Number system || chapter two BCA TU || Digital logic

Number system

A number system in base r or radius r uses unique symbols for r digits. One or more digits are combined to get a number.

In computers, we are concerned with four kinds of number systems, as follows:

Decimal Number System      ------------      Base 10
Binary Number System         ------------      Base 2
Octal Number System           ------------      Base 8
Hexa-Decimal Number System --------      Base 16


Decimal Number System
It consists of 10 digits (0-9).

Binary Number System
It consists of 2 digits (0-1).

Octal Number System 
It consists of 8 digits (0-7).


Hexa-Decimal Number System
It consists of 16 digits (0-9 and A-E) whereas from digit 10 it is named after alphabets A that means A is number 10 and E is number 15.

Conversion of Decimal to Binary, Octal, Hexa-decimal

binary-divide by 2, for fraction multiply by 2
octal-divide by 8, for fraction multiply by 8, and,
hexadecimal-divide by 16, for fraction by 16.

Conversion of Binary to Decimal, Octal, Hexadecimal

decimal-multiply by 2 with the power of their descending position and for fraction with power of their ascending position but with negative sign in powers.
 octal-multiply by 8 with the power of their position and for fraction with the power of negative sign.
hexadecimal-multiply 16 with the power in it and negative for fraction.

Conversion of Binary to Octal, Hexa-Decimal

8= that means while converting binary digits to octal we have to arrange three digits of binary number starting from last position to get octal number.
 16= that means while converting binary digits to hexadecimal we have to arrange four digits of binary number starting from last position to get octal number.
Conversion of Octal, Hexadecimal to Binary
As said above each number of octal number equals to three digits of binary number i.e. final binary number will be combination of each number formed while converting.
Convert each hexadecimal digit to four digits of binary number and the combination of all digits will be the final conversion.

Radix Complement 

The r's complement of "n" digit, number "N" in base "r" is defined as -N for N=0 and 0 for N=0.
Comparing with (r-1)'s complements the r's complement is obtained by adding 1 to the (r-1)'s complement since -N= [(-1)-N] +1.

Binary Arithmetic
Binary Addition
Binary Subtraction
Binary Multiply
Binary Divide

Binary Addition

The Addition of the Binary numbers involve the following steps:
Start addition by adding the bits in unit column (the right-most column).
The result of adding of a column is a sum with or without a carry.
Write the sum in the result of that column.
If a carry is present, the carry is carried over to the addition of the next left column.

Binary Subtraction

The steps for performing subtraction of the binary numbers are as follows: -
Start subtraction by subtracting the bit in the lower row from the upper row, in the unit column.
Use the binary subtraction rules. If the bit in upper row is less than lower row, borrow 1 from the upper row of the next column (on the left side).
Rule of subtraction
1-1=0              0-1=1 (borrowed 1 from upper one)



Binary Multiply
Rules for Binary Multiply: 1*0=0   1*1=1


Binary Division
Binary number is divided same as dividing decimal number.


Binary Coded Decimal (BCD)
In this code each digit is responsible by a 4-bit binary number. The positional weight is assigned to the binary digits in BCD code are 8-4-2-1 with 1 corresponding to LSB and corresponding MSB.

Positional 8 4 2 1
Weight

   While converting from BCD to binary, convert same as converting binary.
0 1 2 3 4 5 6 7 8 9
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Invalid BCD Codes

with 4-bits we can represent total 16 numbers (0000 to 1111) in BCD only first ten codes on used (0000 to 1001)
Therefore, remaining six-codes (1010 to 1111) are invalid in BCD.
The binary coding schemes that are mostly used are: -
Extended Binary Code Decimal Interchange Code (EBCDIC)
American Standard Code for Information Interchange (ASCII)
Unicode

EBCDIC

The Extended Binary Coded Decimal Interchange Code (EBDIC) uses 8 bits (4 bits for zone, 4 bits for digit) to represent a symbol in the data.
EBCDIC allows = 256 combinations of bits.
256 unique symbols are represented using EBCDIC code. It represents decimal numbers (0-9) lower case letters (a-z), upper case letters (A-Z), Special characters, and Control characters (printable and non-printable, e.g., for cursor movement, printer vertical spacing, etc.
 EBCDIC codes are mainly used in the mainframe computers.

