Showing posts with label EQUATION SOLUTION IN DIGITAL WORLD. Show all posts
Showing posts with label EQUATION SOLUTION IN DIGITAL WORLD. Show all posts

Thursday, April 15, 2021

HOW DO COMPUTERS UNDERSTAND BINARY LOGIC

 

HOW DO COMPUTER UNDERSTAND THE BINARY LOGIC?



 

As we have shown in the previous post LANGUAGE THAT COMPUTER AS WELL AS HUMAN UNDERSTAND I am going to dig deep of that topic in this post.  I will try to explain how computer understand the binary logic.  So lets begin.  I am going to write about binary logic, that most of the computer manufacturers and developers use.

                

Binary logic deals with variables that take on two discrete values and with operations that assume logical meaning.  The two values the variables take may be called by different names (e.g. true and false, yes and no, etc.), but for our purpose it is convenient to think in terms of bits and assign the values of 1 and 0. 

 

Binary logic is used to describe, in a mathematical way, the manipulation and processing of binary information.  It is particularly suited for the analysis and design of digital systems.  For example, the digital logical circuits of many circuits that perform binary arithmetic are circuits whose behavior is most conveniently expressed by means of binary variables and logical operations.  The binary logic to be introduced in this section is equivalent to an algebra called Boolean algebra.

 

Binary logic consists of binary variables and logical operations.  The variables are designated by letters of the alphabet such as A, B, C, x, y, z, etc., with each variable having two and only two distinct  values : 0 and 1.  There are basic logic operations: AND, OR and NOT.

 

·        AND: This operation is represented by a dot or by the absence of an operator.  For example, x.y = z or xy=z is read “x AND y is equal to z”.  The logical operation  AND interpreted  to mean and z = 1 if and only if x = 1 and y = 1 otherwise z = 0. (Remember that x, y and z are binary variables and can be equal to either 1 or 0 nothing else).

·        OR : This operation is shown by addition symbol.  For example,  x + y = z is read “ x OR  y is equal to z”      meaning that z = 1 if x=1 or y=1 or both x=1 or if both x=1 and y = 1.  If both x = 0 , then y = 0 then z = 0.

·        NOT : This operation is presented by  a prime (sometimes by a bar).  For example , x’ = z (or x not equal to z meaning that x is what z is not) .  In other words, if x = 1, and z = 0 .  But if x=0 then z = 1.

 

Binary logic resembles binary arithmetic and the operations “AND” and “OR” have some similarities to multiplication and additions, respectively.  In fact, the symbols used for AND and OR are the same as those used for multiplication and addition.  However, binary logic should not be confused with binary arithmetic.  One should realize that an arithmetic variable designates a number that may consist of many digits.  A logic variable is either a one or zero.  For example, in binary arithmetic we have 1 + 1 = 1 (read “one plus one equal to 2” while in binary logic we have 1 + 1 = 1 (read “ one or one equal to one”

 

For each combination of the values of x and y there is a value of z specified by the definition of the logical operation.  These definations may be listed in compact form using truth tables.  A truth table is a table of all possible combination of the variables showing the relations between the balues that the variables may take and the result of the operation.  For example, the truth tables for he operations AND and OR with variables x and y are obtained by listing all possible values that the variable may have when combined in pairs.  The result of the operation for each combination is when listed in a separate row.  The truth tables for “AND” , “OR” and “NOT” are as under.

 

                                    AND                        

X

y

x.y

0

0

0

0

1

0

1

0

0

1

1

1

 

                                     OR

 

X

y

x +  y

0

0

0

0

1

1

1

0

1

1

1

1

 

 

                                  

 

                      NOT

X

x’

0

1

0

1

1

0

1

0

 

Monday, March 9, 2020

SUBTRACTION IN COMPUTERS


SUBTRACTION IN COMPUTERS



HOW DO COMPUTERS SUBTRACT IN ANY EQUATION?
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Computer is one of the best inventions in the whole world. Now in the era of electronics, computers are shrinking in the size.  The dream of having pocket computers which was shown in many Hollywood as well as many film industry is now a reality.

 Many unimaginable discoveries and inventions are done with the help of computers. As in this age the  of semiconductors and embedded systems we are achieving what was a dream, about 60-70 years ago.

Now to achieve such a dream, the hard-work and dedication needed was given by many scientists whose aim was to uplift the society.  Now if we want to invent something we need good hold on arithmetic computation.  This computation was understood and logically implemented by many scientists and engineers. 

We use Complements  in digital computers for almost  simplifying the subtraction operation and for logical manipulations and computations. 

There are two types of complement for each base-r system:

(1) the r’s complement
(2) the (r-1)’s complement

When the value of the base is substituted, the two types receive the names 2’s and 1’s complementary for binary numbers, or 10’s and 9’s complement for decimal numbers.

Now let us see how this happens

The r’s complement

Given a positive number N in base r with an integer part of n digits, the r’s complement of N is defined as rn – N for N#0 and 0 for N=0.  The following numerical example will help clarify the definition.

The 10’s complement of (89654)10 is 105- 89654 = 10346.
The number of digits in the number is n = 5.
The 10’s complement of (0.9287)10 is 1 - 0.9287 = 0.0713
No integer part 10n = 100 is 1

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