithm on the right, one does the same thing, but with **polynomials**. One finds the polynomial of biggest **degree** whose product with the divisor is equal to the dividend or less. In the case of polynomials, "less" is measured by the degree of the polynomial remainder rather than its numerical value. Had the dividend been x^{2} - 4, the quotient would still have been x - 1, with a remainder of -3, because any other quotient would have left a remainder whose degree was greater than or equal to that of the divisor.

These last two examples point out another way in which division is a less versatile operation than multiplication. If one is working with **integers**, one can always multiply two of them an have an integer for a result. That is not so with division. Although 3 and 4 are integers, their quotient is not. Likewise, the product of two polynomials is always a polynomial, but the quotient is not. Occasionally it is, as in the example above, but had one tried to divide x^{2} - 4 by x + 1, the best one could have done would have been to find a quotient and remainder, in this case a quotient of x - 1 and a remainder of -3. Many sets that are closed with respect to multiplication (i.e., multiplication can always be completed without going outside the set) are not closed with respect to division.

One number that can never, ever be used as a divisor is **zero**. The definition of division says that (a/b)b = a, but the multiplicative property of zero says that (a/b) × 0 = 0. Thus, when one tries to divide a number such as 5 by zero, one is seeking a number whose product with 0 is 5. No such number exists. Even if the dividend were zero as well, division by zero would not work. In that case one would have (0/0)0 = 0, and 0/0 could be any number whatsoever.

Unfortunately division by zero is a trap one can fall into without realizing it. If one divides both sides of the equation x^{2} - 1 = 0 by x - 1, the resulting equation, x + 1 = 0, has one root, namely -1. The original equation had two roots, however, -1 and 1. Dividing by x - 1 caused one of the roots to disappear, specifically the root that made x - 1 equal to zero.

The division algorithm shown above on the left converts the quotient of two numbers into a decimal, and if the division does not come out even, it does so only approximately. If one uses it to divide 2 by 3, for instance, the quotient is 0.33333... with the 3s repeating indefinitely. No matter where one stops, the quotient is a little too small. To arrive at an exact quotient, one must use fractions. Then the "answer," a/b, looks exactly like the "problem," a/b, but since we use the bar to represent both division and the separator in the **ratio** form of a **rational number**, that is the way it is.

The algorithm for dividing rational numbers and leaving the quotient in ratio form is actually much simpler. To divide a number by a number in ratio form, one simply multiplies by its **reciprocal**. That is, (a/b) ÷ (c/d) = (a/b)(d/c).

## Resources

### Books

Bittinger, Marvin L., and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Eves, Howard Whitley. *Foundations and Fundamental Concepts of Mathematics.* NewYork: Dover, 1997.

Grahm, Alan. *Teach Yourself Basic Mathematics.* Chicago: McGraw-Hill Contemporary, 2001.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

J. Paul Moulton

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