ASCII
The American Standard Code for Information interchange (ASCII) is widely used in computers of all types.
ASCII codes are two types- ASCII-7 and ASCII-8.
 ASCII-7 is a 7-bit standard ASCII code. In ASCII-7, the first 3 bits are the zone bits and the next 4 bits are for the digits. ASCII-7 allows = 128 combinations. 128 unique symbols are represented using ASCII-7 has been modified by IBM to ASCII-8
ASCII-8 is an extended version of ASCII-7. ASCII-8 allows =256 combinations. ASCII-8 represents unique symbols. ASCII-8 is an 8-bit code having 4 bits for zone and 4 bits for the digits. ASCII is widely to represent data in computers.
Code 0-31 used for actions like Carriage Return (CR), Bell (BEL), etc.
Code 48-57 for numeric
Code 65-90 for uppercase letters
Code 97-122 for lowercase letters
Code 128-255 are extended ASCII codes.

Unicode
It is a universal character encoding standard for the representation of text which includes letters, numbers and symbols in multi-lingual environments. The Unicode Consortium based in California developed the Unicode standard.
It uses 32 bits to represent a symbol in the data.
It allows =4164895296 (~ 4 billion) combinations.
It can uniquely represent any character or symbol present in any language. In addition to the letters; mathematical and scientific symbols are also represented in Unicode.
It is compatible with the ASCII-8 codes. the first 256 codes are identical to the ASCII-8 codes.

Gray Code
It is also known as Reflected Binary Code is defined as an ordering of the binary number system such that each incremental value can only differ by one bit. I n gray code, whole traversing from one step to another step only one bit in the group changes. That is to say that two adjacent code numbers differ from each other by only one bit.
Gray code is the most popular of the unit distance code, but it is not applicable for Arithmetic Operations. Gray Code has some application is Analog to digital Convertors, as well as used for error correction in Digital Communication.

Excess-3 Code
It is basically a binary code which is made by adding 3 with equivalent decimal of binary number and again converting it into binary number. So, if we consider any binary number we have to first convert it into decimal number then add 3 with it and then convert into binary.

Self-complementing Property:
Excess-3 code is non-weighted and self-complementary code. A self-complementary binary code is always complemented themselves. The compliment of a binary number can be obtained from that number by replacing 0's with 1's and 1's with 0's. The sum of binary number and its complement is always equal to decimal 9. In other words, the 1's complement of an excess-3 code is the excess-3 code for the 9's complement of the corresponding decimal number.

Alpha-numeric Codes
A binary bit can represent only two symbols '0' and '1'. But it is not enough for communication between two computers because there we need many more symbols for communication.
These symbols are required to represent:
26 numbers with capital and small letters
Number 0 to 9
Punctuation marks
Alpha-numeric codes represent numbers and alphabetic characters. They also represent other characters such as punctuation symbols and instructions for conveying information.

Boolean algebra and logic gates
The most common postulates used to formulate various algebraic structures are:
Associative law: A binary operator * on a set S is said to be associative whenever:
  (x*y) *z=x*(y*z)
Commutative law: A binary operator * on a set S is said to be commutative whenever:
x*y=y*x
Identity element: A set S is said to have an identity element with respect to a binary operation * on S if there exists an element eS with the property:
e*x=x*e=x for every xS
Inverse: A set having the identity element e with respect to a binary operator * is said to have an inverse whenever, for every xS, there exists an element yS such that:
x*y=e
Distributive law: If * and · are two binary operators on set S, * is said to be distributive over* whenever:
x*(y*z) =(x*y) *(y*z)
Theorem 1(a): x+x=x
Theorem 1(b):x*x=x
Theorem 2(a):x+1=x
Theorem 2(b):x*0=0
Theorem 3     :(x') '=x
Theorem 6(a):x+yx=x
Theorem 6(b):x(y+x)=x