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 happens  in 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.

The corresponding octal digit is then assigned to each group.  The following  example illustrates the procedure.

 10 110 001 101 011 .  111 100 000 110   = (26153.406)₂

  2     6     1     5      3        7      4       0     6

Conversion from binary to hexadecimal is similar,  except that the binary number is divided into group of four digits:

 

10 1100 0110 1011 .  1111 0010    =  (2C6B.F2)₁₆

  2     C       6       B            F        2

The corresponding hexadecimal (or octal) digit for each group of binary digits is easily remembered after studying the values. 

Conversion from octal or hexadecimal to binary is done by procedure reverse to the above.  Each octal digit  is converted to three bit binary equivalent .  Similarly, each hexadecimal digit  is converted to its four-digit binary equivalent. 

Binary numbers are difficult to work with because they require three or four times as many digits as their decimal equivalent .  For example, the binary number 111111111111 is equivalent to decimal number  4095.  However, digital computers use binary numbers and its sometimes necessary for human user to communicate directly to with the machine by means of binary numbers.

One scheme that retains binary  system  in the computer but reduces the number of digits human must consider utilizes the relationship between  binary number system and octal and hexadecimal system.  By this method, the human thinks of the number of octal and hexadecimal  numbers  and performs required conversion by inspection when direct communication by the machine is necessary.  Thus the binary numbers 111111111111 is 12 digit and is expressed in octal as 7777 (four digits) and while it is expressed in hexadecimal as FFF (3 digits).

During communication between people (about binary numbers in the computer), the octal or hexadecimal representation is more desirable because it can be represented in third or quarter of number of digits required for equivalent binary number.

When the human communicates with the machine (through console switches or indicator lights or by means of programs written in machine language),  the conversion from octal or hexadecimal  to binary and vice versa is done by inspection by the human user.

                                                                                   Thank you for reading

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