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ARITHMETIC CALCULATION IN COMPUTERS

ARITHMETIC OPERATION AND CONVERSION DONE BY COMPUTER
As we have seen in previous post that arithmetic computation is an important thing in calculation, if a person doesn’t know arithmetic computations,he will be doomed to be in trouble during calculations that happensin everyday life.So it is necessary to learn arithmetic computation.So in this post I am going to take one step further.I am going to write about octal and hexadecimal numbers.
Now let us discuss some arithmetic conversion.Let us see what is octal and hexadecimal numbers used by digital computer.
The conversion from and to binary, octal and hexadecimal plays an important part in digital computers.Since 23 = 8 and 24 = 16 each octal digit corresponds to three binary digits and each four binary digit corresponds to one hexadecimal digit.
The conversion of from binary to octal is easily accomplished by partitioning the binary into group of three digit each, starting from binary point and preceding to the left or to the right…

INTEGRATED CIRCUITS WITH A SOFTWARICAL TWIST

LOGICAL DESIGN OF FUNCTION OF INTEGRATED CIRCUITS
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A binary variable or called as two value variables can take the value of 0 or 1.  A Boolean functions or named as two variable function is an expression formed with binary variables, the two binary operators OR and AND, the unary operator NOT, parenthesis, and equal sign.  For a given value of the variables to made it as good and thorough, the function can be either 0 or 1.  Consider, for example, the Boolean function:

                                                 F1 = xyz’

The function as shown above F1 is equal to 1 if x = 1 and y = 1 and z’ =  1; otherwise F1 = 0.  The above is an example of a Boolean functions represented as an algebraic expression.  A Boolean function may be represented in a truth table.  To represent a function of a truth table we need a list of 2^n combination of 1s and 0s of the n binary variables and a column showing the combinations for a function equal to 0 or 1.
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As shown in Table below, there are eight possible distinct combination for assigning bits to three variables.  The column labeled F1 when x =1, y = 1, and z=0.  It is equal to zero otherwise.  Consider the function
  F2 = x + y’z

F2 = 1 if x = 1 or if y = 0, while z = 1.  In another table , x = 1 in the last four rows and yz = 01 in rows 001 and 101.  The later combination applies also for x = 1.  Therefore there are five combination that make F2 = 1. As a third example, consider the function:

F3 = x’y’z’ + x’yz + xy’

This is shown in table below with four 1’s four 0’s.  F4 is same as F3 and is considered below

Truth tables for F1 = xyz’,  F2 = x + y’z ,  F3 = x’y’z + x’yz + xy’
and F4= xy’ + x’z

X
Y
Z
F1
F2
F3
F4
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
1
0
0
0
0
0
0
1
1
0
0
1
1
1
0
0
0
1
1
1
1
0
1
0
1
1
1
1
1
0
1
1
0
0
1
1
1
0
1
0
0

Any Boolean function or a Boolean equation can be represented in a truth table or any mathematical drawing .  The numbers of rows in the table is 2n where n is the number of binary variables  in the function.  The 1’s and 0’s combination for each row is easily obtained from the binary numbers by counting 0 to    2n – 1.  For each row of the table, there is a value for the function equal to either 1 or 0.  The question now arises, is an algebraic expression of a Boolean function is unique ? In other words, is it possible to find two algebraic expression that specify the same function?  The answer to this question is yes.  As a matter of fact, the manipulation of Boolean algebra is applied mostly to the problem of finding simpler expression for the same function.

Consider for example the functions:
                                F4 = x y’ + x’ z
From table above, we find that F4 is same as F3  since both have identical 1’s  and 0’s for each combination of values of the three binary variables.  In general or in simply terms, two functions of n binary variables are said to be equal if they have the same value for all possible 2n combinations of the n variables.

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