# 6

# Sequential Division

Unlike addition, subtraction and multiplication, division does not, in general, produce an exact answer, because the dividend is not necessarily a multiples of the divisor. A variety of division algorithms are to be described in the rest of this chapter.

## 6.1 SUBTRACT-AND-SHIFT APPROACH

Let's look at an example of pencil-and-paper division in Figure 6.1.

Here 2746 is referred to as the *Dividend*, 32 is the *Divisor*, 85 the *Quotient* and 26 the *Remainder*. After 274 – 256 is performed in the first step, 18 is obtained as a *partial remainder.* By pulling down 6, 18 becomes 180 (+6). On digital computers, most of the division operations are performed in a recursive procedure represented by the following formula:

where *j* = 0, 1, · · ·, *n* − 1 is the *recursion index*, *R*^{(j)} is the partial remainder in the *j*th iteration. The initial partial remainder *R*^{(0)} equals the dividend, and *R*^{(n)} is the *final remainder.* The quotient is determined digit by digit in the recursive procedure, with *q*_{j+1} being the (*j* + 1)th *quotient digit. D* is the divisor, and *r* is the radix. In the above example *r* = 10.

From the recursion formula, we have

Let's look at ...

Get *Arithmetic and Logic in Computer Systems* now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.