ARITHMETIC CIRCUITS BASIC BUILDING BLOCKS TUTORIALS

In this article, we will discuss those combinational logic building blocks that can be used to perform addition and subtraction operations on binary numbers. Addition and subtraction are the two most commonly used arithmetic operations, as the other two, namely multiplication and division, are respectively the processes of repeated addition and repeated subtraction

We will begin with the basic building blocks that form the basis of all hardware used to perform the aforesaid arithmetic operations on binary numbers. These include half-adder, full adder, half-subtractor, full subtractor and controlled inverter.


A half-adder is an arithmetic circuit block that can be used to add two bits. Such a circuit thus has two inputs that represent the two bits to be added and two outputs, with one producing the SUM output and the other producing the CARRY.

The Boolean expressions for the SUM and CARRY outputs are given by the equations
SUM S = A B+A B
CARRY C = A B


A full adder circuit is an arithmetic circuit block that can be used to add three bits to produce a SUM and a CARRY output. Such a building block becomes a necessity when it comes to adding binary numbers with a large number of bits.

The full adder circuit overcomes the limitation of the half-adder, which can be used to add two bits only. Let us recall the procedure for adding larger binary numbers. We begin with the addition of LSBs of the two numbers. We record the sum under the LSB column and take the carry, if any, forward to the next higher column bits.


A half-subtractor is a combinational circuit that can be used to subtract one binary digit from another to produce a DIFFERENCE output and a BORROW output. The BORROW output here specifies whether a ‘1’ has been borrowed to perform the subtraction.


A full subtractor performs subtraction operation on two bits, a minuend and a subtrahend, and also takes into consideration whether a ‘1’ has already been borrowed by the previous adjacent lower minuend bit or not.

As a result, there are three bits to be handled at the input of a full subtractor, namely the two bits to be subtracted and a borrow bit designated as Bin . There are two outputs, namely the DIFFERENCE output D and the BORROW output Bo. The BORROW output bit tells whether the minuend bit needs to borrow a ‘1’ from the next possible higher minuend bit.


A controlled inverter is needed when an adder is to be used as a subtractor. As outlined earlier, subtraction is nothing but addition of the 2’s complement of the subtrahend to the minuend. Thus, the first step towards practical implementation of a subtractor is to determine the 2’s complement of the subtrahend. And for this, one needs firstly to find 1’s complement. A controlled inverter is used to find 1’s complement.

LOGIC GATES BASIC INFORMATION AND TUTORIALS

The logic gate is the most basic building block of any digital system, including computers. Each one of the basic logic gates is a piece of hardware or an electronic circuit that can be used to implement some basic logic expression.

While laws of Boolean algebra could be used to do manipulation with binary variables and simplify logic expressions, these are actually implemented in a digital system with the help of electronic circuits called logic gates.

The three basic logic gates are the OR gate, the AND gate and the NOT gate.

OR GATE
An OR gate performs an ORing operation on two or more than two logic variables. The OR operation on two independent logic variables A and B is written as Y = A+B and reads as Y equals A OR B and not as A plus B.

An OR gate is a logic circuit with two or more inputs and one output. The output of an OR gate is LOW only when all of its inputs are LOW. For all other possible input combinations, the output is HIGH. This statement when interpreted for a positive logic system means the following.

The output of an OR gate is a logic ‘0’ only when all of its inputs are at logic ‘0’. For all other possible input combinations, the output is a logic ‘1’. Figure 4.3 shows the circuit symbol and the truth table of a two-input OR gate. The operation of a two-input OR gate is explained by the logic expression Y = A+B


AND GATE
An AND gate is a logic circuit having two or more inputs and one output. The output of an AND gate is HIGH only when all of its inputs are in the HIGH state. In all other cases, the output is LOW.

When interpreted for a positive logic system, this means that the output of the AND gate is a logic ‘1’ only when all of its inputs are in logic ‘1’ state. In all other cases, the output is logic ‘0’.

The AND operation on two independent logic variables A and B is written as Y = A B and reads as Y equals A AND B and not as A multiplied by B. Here, A and B are input logic variables and Y is the output.

NOT GATE
A NOT gate is a one-input, one-output logic circuit whose output is always the complement of the input. That is, a LOW input produces a HIGH output, and vice versa.

When interpreted for a positive logic system, a logic ‘0’ at the input produces a logic ‘1’ at the output, and vice versa. It is also known as a ‘complementing circuit’ or an ‘inverting circuit’.

The NOT operation on a logic variable X is denoted as X or X . That is, if X is the input to a NOT circuit, then its output Y is given by Y = X or X and reads as Y equals NOT X. Thus, if X = 0 Y = 1 and if X = 1 Y = 0.

BINARY SYSTEM BASICS AND TUTORIALS

WHAT IS BINARY NUMBER SYSTEM? A TUTORIAL ON BINARY NUMBER SYSTEM

The binary number system is a radix-2 number system with ‘0’ and ‘1’ as the two independent digits. All larger binary numbers are represented in terms of ‘0’ and ‘1’. The procedure for writing higher order binary numbers after ‘1’ is similar to the one explained in the case of the decimal number system.

BOOLEAN LOGIC ALGEBRA AND LOGIC GATES BASICS AND TUTORIAL

BOOLEAN LOGIC AND LOGIC GATES TUTORIALS

Machines of all types, including computers, are designed to perform specific tasks in exact well defined manners. Some machine components are purely physical in nature, because their composition and behavior are strictly regulated by chemical, thermodynamic, and physical properties.

For example, an engine is designed to transform the energy released by the combustion of gasoline and oxygen into rotating a crankshaft. Other machine components are algorithmic in nature, because their designs primarily follow constraints necessary to implement a set of logical functions as defined by human beings rather than the laws of physics.