>> Introduction

>> Binary Number System

>> Complements

>> 2's Complement vs 1's Complement

>> Binary Logic

>> Logic Gates

Introduction

The fundamental idea of digital systems is to represent data in discrete form (Binary: ones and zeros) and processing that information. Digital systems have led to many scientific and technological advancements. Calculators, computers, are the examples of digital systems, which are widely used for commercial and business data processing. The most important property of a digital system is its ability to follow a sequence of steps to perform a task called program, which does the required data processing. The following diagram shows how a typical digital system will look like.

Representing the data in ones and zeros, i.e. in binary system is the root of the digital systems. All the digital system store data in binary format. Hence it is very important to know about binary number system. Which is explained below.

Binary Number System

The binary number system, or base-2 number system, is a number system that represents numeric values using two symbols, usually 0 and 1. The base-2 system is a positional notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all computers. Suppose we need to represent 14 in binary number system.

14 - 01110 - 0x2^{4} + 1x2^{3} + 1x2^{2} + 1x2^{1} + 0x2^{0}

similarly,

23 - 10111 - 1x2^{4} + 0x2^{3} + 1x2^{2} + 1x2^{1} + 1x2^{0}

Complements

In digital systems, complements are used to simplify the subtraction operation. There are two types of complements they are:

The r's Complement

The (r-1)'s Complement

Given:

- N a positive number.
- r base of the number system.
- n number of digits.
- m number of digits in fraction part.

The r's complement of N is defined as r

^{n} - N for N not equal to 0 and 0 for N=0.

The (r-1)'s Complement of N is defined as r

^{n} - r

^{m} - N.

Subtraction with r's complement:The subtraction of two positive numbers (M-N), both are of base r. It is done as follows:

1. Add M to the r's complement of N.

2. Check for an end carry:

(a) If an end carry occurs, ignore it.

(b) If there is no end carry, the negative of the r's complement of the result obtained in step-1 is the required value.

Subtraction with (r-1)'s complement:The subtraction of two positive numbers (M-N), both are of base r. It is done as follows:

1. Add M to the (r-1)'s complement of N.

2. Check for an end carry:

(a) If an end carry occurs, add 1 to the result obtained in step-1.

(b) If there is no end carry, the negative of the (r-1)'s complement of the result obtained in step-1 is the required value.

For a binary number system the complements are: 2's complement and 1's complement.

2's Complement vs 1's ComplementThe only advantage of 1's complement is that it can be calculated easily, just by changing 0s into 1s and 1s into 0s. The 2's complement is calculated in two ways, (i) add 1 to the 1's complement of the number, and (ii) leave all the leading 0s in the least significant positions and keep first 1 unchanged, and then change 0s into 1s and 1s into 0s.

The advantages of 2's complement over 1's complement are:

(i) For subtraction with complements, 2's complement requires only one addition operation, where as for 1's complement requires two addition operations if there is an end carry.

(ii) 1's complement has two arithmetic zeros, all 0s and all 1s.

Binary LogicBinary logic contains only two discrete values like, 0 or 1, true or false, yes or no, etc. Binary logic is similar to Boolean algebra. It is also called as boolean logic. In boolean algebra there are three basic operations: AND, OR, and NOT.

AND: Given two inputs x, y the expression x.y or simply xy represents "x AND y" and equals to 1 if both x and y are 1, otherwise 0.

OR: Given two inputs x, y the expression x+y represents "x OR y" and equals to 1 if at least one of x and y is 1, otherwise 0.

NOT: Given x, the expression x' represents NOT(x) equals to 1 if x is 0, otherwise 0. NOT(x) is x complement.

Logic GatesA logic gate performs a logical operation on one or more logic inputs and produces a single logic output. Because the output is also a logic-level value, an output of one logic gate can connect to the input of one or more other logic gates. The logic gate use binary logic or boolean logic. AND, OR, and NOT are the three basic logic gates of digital systems. Their symbols are shown below.

AND and OR gates can have more than two inputs. The above diagram shows 2 input AND and OR gates. The truth tables of AND, OR, and NOT logic gates are as follows.