## 11.1 NATURAL NUMBERS

### 11.1.1 Basic Adder (Ripple-Carry Adder)

The structure of an *n*-digit *ripple-carry adder* is shown in Figure 11.1. The *full adder* (*FA*) cell calculates *q*(*i* + 1) and *z*(*i*) as a function of *x*(*i*), *y*(*i*), and *q*(*i*), according to the iteration body of Algorithm 4.1:

Let *C*_{FA} and *T*_{FA} be the cost and the computation time of an FA cell. The cost and computation time of an *n*-digit basic adder are equal to

**Examples 11.1**

- In base 2 the FA equations (11.1) are
(∨:

*or*function, ⊕ :*xor*function). - In base 10 the FA equations are

The decimal digits can be represented as 4-bit binary numbers (BCD—binary-coded decimal representation) so that a decimal full adder is a 9-input, 5-output binary circuit. It can be implemented using classical methods and tools of combinational logic synthesis. Another option is to decompose it further on. The following algorithm computes (11.4):

s:=x(i)+y(i)+q(i);ifs>9thenz(i):=(s+6) mod 16; q(i+1):=1;elsez(i):=s; q(i + 1):=0; ...

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