Data Representation

1. Number Systems

1.1 Decimal, Binary, Octal, and Hexadecimal Number Systems

• Decimal: Base 10 number system (0-9).
• Binary: Base 2 number system (0, 1).
• Octal: Base 8 number system (0-7).
• Hexadecimal: Base 16 number system (0-9, A-F).

2. Conversion Between Number Systems

2.1 Decimal to Binary Conversion

To convert a decimal number to binary, divide the number by 2 and record the remainders. Reverse the remainders to get the binary representation.

Example: Convert decimal 13 to binary:

13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Binary representation = 1101

2.2 Binary to Decimal Conversion

To convert a binary number to decimal, multiply each bit by 2 raised to the power of its position from the right (starting at 0).

Example: Convert binary 1101 to decimal:

(1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13

2.3 Decimal to Octal Conversion

To convert decimal to octal, divide the number by 8 and record the remainders. Reverse the remainders for the octal result.

Example: Convert decimal 83 to octal:

83 ÷ 8 = 10 remainder 3
10 ÷ 8 = 1 remainder 2
1 ÷ 8 = 0 remainder 1
Octal = 123

2.4 Octal to Decimal Conversion

Multiply each digit by 8 raised to the power of its position from right to left.

Example: Convert octal 123 to decimal:

(1 * 8^2) + (2 * 8^1) + (3 * 8^0) = 64 + 16 + 3 = 83

2.5 Decimal to Hexadecimal Conversion

To convert decimal to hexadecimal, divide by 16 and record the remainders. Reverse them to get the hexadecimal number.

Example: Convert decimal 254 to hexadecimal:

254 ÷ 16 = 15 remainder 14 (E in hexadecimal)
15 ÷ 16 = 0 remainder 15 (F in hexadecimal)
Hexadecimal = FE

2.6 Hexadecimal to Decimal Conversion

Multiply each digit by 16 raised to the power of its position from right to left.

Example: Convert hexadecimal FE to decimal:

(15 * 16^1) + (14 * 16^0) = 240 + 14 = 254

3. Binary Coded Decimal (BCD)

BCD is a binary encoding of decimal numbers where each decimal digit is represented by its 4-bit binary equivalent.

Example: Convert decimal 93 to BCD:

9 = 1001
3 = 0011
BCD = 1001 0011

4. Hamming Code for Error Detection

The Hamming code is used for detecting and correcting errors in data transmission. Redundant bits are added to data bits for error detection.

4.1 Hamming Code Example

For a 4-bit data 1011, the Hamming code adds parity bits to detect and correct a 1-bit error.

5. Alphanumeric Codes

Alphanumeric codes represent letters and digits using binary patterns. Examples include ASCII, EBCDIC, and Unicode.

5.1 ASCII Example

In ASCII, each character is represented by a 7 or 8-bit binary number.

Example:

A = 01000001
B = 01000010

6. Arithmetic Operations

6.1 Binary Addition

Binary addition follows these rules:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1)

Example of Binary Addition:

  1011
+ 0110
------
  10001

6.2 Binary Subtraction

Binary subtraction follows these rules:

0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1)

Example of Binary Subtraction:

  1010
-  0011
-------
   0111

7. 1’s and 2’s Complement Notation

7.1 1’s Complement

The 1’s complement of a binary number is obtained by inverting all bits (changing 1 to 0 and 0 to 1).

Example: 1's complement of 1011 is 0100.

7.2 2’s Complement

The 2’s complement is obtained by adding 1 to the 1’s complement.

Example: 2’s complement of 1011:

1's complement = 0100
Add 1 = 0101

7.3 9’s and 10’s Complement for Decimal Numbers

• 9’s Complement: Subtract each digit from 9.
• 10’s Complement: Add 1 to the 9's complement.

Example: For decimal 456:

9's complement = 999 - 456 = 543
10's complement = 543 + 1 = 544

8. Binary Multiplication and Division

8.1 Binary Multiplication

Binary multiplication is similar to decimal multiplication, but follows the rules for binary arithmetic.

Example:

  101 (5)
×  11  (3)
------
  101
+1010
------
  1111 (15)

8.2 Binary Division

Binary division works similarly to long division in the decimal system.

9. BCD Arithmetic

In BCD arithmetic, binary coded decimal operations are performed on numbers represented in BCD format.

Example of BCD Addition:

  1001 0011 (93 in BCD)
+ 0001 0100 (14 in BCD)
--------------
= 1010 0111 -> adjust to valid BCD (107 in decimal becomes 107)

10. Floating-Point Addition and Subtraction

In floating-point arithmetic, numbers are represented in the form:

N = M * 2^E

where M is the mantissa and E is the exponent.

Example of Floating-Point Addition:

1.5 (1.5 × 10^0) + 2.5 (2.5 × 10^0) = 4.0